A New Integrable Equation with Peakon Solutions

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Legendrian knots and monopoles

a rX iv:mat h /41559v5[mat h.DG ]2Dec25Legendrian knots and Monopoles Tomasz Mrowka Yann Rollin Tomasz Mrowka,MIT,77Massachusetts Avenue,Cambridge MA 02139,USA Yann Rollin,Imperial College,Huxley Building,180Queen’s Gate,London SW72AZ,UK Email:mrowka@ rollin@ Abstract We prove a generalization of Bennequin’s inequality for Legendrian knots in a 3-dimensional contact manifold (Y,ξ),under the assumption that Y is the boundary of a 4-dimensional manifold M and the version of Seiberg-Witten invariants introduced in [KM]is non-vanishing.The proof requires an exci-sion result for Seiberg-Witten moduli spaces;then,the Bennequin inequality becomes a special case of the adjunction inequality for surfaces lying inside M .AMS Classiﬁcation numbers Primary:57R17;57M25;57M27;57R57Secondary:Keywords:Contact structures,Legendrian knots,Bennequin inequality,Ex-cision,Monopoles11IntroductionThis paper is a sequel to[KM]where the Seiberg-Witten invariants where gen-eralized to invariants of connected oriented smooth four-manifolds carrying a contact structure on their boundary.An oriented contact structureξ(or more generally an oriented two planeﬁeld)induces a canonical Spin c-structure sξon Y.In[KM],the Seiberg-Witten invariants were deﬁned for4-manifolds with boundary endowed with a contact structure.The domain of these in-variants is the set Spin c(M and h is an isomorphism between s|Y and sξ.The Seiberg-Witten invariant is a mapsw:Spin c(The Thurston Bennequin invariant and the rotation number of a knot in S3 generalize to invariants of the pair K andσ.The rotation number is only deﬁned up to sign until an orientation of K is chosen.Both invariants arise because a Legendrian knot has a canonical framing obtained by choosing a vectorﬁeld V transverse to the contact distribution along K.To generalize the Thurston-Bennequin invariant choose an arbitrary orientation for K give Σthe compatible orientation.Push K oﬀslightly in the V-direction,thus obtaining a disjoint oriented knot K′.Then pushΣoﬀitself to get a surface Σ′so that its boundary coincides with K′.Since theΣandΣ′are disjoint along their boundary they have a well deﬁned intersection number.The Thurston-Bennequin invariant relative toΣ,tb(K,σ)is deﬁned to be this self-intersection number.IfΣis contained in Y(so that K is null homologous in Y)thentb(K,σ)=lk(K,K′):=tb(K)Notice that tb(K)does not depend on the choice of the initial orientation for K orΣ.The generalization of the rotation number is obtained as follows.After choosing an orientation the contact distribution can be endowed with an almost complex structure Jξ,which is unique up to homotopy.Thereforeξ→Y has a structure of complex line bundle,hence it has a well deﬁned Chern class.The isomorphism h induces and isomorphism of the determinant line L s=det(W+s)of the bundle of positive spinors for the Spin c-structure s withξon the boundary If we also ﬁx an orientation for K then we get a preferred non vanishing tangent vector ﬁeld v and so by the Legendrian property a non-vanishing section of L s Then, the rotation number of K relative toΣis by deﬁnitionr(K,σ,s,h):= c1(L s,v),σwhere c1(L s,v)is the relative Chern class with respect to the trivialization of ξalong K induced by v.Notice that the rotation number depends a priori on the homology class ofΣand on the orientation of K.This deﬁnition coincides with the usual rotation number deﬁned on(R3,ξstd)as the winding number of v inξstd.1.2Main results.With these deﬁnitions and notation in place we can state our generalization of Bennequin’s Inequality.Theorem A Let(Y,ξ)be a3-dimensional closed manifold endowed with a contact structureξand letboundary Y.Suppose we have a Legendrian knot K⊂Y,andΣ⊂M,ξ(s,h)=0we haveχ(Σ)+tb(K,σ)+|r(K,σ,s,h)| 0,whereχdenotes the Euler characteristic.Notice that this result was known before in the case of compact Stein complex surfaces with pseudo-convex boundary(cf.[AM,LM]).Here are two corollaries of Theorem A.A contact manifold(Y,ξ)is called weakly symplecticallyﬁllable,if it is the boundary of a symplectic manifold(M,ξis identically zero for amanifold with an overtwisted contact boundary. Using the fact that swM,ξis not identically zero?Onthe other hand there are contact structures which are tight but not weakly sym-plecticallyﬁllable.Theﬁrst examples are due to Etnyre and Honda[EH]on certain Seifertﬁbered space and later inﬁnite families where discover by Lisca and Stipsicz(see[LS1,LS2]).4All these results rely on an excision property for Seiberg-Witten invariants. Recall that a symplectic cobordism from a contact manifolds(Y,ξ)to(Y′,ξ′)is a compact symplectic manifold(Z=−Y⊔Y′,where Y and Y′have their orientations induced by the contact structures.Y is called the concave end of the cobordism and Y′is called the convex end.In addition,it is required thatωis strictly positive onξandξ′with their induced orientations.By convention the boundary components will always be given in the order concave,convex.A symplectic cobordism,is said to be special if•the symplectic form is given in a collar neighborhood of the concave boundary by a symplectization of(Y,ξ);•the map induced by the inclusioni∗:H1(M,ξ).There is a canonical way to extend (s,h)to a Spin c structure t on M∪Z.The data of h identiﬁes the restriction s|Y with sξwhile sω|Y is identiﬁed canonically with sξ.Together these provides a gluing map and deﬁnes t.Thus we have deﬁned a canonical mapj:Spin c(M′,ξ′).The main technical result of this paper is the following.Theorem D Let Z be a special symplectic cobordism between(Y,ξ)and second contact manifold (Y′,ξ′).Let M∪Y Z onM,ξ◦j=±swM,ξ)we have:sw M,ξ(s2,h2)=swM′,ξ′◦j(s2,h2).5Remark1.2.2—The assumption(1.1)can be reformulated as follow:for any u∈Map(M.In this case the map j is not generally injective anymore.All the cobordisms of interest in this paper,will be shown to verify assump-tion1.1,that is to say1and2-handle surgeries.However,assumption(1.1) may be removed.But,the conclusion(1.2)of Theorem D has to be replaced by sw M,ξ(s,h).(1.3)This generalization is proved by reﬁning the gluing theorem E as explained in remark2.4.3.Indeed if two pairs(s1,h1)and(s2,h2)have the same image under j then we can assume upto ismorphism that s1=s2and h1=uh2 where u is an automorphism of sξwhich extends to an automorphism of sωwhich is the indentity at inﬁnity.A result of Weinstein[W]shows that,a1-handle surgery,or,a2-handle surgery along a Legendrian knot K with framing coeﬃcient−1relative to the canonical framing,on the boundary of M′given byM∪YZ,ω)is a special symplectic cobordism between(Y,ξ)and a contact boundary(Y′,ξ′)obtained by the surgery.The strategy to prove Theorem A is to capΣdoing a Weinstein surgery along K.Bennequin’s inequality is then obtained by applying the adjunction in-equality[KM2]to the resulting closed surfaceΣ′⊂M′,ξ′◦j(s,h)does not vanish.This is true un-der the assumption that swM with con-tact boundary(Y,ξ)developed by Kronheimer and theﬁrst author in[KM].61.3.1An almost K¨a hler coneWe pick a contact one-formηwith kerη=ξ.The contact form determines the Reeb vectorﬁeld R by the properties thatιR dη=0andη(R)=1.Then (0,+∞)×Y is endowed with a symplectic structure called the symplectization of(Y,η),and the symplectic form is deﬁned by the formula1ω=dη(J·,·).2Let M be the manifoldM=M.The Spin c-structure sξinduced by the contact structure on Y is canonically identiﬁed with the restriction to{T}×Y of the Spin c-structure sωinduced by the symplectic formωon C M.Therefore,an element(s,h)∈Spin c(1.3.2Monopole equationsLet A be a spin connection on W and∇A be its covariant derivative.The the Dirac operator is then deﬁned by D g A= i e i·∇A e j,where e j is an oriented orthonormal local frame on M.e i is the dual coframe and acts by Cliﬀord multiplication.It is aﬁrst order elliptic operator of order1between the space of spinorﬁelds D A:Γ(W±)→Γ(W∓).PutΨ=(1,0)∈Λ0,0⊕Λ0,2.Let B be the spin-connection in the spinor bundle W+J=Λ0,0⊕Λ0,2so that∇BΨis a section of T∗X⊗Λ0,2.The Dirac operator D can on W J is obtained by replacing∇A withˆ∇in the deﬁnition.ForΦ∈Γ(W+J),we have explicitly D canΦ=√M.The domain of the Seiberg-Witten equations on(M,g,s)is the space of con-ﬁgurationsC={(A,Φ)∈Conn(W)×Γ(W+)},where Conn(W)is the space of spin connections on W.Recall that Conn(W) is an aﬃne space modeled onΓ(iΛ1):for two spin connections A and˜A,we have˜A=A+a⊗id|W,where a is a purely imaginary1-form.We will simply write the above identity ˜A=A+a in the sequel.We introduce the curvature formF A(X,Y)=12FˆA8whereˆA is the unitary connection induced by A on the determinant line bundle1 is its usual curvature form.The Seiberg-Witten equations areL,and FˆAF+A−{Φ⊗Φ∗}0=F+B−{Ψ⊗Ψ∗}0+̟(1.5)D AΦ=0,(1.6) where̟is a self-dual purely imaginary2-form on M and{·}0represents the trace free part of an endomorphism;theﬁrst equation makes sense since purely imaginary self-dual2-forms are identiﬁed with the traceless endomorphisms of W+via Cliﬀord multiplication.The conﬁguration(B,Ψ)is clearly solution of the equations over C M.The space of solutions Z̟is acted on by a gauge group G=Map(M,S1)and the action is deﬁned on C byu·(A,Φ)=(A−u−1du,uΦ)∀u∈G.(1.7) The moduli space M̟(M,g,s)=Z̟/G,for suitable generic̟,is a compact smooth manifold of dimensiond= e(W+,Ψ),[M,C M] ,where e(W+,Ψ)is the relative Euler class of W+.If d=0,then the Seiberg-Witten invariant is always0;if d=0,the Seiberg-Witten invariant is the number of points of M̟(M,s)counted with signs.Following[KM]there is a trivial bundle over the conﬁguration space(the determinant line bundle of the appropriate deformation operator)which is identiﬁed with the orientation bundle of moduli space.Furthermore a trivialization of this determinant line bundle for one relative Spin c-structure determines a trivialization for all others in a canonical manner and determines a consistent orientation.Thus the set of consistent orientations is a two element set.In particular unlike the closed case the sign of the invariant cannot be pined down by a homology orientation rather the ratio of the signs of the values of the invariant for diﬀerent relative Spin c-structures is well deﬁned.It turns out that this number depends only onM,ξ); this explains the notation sw1Alternatively F A can be viewed as the curvature of the unitary connection induced by A on the virtual line bundle L1/29AcknowledgmentsThis paper obviously owes a large intellectual debt to Peter Kronheimer.The second author thanks the MIT where most of this work was done.He also thanks the Institute for Advanced Studies where he was hosted on several occa-sions.Theﬁrst author was partially supported by NSF grants DMS-0111298, DMS0206485,and FRG-0244663.The second author was partially supported by NSF grant DMS-0305130.2ExcisionThe goal of this section is to prove Theorem D.The strategy is to show,in a more general setting,that the Seiberg-Witten moduli spaces associated to M′are diﬀeomorphic for a suitable choice of metrics and perturbations. This is achieved thanks to a gluing technique in Theorem E.2.1Families of AF AK manifoldsWe set-up the analytical framework for our gluing problem.Theﬁrst step is to construct suitable families of asymptoticallyﬂat almost K¨a hler metrics.LetM. Hence we have obtained(M,g,ω,J)where g is Riemannian metric and M splits asM=(i)There is a constantκ>0,such that the injectivity radius satisﬁesκinj(x)>σ(x)for all x∈M.(ii)For each x∈M,let e x be the map e x:v→exp x(σ(x)v/κ)andγx be the metric on the unit ball in T x M deﬁned as e∗x g/σ(x)2.Then,these metrics have bounded geometry in the sense that all covariant derivatives of the curvature are bounded by some constants independent of x. (iii)For each x∈M\K,let o x similarly be the symplectic form e∗zω/σ(z)2on the unit ball.Then o x similarly approximates the translation-invariant form,along with all its derivatives.(iv)For allε>0,the function e−εσis integrable on M.(v)The symplectic formωextends as a closed form on K.Convention:When the end of M has a structure of symplectic cone as in(2.1) we chooseσ(t,y)=t on the end C M and extend it arbitrarily with the condition 0<σ(x) T onin this section,and the property(2.2)will be a consequence of the as-sumption(1.1).This property will be used for the compactness results of Section2.2.4.However this assumption is not essential as explained in remark2.2.3.•If we scale the metric byλ2,and accordingly scale T,σintoλT,λσ, then the constantκand the constants controllingγx and o x remain un-changed.In particular,we may always assume T=1in Deﬁnitions2.1.1 and2.1.2.•In Deﬁnition2.1.2,the contact formηcan be replaced by any other1-form representing the same contact structure.This is shown in the next lemma which allows us to modify the symplectic formωZ near the sharp end of the cone C Z.Lemma2.1.4Let Y be a compact3-manifold endowed with a contact struc-tureξ.Letη1andη2be two1-forms such that kerηj=ξ.Then,for every ε>0,there existα∈(0,ε)and a symplectic formωon(0,+∞)×Y such that •ω=1d(t2η2)on(ε,+∞)×Y.2Proof We writeη2=eµη1whereµis a real function on Y.We consider the exact2-form1ω=t3(2+tµf′)dt∧η∧dη.2Let c be the minimum of the functionµon Y.If c 0,we just require that f is an increasing function of t.If c<0,a suﬃcient additional condition for havingω2>0is2f′(t)<−ln(2etc],2e•f1(t)=f0(t)for t∈[ε2],12•f1(t)=1for t∈[ε4e ,ε]in such a way that we havef′ f′1on each interval where f′1is deﬁned.Therefore,the condition f′<−2MThanks to Remarks2.1.3,we may assume from now on that T=1and that the contact formηis the same in the deﬁnition of the almost K¨a hler cone C M=(T,∞)×Y and in the deﬁnition of the AFAK end2.1.2.We identify an annulus in C M⊂M with an annulus in C Z⊂Z using the dilation mapM⊃C M≃(1,+∞)×Y⊃(1,τ)×Yντ−→(1/τ,1)×Y⊂(0,1)×Y≃C Z⊂Z(t,y)−→(t/τ,y)and deﬁne the manifold Mτas the union of M∩{σM<τ}and Z∩{σZ>1/τ} and with the identify along the annuli given by the dilationντ.The operation of connected sum along Y we just deﬁned is represented in theﬁgure below.The gray regions represent the annuli,the arrows suggest that they are identiﬁed by a dilation,and the dashed regions are the parts of M and Z that are taken oﬀfor the construction of Mτ.Figure1:Construction of AFAK MτNowντ∗ωCM =τ2ωZ andσZ◦ντ=σM/τ,hence,if we scaleωZ byτ2andσZ byτ,all the structure will match on the annuli.In conclusion Mτcarries an almost K¨a hler structure(ωτ,Jτ)deﬁned outside the compact setRemark—Every compact set of K⊂M is also,by deﬁnition,a compact set of Mτprovidedτis large enough.Similarly,the structures gτ,στ,Jτ,ωτare equal on every compact set whenτis large enough.The following lemma is satisﬁed by construction.Lemma2.1.6The manifolds with almost K¨a hler structure deﬁned outside a compact set and a proper function(Mτ,gτ,Jτ,στ)satisfy Deﬁnition2.1.1uniformly,in the sense that the constantκ,ε,the bounds onγx,o x and on theintegral of e−εστcan be chosen independently ofτ.Remarks•A simple consequence of the lemma is the following:for all ε>0,there exists T k large enough such that,for everyτ,the pull-backof the almost K¨a hler structure on a unit balls in Mτ∩{στ T k},via the exponential map v→exp x v,on the tangent space T x Mτ,isε-close in.C k-norm to the euclidean structure(gτ,ωτ,Jτ)|Tx Mτ•For all the remainder of Section2,M and Mτwill denote the the man-ifolds that we just constructed at Section2.1.5together with their addi-tional structures(identiﬁcation of C M with an almost K¨a hler cone,proper functionσ,almost K¨a hler structure and Riemannian metric.2.1.7Spin c-structures on AF AK manifoldsSimilarly to the case of a manifold with a contact boundary,we deﬁne the space Spin c(X,ω)for an AFAK manifold X as the set of equivalence classes of pairs(s,h),where s is a Spin c-structure on X and h is an isomorphism, deﬁned outside a compact set K1⊂X,between sω|X\K1and s|X\K1.As we saw in Section1.3.1,there is a well deﬁned identiﬁcation of Spin c(M,ξ)→Spin c(Mτ,ωτ)when Mτis obtained by adding an AFAK end Z to the end ofand the transition map from W to W Jis given by h over the annulus{1<τστ<τ}∩Mτ⊂C M(where W Jτ=W J).At the level of equivalence classes of Spin c-structures,this procedure deﬁnes an identiﬁcation of Spin c(M,ω)with Spin c(Mτ,ωτ).We stress the fact that Wτis identiﬁed with the canonical spinor bundle W Jτfor it is,by construction,equal to it on Mτ∩{στ>1}.Moreover,the spinorbundles Wτrestricted to any compact set K⊂M are all identiﬁed providedτis large enough.In Section1.3.2we deﬁned a conﬁguration(B,Ψ)for the spinor bundle W→M.Similarly,we can deﬁne a conﬁguration(Bτ,Ψτ)for the spinor bundle Wτ→Mτ:outsideM⊂Mτ.In conclusion we have constructed a family of spinor bundles Wτ→Mτ, identiﬁed with W JoutsideτM,ξ),we consider the Spin c-structure induced on M and j(s,h)on Mτ.The Seiberg-Witten equations were intro-duced in Section1.3.2on M,which has an end modeled on an almost K¨a hler cone C M.The equations are given on Mτin the same way byF+A−{Φ⊗Φ∗}0=F+B−{Ψ⊗Ψ∗}0+̟τ(2.4)D AΦ=0,(2.5)15whereΦis a section of W+τ,A is a spin connection on Wτand̟τis a perturbation inΓ(iΛ+Mτ).As in the case of an almost K¨a hler conical end,the almost K¨a hler structure deﬁned outside2(¯∂⊕¯∂∗)and we have D gτB=D can,for B the spin connection deduced from the Levi-Civita connection on Mτand the Chern connection on Lτ≃L Jτ.Therefore(B,Ψ)solves the Seiberg-Witten equations restricted to Mτ\+1 ,(2.6)N0where N0is any number with N0 1;N0will beﬁxed later on to make the derivatives ofχτas small as required in our constructions.We deﬁne a cut-oﬀfunction on Mτby the formulaχτ(στ).By a slight abuse of notation,the latter function will be denotedχτas well.For a given perturbation of Seiberg-Witten equation̟on M,the perturbation of the equations on Mτis deﬁned by̟τ=χτ̟.(2.7) 2.2.2Linear theoryThe Study of Seiberg-Witten equations requires introducing suitable Sobolev spaces rather than using naive smooth objects deﬁned in Section1.3.2.We recall very quickly the results of[KM]in this section.The conﬁguration(B,Ψ)is a solution of Seiberg-Witten equations on the al-most K¨a hler end of Mτ;hence we study solutions(A,Φ)with the same asymp-totic behavior.We introduce the conﬁguration spaceC l(Mτ)={(A,Φ)∈Conn(Wτ)×Γ(W+τ)/A−B∈L2l(gτ),andΦ−Ψ∈L2l(gτ,B)},16the gauge groupG l(Mτ)={u:Mτ→C/|u|=1,and1−u∈L2l+1(gτ)},acting on C l by u·(A,Φ)=(A−u−1du,uΦ),and,for someﬁxedε0>0,the perturbation spaceN(Mτ)=e−ε0στC r(i su(W+τ)),equipped with the norm̟ Nτ= eε0στ̟ C r(gτ).The L2k(gτ)-norm is the usual L2norm with k derivatives on Mτdeﬁned using the metric gτ.In order to deﬁne a similar norm on the spinorﬁelds,a unitary connection A is needed.We putφ 2L2l(gτ,A)= Mτ |φ|2+|∇Aφ|2+···+|∇l Aφ|2 vol gτ,and deﬁne L2l(gτ,A)as the completion of the space of smooth sections for this norm.For two diﬀerent connections,A and A′with A−A′∈L2l(gτ), Sobolev multiplication theorems show that the norms L2l(A,gτ)and L2l(A′,gτ) are commensurate.Remark—For any pairτ,τ′,it is easy to construct a diﬀeomorphism f: Mτ→M′τcovered by an isomorphism F between Wτand Wτ′which are a dilations near inﬁnity.Therefore F∗C l(Mτ′)=C l(Mτ)and f∗G l(Mτ′)= G l(Mτ).In this sense,the spaces C l and G l are in fact independent ofτ. However,the fact that the norms depend onτwill become crucial for analyzing the compactness properties of the family of moduli spaces on Mτ.Of course the choice of l 2is actually perfectly arbitrary thanks to elliptic regularity.However it will be chosen with l 4so that we have the inclusion L2l⊂C1.The Sobolev multiplication theorem shows that G l is a Hilbert Lie group acting smoothly on the Hilbert aﬃne space C l.Furthermore action of G l is free:if we have u·(A,Φ)=(A,Φ),then du=0hence u must be constant. Now u−1∈L2l+1therefore u=1.Let Zτl be the space of conﬁgurations(A,Φ)∈C l which verify the Seiberg-Witten equations(2.4)on Mτwith perturbation̟τ.Then Zτl is invariant under the gauge group action and we deﬁneM l(Mτ)=Zτl/G l.17We drop the reference to the indexτat the moment,for simplicity of notations. All of what we say in the rest of Section2.2.2holds for M and Mτor indeed any AFAK manifold.The linearized action of the gauge group at an arbitrary conﬁguration(A,Φ)∈C l is given by a diﬀerential operatorδ1,(A,Φ):L2l+1(i R)−→L2l(iΛ1)⊕L2l,B(W+)v−→(−dv,vΦ)and its formal adjoint is given byδ∗1,(A,Φ)(a,φ)=−d∗a+i Im Φ,φ ;notice that with our convention,the Hermitian product ·,· is anti-complex linear in theﬁrst variable.A tangent vector(a,φ)is L2-orthogonal to the orbit through(A,Φ)if and only ifδ∗1(a,φ)=0;the orbit space C l/G l is a smooth Hilbert manifold,and its tangent space at(A,Φ)is identiﬁed with kerδ∗1,(A,Φ).The linearized Seiberg-Witten equations at(A,Φ)are given as well by a diﬀer-ential operatorδ2,(A,Φ):L2l+1(iΛ1)⊕L2l+1,B(W+)→L2l(i su W+)⊕L2l,B(W−)(a,φ)→(d+a−{Φ⊗φ∗+φ⊗Φ∗}0,D Aφ+a·Φ).Remark—There is a slight inconsistency in the conventions of[KM].The linear theory studied there is exactly the one presented in the current paper. However,this is not the one of the equations written in[KM]where F A isreplaced by the curvature FˆA of the unitary connectionˆA induced by A onthe determinant line bundle L.Since F A=2FˆA ,the corresponding linearizedequations should beδ2,(A,Φ)(a,φ)=(2d+a−{Φ⊗φ∗+φ⊗Φ∗}0,D Aφ+a·Φ). At at solution of Seiberg-Witten equations(A,Φ),the operators verifyδ2,(A,Φ)◦δ1,(A,Φ)=0,so we have an elliptic complex0→L2l+2(i R)δ1,(A,Φ)−→L2l+1(iΛ1)⊕L2l+1,B(W+)δ2,(A,Φ)−→L2l(i su W+)⊕L2l,B(W−)−→0; H0=0since the action of the gauge group is free,H1=kerδ2/Imδ1is the virtual tangent space to the Seiberg-Witten moduli space at(A,Φ),and H2= cokerδ2is the obstruction space.Equivalently H1can be viewed as the kernel of the elliptic operator18D(A,Φ)=δ∗1,(A,Φ)⊕δ2,(A,Φ).(2.8) Facts:•The moduli space M l is compact.•By elliptic regularity,M l=M l+1=M l+2=···=M r+1;this is precisely why the choice of l does not matter.So the moduli space is simply referred to by M.•By Sard-Smale theory,we may always assume that H2=0after choosinga suitable generic perturbation̟τ.Then M is unobstructed;it is asmooth manifold of dimension equal to the virtual dimensiond= e(W+,Ψ),[M be a manifold with a contact boundary(Y,ξ)and an element(s,h)∈Spin c(M with Z or Z′along Y,together with the Riemannian metrics gτand g′τand the particular Seiberg-Witten equations with perturbation̟τconstructed in Section2.2.1.Then,forτlarge enough,the moduli spaces M(Mτ)and M(M′τ)are generic, and there is a diﬀeomorphismG:M̟τ(Mτ)→M̟τ(M′τ).Furthermore there is a canonical identiﬁcation of the set of consistent orienta-tions for Mτand M′τ.Using this canonical identiﬁcation the above diﬀeomor-phism becomes orientation preserving.Remark 2.2.3—If we remove the assumption(2.2)for the cobordisms Z and Z′,then the extension maps j:Spin c(M,ξ)→Spin c(M′τ,ω′τ)are not injective in general.Then we may still prove a generalization of Theorem E:assume that Z is just the symplectization of(Y,ξ)and that Z′is an AFAK end as before,but without19assuming the property(2.2).To discuss the generalization we need to make the notation more precise.We denote by M̟τ(Mτ,j(s,h))the moduli space for some choice of(s,h)∈Spin c(M,ξ) j′(s,h)=(s′,h′)M̟τ(Mτ,j(s,h)).(2.9)Then the conclusion of Theorem E is the same if we replace M̟τ(M′τ)byM̟τ(M′τ,(s′,h′))and M̟τ(Mτ)by M̟τ(Mτ,(s′,h′)).In particular,there isan orientation preserving diﬀeomorphismG: M̟τ(Mτ,(s′,h′))→M̟τ(M′τ,(s′,h′)).The rest of Section2is devoted to construct the map G by a gluing technique and to show that it is a diﬀeomorphism.2.2.4CompactnessIn this section,we reﬁne the result of compactness for oneﬁxed moduli space M(Mτ),by showing that a sequence of solutions(Aτ,Φτ)of Seiberg-Witten equations on Mτconverge in some sense asτ→∞,up to extraction of a sub-sequence,and modulo gauge transformations,to a solution of Seiberg-Witten equations on M.We review the arguments proving the compactness of one particular moduli space in[KM]and explain how to apply them to the family Mτ.Lemma2.2.5There exist constantsκ1,κ2such that for everyτand every solution of Seiberg-Witten equations(A,Φ)on Mτwe have the estimateΦ C0 κ1+κ2 ̟τ C0.Proof By construction of the moduli space,Φ−Ψ∈L23;now the Sobolev inclusion C0⊂L23together with Lemma2.1.6tells us that the pointwise norm |Φ−Ψ|C0→0near inﬁnity on Mτ.Hence,either|Φ| 1,and we are done,either this is not true,and|Φ|must have a local maximum at a point x∈Mτ.We apply the maximum principle20at x:14|Φ2|where the last identity follows from the Lichnerowicz ing Seiberg-Witten equations,we have0 −14|Φ|2.The lemma follows from the fact that the pointwise norm of s,F B,Ψis bounded independently ofτby Lemma2.1.6.As we saw,the AFAK structure induces a Chern connectionˆ∇and and a spin connection B on the bundle Wτrestricted to Mτ\Mbyˆ∇AΦ:=ˆ∇Φ+a⊗Φ;in particularˆ∇B:=ˆ∇with these notations.Notice that∇B=ˆ∇B unless the almost complex structure is integrable.Proposition2.2.6There exists a compact K⊂M large enough andδ>0 such that for every integer k there is a constant c k>0so that,for everyτlarge enough and every solution of Seiberg-Witten equations(A,Φ)on Mτ,we have the pointwise estimate on Mτ\K|1−|β|2−|γ|2|,|γ|,|ˆ∇AΦ|,|ˆ∇2AΦ|,······,|ˆ∇k AΦ| c k e−δστ,(2.10) whereΦ=(β,γ)∈Λ0,0⊕Λ0,2.Remark—The quantities(2.10)controlled by the lemma are gauge invari-ant.Proof The lemma was proved in[KM,Proposition3.15]forτﬁxed.It is readily checked that it extends as stated for the family Mτ.We recall what the ingredients of the proof are.A conﬁguration(A,Φ)on Mτhas an energy which is a gauge invariant quantity deﬁned byEτ(A,Φ)= Mτ\K0 (|β|2+|γ|2−1)2+|γ|2+|ˆ∇AΦ|2 vol gτ, where all the norms,connections are taken with respect to the structures deﬁned on Mτand K0is a compact in M containingLemma2.2.7There exist a compact K0⊂M large enough,and some con-stantsκ3andκ4,such that for everyτlarge enough and every solution of Seiberg-Witten equations(A,Φ)on Mτ,we haveEτ(A,Φ) κ3+κ4 ̟τ 2Nτ.Proof The proof of this lemma is the same proof than for[KM,Lemma3.17]. The fact thatκ3andκ4do not depend onτis insured by Lemma2.1.6.If we look carefully at the proof,using the notations of[KM,page232],we read the claim that K3da∧ωcan be controlled,for K3a compact domain large enough,ωa closed form extending the symplectic form on the whole manifold as in Deﬁnition2.1.1,and a=A−B decaying exponentially fast.No explanation of this is given and we provide one now.Pick an arbitrarily smallε>0.We haveK3da∧ω ε2ε K3|ω|2 ε2ε K3|ω|2.Using the exponential decay of a,we have as in the case of a compact manifoldM|da|2=2 M|d+a|2.ThereforeK3da∧ω ε M\K3|d+a|2+ε K3|d+a|2+1。

Edge solitons in the QHE

1
Reduction of Chern-Simons
The Landau-Ginzburg theory of the Quantum Hall Eﬀect [1] uses the Chern-Simons Lagrangian in (2 + 1) dimensions, 1 1 ǫµνρ Fµν Aρ + iφ⋆ Dt φ − |Dφ|2 − V (φ), (1) 4κ 2 where the scalar ﬁeld φ is the order parameter and Aµ is the statistical gauge ﬁeld; Dµ = ∂µ − iAµ is the covariant derivative. The constant κ is interpreted as the Hall conductivity. The second-order ﬁeld equations are not integrable [2]; they admit integrable reductions, though. The simplest of these is when time-dependence is eliminated; then, for a judicious choice of the self-interaction potential V (φ), the system admits ﬁniteenergy vortex solutions [3]. Here we focus our attention to another, space-like reduction [4]. Assuming independence from one spacelike coordinate and adding a suitable kinetic term yields in fact, after elimination of the gauge ﬁeld using its equation of motion, L= 1 L = iφ⋆ ∂t φ − |(∂x − iκ2 ρ)φ|2 − V, 2 (2)

Statistical mechanics of two-dimensional vortices and stellar systems

a r X iv:c o n d -ma t/212223v1[c ond-m at.stat-m ec h]1D ec22Statistical mechanics of two-dimensional vortices and stellar systems Pierre-Henri Chavanis Laboratoire de Physique Quantique,Universit´e Paul Sabatier,118,route de Narbonne 31062Toulouse,France Abstract.The formation of large-scale vortices is an intriguing phenomenon in two-dimensional turbulence.Such organization is observed in large-scale oceanic or atmo-spheric ﬂows,and can be reproduced in laboratory experiments and numerical simula-tions.A general explanation of this organization was ﬁrst proposed by Onsager (1949)by considering the statistical mechanics for a set of point vortices in two-dimensional hydrodynamics.Similarly,the structure and the organization of stellar systems (glob-ular clusters,elliptical galaxies,...)in astrophysics can be understood by developing a statistical mechanics for a system of particles in gravitational interaction as initiated by Chandrasekhar (1942).These statistical mechanics turn out to be relatively similar and present the same diﬃculties due to the unshielded long-range nature of the in-teraction.This analogy concerns not only the equilibrium states,i.e.the formation of large-scale structures,but also the relaxation towards equilibrium and the statistics of ﬂuctuations.We will discuss these analogies in detail and also point out the speciﬁcities of each system.1Introduction Two-dimensional ﬂows with high Reynolds numbers have the striking property of organizing spontaneously into coherent structures (the vortices)which dominate the dynamics [93](see Fig.1).The robustness of Jupiter’s Great Red Spot,a huge vortex persisting for more than three centuries in a turbulent shear between two zonal jets,is probably related to this general property.Some other coherent structures like dipoles (pairs of cyclone/anticyclone)and sometimes tripoles have been found in atmospheric or oceanic systems and can persist during several days or weeks responsible for atmospheric blocking.Some astrophysicists invoke the existence of organized vortices in the gaseous component of disk galaxies in relation with the emission of spiral density waves [99].It has also been proposed that planetary formation might have begun inside persistent gaseous vortices born out of the protoplanetary nebula [5,121,15,60,33](see Fig.2).As a result,hydrodynamical vortices occur in a wide of geophysical or astrophysical situations and their robustness demands a general understanding.Similarly,it is striking to observe that self-gravitating systems follow a kindof organization despite the diversity of their initial conditions and their en-vironement [9](see Fig.3).This organization is illustrated by morphological classiﬁcation schemes such as the Hubble sequence for galaxies and by simple2Pierre-Henri Chavanisrules which govern the structure of individual self-gravitating systems.For ex-ample,elliptical galaxies display a quasi-universal luminosity proﬁle described by de Vaucouleur’s R1/4law and most of globular clusters are well-ﬁtted by the Michie-King model.On the other hand,theﬂat rotation curves of spiral galaxies can be explained by the presence of a dark matter halo with a density proﬁle decreasing as r−2at large distances.The fractal nature of the interstellar medium and the large scale structures of the universe also display some form of organization.Fig.1.Self-organization of two-dimensional turbulentﬂows into large-scale vortices [93].These vortices are long-lived and dominate the dynamics.The question that naturally emerges is what determines the particular con-ﬁguration to which a self-gravitating system or a large-scale vortex settles.It is possible that their actual conﬁguration crucially depends on the conditions that prevail at their birth and on the details of their evolution.However,in view of their apparent regularity,it is tempting to investigate whether their organi-zation can be favoured by some fundamental physical principles like those of thermodynamics and statistical physics.We ask therefore if the actual states of self-gravitating systems in the universe and coherent vortices in two-dimensional2D vortices and stellar systems3Fig.2.A scenario of planet formation inside large-scale vortices presumably present in the Keplerian gaseous disk surrounding a star at its birth.Starting from a random vorticityﬁeld,a series of anticyclonic vortices appears spontaneously(upper panel). Due to the Coriolis force and to the friction with the gas,these vortices can eﬃciently trap dust particles passing nearby(lower pannel).The local increase of dust concen-tration inside the vortices can initiate the formation of planetesimals and planets by gravitational instability.This numerical simulation is taken from[15].turbulentﬂows are not simply more probable than any other possible conﬁgura-tion,i.e.if they cannot be considered as maximum entropy states.This statistical mechanics approach has been initiated by Onsager[101]for a system of point vortices and by Chandrasekhar[21]in the case of self-gravitating systems.It turns out that the statistical mechanics of two-dimensional vortices and self-gravitating systems present a deep analogy despite the very diﬀerent physical nature of these systems.This analogy was pointed out by Chavanis in[29,32,35] and further developed in[54,30,34,36,47,48].In the following,we will essentially discuss the statistical mechanics of2D vortices and refer to the review of Pad-manabhan[103](and his contribution in this book)for more details about the4Pierre-Henri Chavanisrge-scale structures in the universe as observed with the Hubble space tele-scope.The analogy with Fig.1is striking and will be discussed in detail in this paper. statistical mechanics of self-gravitating systems.We will see that the analogy be-tween two-dimensional vortices and(three-dimensional)self-gravitating systems concerns not only the prediction of the equilibrium state,i.e.the formation of large-scale structures,but also the statistics ofﬂuctuations and the relaxation towards equilibrium.This paper is organized as follows.In Sec.2,we discuss the statistical me-chanics of point vortices introduced by Onsager[101]and further developed by Joyce&Montgomery[70]and Pointin&Lundgren[107]among others(see a complete list of references in the book of Newton[98]).We discuss the existence of a thermodynamic limit in Sec.2.7and make the connexion withﬁeld theory. Statistical equilibrium states of axisymmetricﬂows are obtained analytically in Sec.2.8-2.9.The relation with equilibrium states of self-gravitating systems is shown in Sec.2.10.In Sec.3,we discuss the statistics of velocityﬂuctuations pro-duced by a random distribution of point vortices and use this stochastic approach to obtain an estimate of the diﬀusion coeﬃcient of point vortices.Application to2D decaying turbulence is considered in Sec.3.4.In Sec.4,we describe the relaxation of a point vortex in a thermal bath and analyze this relaxation in terms of a Fokker-Planck equation involving a diﬀusion and a drift.In Sec.5,we develop a more general kinetic theory of point vortices.A new kinetic equation is obtained which satisﬁes all conservation laws of the point vortex system and increases the Boltzmann entropy(H-theorem).We mention the connexion with2D vortices and stellar systems5 the kinetic theory of stars developed by Chandrasekhar[21].In Sec.6,we dis-cuss the violent relaxation of2D vortices and stellar systems.We mention the analogy between the Vlasov and the Euler equations and between the statistical approach developed by Lynden-Bell[90]for collisionless stellar systems and by Kuz’min[83],Miller[95]and Robert&Sommeria[111]for continuous vorticity ﬁelds.The concepts of“chaotic mixing”and“incomplete relaxation”are dis-cussed in the light of a relaxation theory in Sec.6.3.Application of statistical mechanics to geophysicalﬂows and Jupiter’s Great Red Spot are evocated in Sec.6.4.2Statistical mechanics of point vortices intwo-dimensional hydrodynamics2.1Two-dimensional perfectﬂowsThe equations governing the dynamics of an invisicidﬂow are the equation of continuity and the Euler equation:∂ρ∂t +(u∇)u=−16Pierre-Henri Chavaniswhere ∆=∂2xx +∂2yy is the Laplacian operator.In an unbounded domain,thisequation can be written in integral form asψ(r ,t )=−12πz × ω(r ′,t )r −r ′∂t +u ∇ω=0.(10)This corresponds to the transport of the vorticity ωby the velocity ﬁeld u .It is easy to show that the ﬂow conserves the kinetic energyE =u 22 (∇ψ)2d 2r =12 ωψd 2r ,(12)where the second equality is obtained by a part integration with the condition ψ=0on the boundary.Therefore,E can be interpreted either as the kinetic energy of the ﬂow (see Eq.(11))or as a potential energy of interaction between vortices (see Eq.(12)).2.2The point vortex gasWe shall consider the situation in which the velocity is created by a collection of N point vortices.In that case,the vorticity ﬁeld can be expressed as a sum of δ-functions in the formω(r ,t )=N i =1γi δ(r −r i (t )),(13)where r i (t )denotes the position of point vortex i at time t and γi is its circulation.According to Eqs.(9)(13),the velocity of a point vortex is equal to the sum of the velocities V (j →i )produced by the N −1other vortices,i.e.V i = j =i V (j →i )with V (j →i )=−γj |r j −r i |2.(14)2D vortices and stellar systems7 As emphasized by Kirchhoﬀ[79],the above dynamics can be cast in a Hamil-tonian formγi dx i∂y i,γidy i∂x i,(15) H=−12m).Thisis related to the particular circumstance that a point vortex is not a material particle.Indeed,an isolated vortex remains at rest contrary to a material particle which has a rectilinear motion due to its inertia.Point vortices form therefore a very peculiar Hamiltonian system.Note also that the Hamiltonian of point vortices can be either positive or negative(in the case of vortices of diﬀerent signs)whereas the kinetic energy of theﬂow is necessarily positive.This is clearlya drawback of the point vortex model.2.3The microcanonical approach of Onsager(1949)The statistical mechanics of point vortices wasﬁrst considered by Onsager[101] who showed the existence of negative temperature states at which point vortices of the same sign cluster into“supervortices”.He could therefore explain the formation of large,isolated vortices in nature.This was a remarkable anticipation since observations were very scarce at that time.Let us consider a liquid enclosed by a boundary,so that the vortices are conﬁned to an area A.Since the coordinates(x,y)of the point vortices are canonically conjugate,the phase space coincides with the conﬁguration space and isﬁnite:dx1dy1...dx N dy N= dxdy N=A N.(17)This striking property contrasts with most classical Hamiltonian systems con-sidered in statistical mechanics which have unbounded phase spaces due to the presence of a kinetic term in the Hamiltonian.As is usual in the microcanonical description of a system of N particles,we introduce the density of statesg(E)= dx1dy1...dx N dy Nδ E−H(x1,y1,...,x N,y N) ,(18)8Pierre-Henri Chavaniswhich gives the phase space volume per unit interaction energy E.The equilib-rium N-body distribution of the system,satisfying the normalization condition µ(r1,...,r N)d2r1...d2r N=1,is given byµ(r1,...,r N)=1T =dS2π i<jγiγj ln|r i−r j|.(22)2D vortices and stellar systems9 Making the change of variable x=r/R,weﬁnd that g(E,V)=V N g(E′,1) with E′=E+1βV 1+ββV 1+γ2(N−1)γ2.(25)We shall see in Sec.2.8that this negative critical inverse temperature is the minimum inverse temperature that the system can achieve.If,on the other hand,we consider a neutral system consisting of N/2vortices of circulationγand N/2vortices of circulation−γ,weﬁndP=N8π .(26)This result is well-known is plasma physics[113].The critical temperature at which the pressure vanishes is now positiveβc=8π∂t =−Ni=1γ∇δ(r−r i(t))V i.(28)Since V i=u(r i(t),t),we can rewrite the foregoing equation in the form∂ω10Pierre-Henri ChavanisSince the velocity is divergenceless,we obtain∂ω4π i=jγ2 ln|r i−r j|1=−2D vortices and stellar systems 11whereg (r 1,r 2,t )=µ(r 1,...,r N ,t )d 3r 2...d 2r N ,(36)is the two-body distribution function.In the mean-ﬁeld approximation,whichis exact in a properly deﬁned thermodynamic limit with N →+∞(see Sec.2.7),we haveg (r 1,r 2,t )=P (r 1,t )P (r 2,t ).(37)Accounting that N (N −1)≃N 2for large N ,the average energy takes the formE =−12ω ψd 2r =u 2n i !.(40)The logarithm of this number deﬁnes the Boltzmann ing Stirlingformula and considering the continuum limit in which ∆,ν→0,we get the classical formula S =−NP (r )ln P (r )d 2r ,(41)where P (r )is the density probability that a point vortex be found in the sur-face element centered on r .At equilibrium,the system is in the most probablemacroscopic state,i.e.the state that is the most represented at the microscopic level.This optimal state is obtained by maximizing the Boltzmann entropy (41)at ﬁxed energy (39)and vortex number N ,or total circulationΓ=Nγ=ω d 2r .(42)12Pierre-Henri ChavanisWriting the variational principle in the formδS−βδE−αδΓ=0,(43) whereβandαare Lagrange multipliers,it is readily found that the maximum entropy state corresponds to the Boltzmann distributionω =Ae−βγψ,(44) with inverse temperatureβ.We can account for the conservation of angular momentum L= ω r2d2r(in a circular domain)and impulse P= ω yd2r (in a channel)by introducing appropriate Lagrange multipliersΩand U foreach of these constraints.In that case,Eq.(44)remains valid provided that we replace the streamfunctionψby the relative streamfunctionψ′=ψ+Ω2π i<j ln|r i−r j|2D vortices and stellar systems13 where the potential of interaction has been normalized by R.For simplicity, we have ignored the contribution of the images but this shall not aﬀect the ﬁnal results.We now introduce the change of variables x=r/R and deﬁne the functionu(x1,...,x N)=1N2γ2.(49) In terms of these quantities,the density of states can be rewritteng(E)=2πV NNu(x1,...,x N) .(50)The proper thermodynamic limit for a system of point vortices with equal cir-culation in the microcanonical ensemble is such that N→+∞withﬁxedΛ.Wesee that the box size R does not enter in the normalized energyΛ.Therefore,the thermodynamic limit corresponds to N→+∞withγ∼N−1→0and E∼1. This is a very unusual thermodynamic limit due to the non-extensivity of thesystem.Note that the total circulationΓ=Nγremainsﬁxed in this process.For suﬃciently large N,the density of states can be writteng(E)≃ Dρe NS[ρ]δ Λ−E[ρ]) δ 1− ρ(r)d2r ,(51) withS[ρ]=− ρ(r)lnρ(r)d2r,(52) E[ρ]=−114Pierre-Henri Chavaniswhich is the normalization factor in the Gibbs measure1µ(r1,...,r N)=ln Z.(57)βUsing the notations introduced previously,we can rewrite the integral(55)in the formZ(β)=V N N0... N0N i=1d2x i eηu(x1,...,x N),(58)whereβNγ2η=2D vortices and stellar systems15 and phase transitions occur.This is the case,in particular,for the gravitational problem(see Sec.2.10).Finally,the grand canonical partition function is deﬁned byZ GC=+∞ N=0z N2(∇ξ)2−ρ(r)ξ(r)}d2r=e−12(∇ξ)2d2r+√−βγξ(r),we can easily carry out the summation on N to obtainZ GC= Dφe−12(∇φ)2−µ2eφ(r)},(67)T eff=−βγ2,µ2=−zβγ2.(68) Therefore,the grand partition function of the point vortex gas corresponds to a Liouvilleﬁeld theory with an actionS[φ]=12(∇φ)2−µ2eφ(r)}.(69)While the previous description is formally correct if we deﬁne Z and Z GC by Eqs.(55)and(64),it must be noted however that the canonical and grand canon-ical ensembles may not have a physical meaning for point vortices.In particular, it is not clear how one can impose a thermal bath at negative temperature. On the other hand,the usual procedure to derive the canonical ensemble from the microcanonical ensemble rests on a condition of additivity which is clearly lacking in the present case.2.8Axisymmetric equilibrium states in a diskLet us consider a collection of N point vortices with circulationγconﬁned within a disk of radius R.At statistical equilibrium,the streamfunctionψis solution of the Boltzmann-Poisson equation(45).If we work in a circular domain,we must in principle account for the conservation of angular momentum.This can lead to bifurcations between axisymmetric and oﬀ-axis solutions[117].We shall,16Pierre-Henri Chavanishowever,ignore this constraint for the moment in order to obtain analytical ex-pressions for the thermodynamical parameters.This is a suﬃcient approximation to illustrate the structure of the problem,which is our main concern here.If we conﬁne our attention to axisymmetric solutions,the Boltzmann-Poisson equation(45)can be written1dr r dψξddξ =λe−φ,(72)φ(0)=φ′(0)=0,(73)withλ=1ifβ<0andλ=−1ifβ>0.It turns out that this equation can be solved analytically as noticed by a number of authors.With the change of variables t=lnξandφ=2lnξ−z,Eq.(72)can be rewrittend2zdz(λe z).(74)This corresponds to the motion of aﬁcticious particle in a potential V(z)=λe z. This equation is readily integrated and,returning to original variables,weﬁnally obtaine−φ=18ξ2)2.(75)From the circulation theorem(6)applied to an axisymmetricﬂow,we have−dψ2πr.(76) whereΓ(r)= r0ω(r′)2πr′dr′is the circulation within r.Taking r=R and introducing the dimensionless variables previously deﬁned,we obtainη≡βγΓπR2(η+4)1η+4r22D vortices and stellar systems 17At positive temperatures (η>0),the vorticity is an increasing function of the distance and the vortices tend to accumulate on the boundary of the domain (Fig.4).On the contrary at negative temperatures (η<0),the vorticity is a decreasing function of the distance and the vortices tend to group themselves in the core of the domain to form a “supervortex”(Fig.5).These results are consistent with Onsager’s prediction [101].We also conﬁrm that statistical equilibrium states only exist for η>ηc =−4,as previously discussed.At this critical temperature,the central vorticity becomes inﬁnite and the solution tends to a Dirac peak:ω (r )→Γδ(r ),for η→ηc =−4.(79)00.20.40.60.81r/R1234567<ω>/ω∗η>0η=0η=20Fig.4.Statistical equilibrium states of point vortices at positive temperatures (ω∗=Γ/πR 2).The vortices are preferentially localized near the wall.The energy deﬁned by Eq.(39)can be written in the dimensionless formΛ≡2πE2η2 α0φ′(ξ)2ξdξ.(80)The integral can be carried out explicitly using Eq.(75).Eliminating αbetweenEqs.(80)and (77),we ﬁnd that the temperature is related to the energy by the equation of state Λ=1ηln418Pierre-Henri Chavanis00.20.40.60.81r/R246810<ω>/ω∗η<0η=−2η=−3η=−3.5Fig.5.Statistical equilibrium states of point vortices at negative temperatures showing a clustering.For η=−4,the vortices collapse at the center of the domain and the vorticity proﬁle is a Dirac peak.0.250.50.751Λ=2πE/Γ2−10−8−6−4−20246810η=βγΓ/2πΛ0=1/8ηc =−4Fig.6.Equilibrium phase diagram (caloric curve)for point vortices with equal circula-tion conﬁned within a disk.For simplicity,the angular momentum has not been taken into account (Ω=0).2D vortices and stellar systems1900.250.50.751Λ=2πE/Γ2−7−6−5−4−3S /NΛ0=1/8η=0η>0η<0ηc =−4Fig.7.Entropy vs energy plot for a system of point vortices with equal circulation conﬁned within a disk.The entropy (41)can also be calculated easily from the above results.Within an unimportant additive constant,it is given byS ηln 4−1+820Pierre-Henri Chavanisaccount,the density of states is given byg (E,L )=δ E −H (r 1,...,r N ) δ L −N i =1γr 2i N i =1d 2r i ,(83)and the angular velocity of the ﬂow byΩ=2T∂SL ,oneﬁnds the exact resultΩ=2N8π.(85)Therefore,the vorticity ﬁeld is determined by the Boltzmann distributionω =Ae −βγψ′,(86)where ψ′is the relative streamfunctionψ′≡ψ+Ω4βL(4+η)r 2.(87)For η=0,one hasω =γNΓLr 2.(88)For large r ,the asymptotic behavior of Eq.(86)isω ∼14L(4+η)r 2(r →+∞),(89)where we have used ψ∼−(Γ/2π)ln r at large distances.From Eq.(89),one sees that η≥−4is required for the existence of an integrable solution.Inserting the relation (86)in the Poisson equation (7),we get−∆ψ′=Ae −2πη2πLη(4+η).(90)With the change of variablesξ=γNN 2γ2−2πηdξ2+1dξ=2πηe φ−(4+η),(92)2D vortices and stellar systems21 and the vorticity(86)becomesω =N2γ22πL,if r≤(2L/Γ)1/2,(94)and ω =0otherwise.This vortex patch is the state of minimum energy at ﬁxed circulation and angular momentum.Forη→−4,one has approximatelyω =N2γ2(1−AπηγN4L(4+η)r2.(95)Theﬁrst factor is an exact solution of Eq.(92)with the second term on the right hand side neglected(see Sec.2.8).The second factor is a correction for large r, in agreement with the asymptotic result expressed by Eq.(89).The parameter A tends to inﬁnity asη→−4and is determined from the condition ω d2r=Γby the formulaπA+ln(πA)=−C−ln 1+η|r j−r i|3,(97)where F(j→i)is the force created by star j on star i.The force can be written as the gradient F=−∇Φof a gravitational potentialΦwhich is related to the stellar densityρ(r,t)=Ni=1mδ(r−r i),(98)by the Poisson equation∆Φ=4πGρ.(99)22Pierre-Henri ChavanisFurthermore,the equations of motion(Newton’s equations)can be put in the Hamiltonian formm d r i∂v i,md v i∂r i,H=1|r i−r j|.(100)In the analogy between stellar systems and two-dimensional vortices,the star densityρplays the role of the vorticityω,the force F the role of the velocity V and the gravitational potentialΦthe role of the streamfunctionψ.The crucial point to realize is that,for the two systems,the interaction is a long-range un-shielded Coulombian interaction(in D=3or D=2dimensions).This makes the connexion between point vortices and stellar systems deeper than between point vortices and electric charges for example.In particular,point vortices can organize into large scale clusters,like stars in galaxies,while the distribution of electric charges in a neutral plasma is uniform.There are,on the other hand, fundamental diﬀerences between stars and vortices.In particular,a star creates an acceleration while a vortex creates a velocity.On the other hand,the gravi-tational interaction is attractive and directed along the line joining the particles while the interaction between vortices is rotational and perpendicular to the line joining the vortices.Despite these important diﬀerences,the statistical mechanics of2D vortices and stellar systems are relatively similar.Like the point vortex gas,the self-gravitating gas is described at statistical equilibrium by the Boltzmann distri-butionρ =Ae−βΦ,(101) obtained by maximizing the Boltzmann entropy atﬁxed mass M and energy E. Its structure is therefore determined by solving the Boltzmann-Poisson equation∆Φ=4πGAe−βΦ,(102) where A andβ>0have to be related to M and E.This statistical mechanics approach has been developed principally for globular clusters relaxing towards equilibrium via two-body encounters[9].It is clear that the Boltzmann-Poisson equation(102)is similar to the Boltzmann-Poisson equation(45)for point vor-tices at negative temperatures.The density proﬁle determined by these equations is a decreasing function of the distance,which corresponds to a situation of clus-tering(see Figs.5and8).The similarity of the maximum entropy problem for stars and vortices,and the Boltzmann-Poisson equations(102)(45),is aﬁrst manifestation of the formal analogy existing between these two systems.However,due to the diﬀerent dimension of space(D=3for stars instead of D=2for vortices),the mathematical problems diﬀer in the details.First of all,the density proﬁle determined by the Boltzmann-Poisson equation(102)in D=3decreases like r−2at large distances leading to the so-called inﬁnite mass2D vortices and stellar systems 23−50510ln(ξ)−20−15−10−50ln (e −ψ)singularsphereξ−2Fig.8.Density proﬁle of the self-gravitating gas at statistical equilibrium.The dashed line corresponds to the singular solution ρ=1/2πGβr 2.problem since M = +∞0ρ4πr 2dr →+∞[19].There is no such problem forpoint vortices in two dimensions:the vorticity decreases like r −4,or even more rapidly if the conservation of angular momentum is accounted for,and the totalcirculation Γ= +∞0ω2πrdr is ﬁnite.The inﬁnite mass problem implies thatno statistical equilibrium state exists for open star clusters,even in theory.A system of particles in gravitational interaction tends to evaporate so that the ﬁnal state is just two stars in Keplerian orbit.This evaporation process has been clearly identiﬁed in the case of globular clusters which gradually lose stars to the beneﬁt of a neighboring galaxy.In fact,the evaporation is so slow that we can consider in a ﬁrst approximation that the system passes by a succession of quasiequilibrium states described by a truncated isothermal distribution function (Michie-King model)[9].This justiﬁes the statistical mechanics approach in that sense.Another way of solving the inﬁnite mass problem is to conﬁne the system within a box of radius R .However,even in that case,the notion of equilibrium poses problem regarding what now happens at the center of the conﬁguration.The equilibrium phase diagram (E,T )for bounded self-gravitating systems is represented in Fig.9.The caloric curve has a striking spiral behavior parametrized by the density contrast R =ρ(0)/ρ(R )going from 1(homogeneous system)to +∞(singular sphere)as we proceed along the spiral.There is no equilibrium state below E c =−0.335GM 2/R or T c =GMm24Pierre-Henri ChavanisΛ=−ER/GM 20.51.52.5η=βG M /R Fig.9.Equilibrium phase diagram for self-gravitating systems conﬁned within a box.For suﬃciently low energy or temperature,there is no equilibrium state and the system undergoes gravitational collapse.−1−0.500.51 1.5Λ=−ER/GM 200.511.522.533.5η=βG M /R µ=105µ=104µ=103µ=102µ=10Fig.10.Equilibrium phase diagram for self-gravitating fermions [41].The degeneracy parameter µplays the role of a small-scale cut-oﬀǫ∼1/µ.For ǫ→0,the classical spiral of Fig.9is recovered.2D vortices and stellar systems25 mass goes to zero.Therefore,the singularity contains no mass and this process alone cannot lead to a black hole.Since the T(E)curve has turning points,this implies that the microcanon-ical and canonical ensembles are not equivalent and that phase transitions will occur[103].In the microcanonical ensemble,the series of equilibria becomes un-stable after theﬁrst turning point of energy(MCE)corresponding to a density contrast of709.At that point,the solutions pass from local entropy maxima to saddle points.In the canonical ensemble,the series of equilibria becomes unsta-ble after theﬁrst turning point of temperature(CE)corresponding to a density contrast of32.1.At that point,the solutions pass from minima of free energy (F=E−T S)to saddle points.It can be noted that the region of negative speciﬁc heats between(CE)and(MCE)is stable in the microcanonical en-semble but unstable in the canonical ensemble,as expected on general physical grounds.The thermodynamical stability of isothermal spheres can be deduced from the topology of theβ−E curve by using the turning point criterion of Katz[75]who has extended Poincar´e’s theory of linear series of equilibria.The stability problem can also be reduced to the study of an eigenvalue equation associated with the second order variations of entropy or free energy as studied by Padmanabhan[102]in the microcanonical ensemble and by Chavanis[37]in the canonical ensemble.This study has been recently extended to other statisti-cal ensembles[44]:grand canonical,grand microcanonical,isobaric....The same stability limits as Katz are obtained but this method provides in addition the form of the density perturbation proﬁles that trigger the instability at the critical points.It also enables one to show a clear equivalence between thermodynamical stability in the canonical ensemble and dynamical stability with respect to the Navier-Stokes equations(Jeans problem)[37,44].These analytical methods can be extended to general relativity[38].It must be stressed,however,that the statistical equilibrium states of self-gravitating systems are at most metastable: there is no global maximum of entropy or free energy for a classical system of point masses in gravitational interaction[2].Phase transitions in self-gravitating systems can be studied in detail by in-troducing a small-scale cut-oﬀǫin order to regularize the potential.This can be achieved for example by considering a system of self-gravitating fermions (in which case an eﬀective repulsion is played by the Pauli exclusion principle) [62,8,52,41,43]or a hard spheres gas[3,103,120].Other forms of regularization are possible[59,128,45].For these systems,there can still be gravitational collapse but the core will cease to shrink when it feels the inﬂuence of the cut-oﬀ.The result is the formation of a compact object with a large mass:a“fermion ball”or a hard spheres“condensate”.The equilibrium phase diagram of self-gravitating fermions is represented in Fig.10and has been discussed at length by Chavanis [41]in the light of an analytical model.The introduction of a small-scale cut-oﬀhas the eﬀect of unwinding the classical spiral of Fig.9.For a small cut-oﬀǫ≪1,the trace of the spiral is still visible and the T(E)curve is multivalued (Fig.11).This can lead to a gravitationalﬁrst order phase transition between a gaseous phase with an almost homogeneous density proﬁle(upper branch)and a。

化学英语考试题及答案

化学英语考试题及答案一、选择题（每题2分，共20分）1. What is the chemical symbol for oxygen?A. OB. OHC. H2OD. O22. Which of the following is not a basic unit of a chemical element?A. AtomB. MoleculeC. ProtonD. Electron3. What is the law of conservation of mass?A. Mass is neither created nor destroyed in a chemical reaction.B. Mass is always created in a chemical reaction.C. Mass is always destroyed in a chemical reaction.D. None of the above.4. What is the correct chemical formula for water?A. H2OB. H2O2C. O2HD. OH-5. Which of the following is a noble gas?A. Oxygen (O)B. Helium (He)C. Carbon (C)D. Nitrogen (N)6. What is the valency of hydrogen in the compound HCl?A. +1B. -1C. +2D. -27. Which of the following is a compound?A. Oxygen gas (O2)B. Carbon dioxide (CO2)C. Hydrogen gas (H2)D. Nitrogen gas (N2)8. What is the common name for the chemical NaCl?A. SodiumB. ChlorineC. Table saltD. Baking soda9. What is the atomic number of carbon?A. 6B. 16C. 12D. 810. What is the process of a substance changing from a solidto a liquid state called?A. SublimationB. VaporizationC. MeltingD. Condensation二、填空题（每空1分，共10分）11. The chemical equation for the combustion of methane is__________.Answer: CH4 + 2O2 → CO2 + 2H2O12. The symbol for the element with atomic number 17 is__________.Answer: Cl13. The process of a substance changing from a liquid to agas is known as __________.Answer: Vaporization14. The chemical formula for calcium carbonate is __________. Answer: CaCO315. The valency of chlorine in the compound KCl is __________. Answer: -116. The noble gas that is used in balloons is __________.Answer: Helium (He)17. The atomic number of oxygen is __________.Answer: 818. The process of a substance changing from a gas to aliquid is known as __________.Answer: Condensation19. The chemical symbol for the element with the highest atomic mass is __________.Answer: Og (Oganesson)20. The law that states the total mass of the reactants equals the total mass of the products in a chemical reaction is known as the __________.Answer: Law of Conservation of Mass三、简答题（每题5分，共20分）21. Explain the difference between a pure substance and a mixture.Answer: A pure substance consists of a single type of particle, while a mixture is composed of two or moredifferent substances that are not chemically combined.22. What is an acid and what are its properties?Answer: An acid is a substance that donates hydrogen ions (H+) when dissolved in water. Properties of acids include a sour taste, the ability to change the color of certain indicators, and the ability to react with bases to form salts and water.23. Describe the process of photosynthesis.Answer: Photosynthesis is the process by which green plants and some other organisms use sunlight to synthesize foods with the help of chlorophyll pigments. It involves theconversion of carbon dioxide and water into glucose and oxygen in the presence of light energy.24. What is a chemical equilibrium and how is it represented? Answer: Chemical equilibrium is the state in a reversible reaction where the rate of the forward reaction equals the rate of the reverse reaction, resulting in no net change in the concentrations of reactants and products. It is represented by the equilibrium constant expression, Kc, which is the ratio of the concentrations of products to reactants raised to their respective stoichiometric coefficients.四、计算题（每题5分，共10分）25. If 10 grams of sodium bicarbonate (NaHCO3) are dissolved in water and reacted with an excess of hydrochloric acid (HCl), calculate the volume。

Aguidetomanipulatingequations.

Manipulating EquationsIn derivations in class and on the homework assignments,we often need to start with one set of equations and try to end up with another equation,which tells us the quantity we are trying toﬁnd or describes some new physical principle.Here are some reminders on the basics of manipulating algebraic equations.If we start with an equation a=b,we can add the same thing to both sides:a+c=b+c. Furthermore,if we have a second equation c=d,then a+c=b+d,since we are still adding the same thing to both sides.If we have an equation a=b,we can multiply both sides of the equation by the same quantity or divide both sides of the equation by the same quantity:ac=bc or a/c=b/c.The one thing you have to be careful of here is not to divide by zero.(I will use ac and a×c interchangeably to mean“a times c”.a/c=ab×cbd=ab.Often this kind of rearrangement will allow you to divide numbers that are more convenient to divide(e.g.,because they have the same units).Dividing by a fraction is the same as multiplying by the reciprocal of that fraction(where you switch the numerator and denominator),for examplea/bb×dbc.If you have the same quantity in the numerator and denominator,you can cancel it outabc and a×ba=√a×b×c=√b×√b,since the square-root of x2is x and the square-root of a2is a. OVERPhysical quantities usually come with units,and these units multiply and divide and get raised to powers along with the numbers attached to them.For example,one meter divided by one second is1m/sec;one watt divided by one square-meter is1watt/m2.If you divide two numbers with the same units,then the units cancel out leaving you with a pure number,e.g.,(20m/10m)=2.Often you can make your life easier by setting up ratios so that units cancel out in this way as much as possible.Sometimes a homework question will ask you to produce an equation as an answer,and some-times it will ask you to produce a number with units(e.g.,the distance to the Andromeda galaxy, or the value of Hubble’s constant).In the second case,you want toﬁrst get an equation for the quantity you are trying toﬁnd,then substitute values to get your number.It is always easiest to work with symbols(f,L,d,r,θ,etc.)for as long as you can,and only substitute numbers at the end;otherwise you do a lot of unnecessary work multiplying numbers and it is easy to make a mistake along the way.With these rules in mind,let’s investigate a couple of applications of ourﬁrst equation,f= L/4πd2.(1)The luminosity of the sun is3.9×1026watts,and theﬂux of the sun at the earth’s surface is 1380watts/m2.What is the distance from the earth to the sun,in meters?Since we want a distance,let’sﬁrst solve the equation for distance by multiplying both sides by d2,dividing by f,and taking the square root of both sides.f=L4πf=⇒d= 4πf.Now substitute values:d= 4π×1380watts/m2=1.5×1011m.Note that the watts in the numerator cancelled the watts in the denominator,that1/m2in the denominator becomes m2in the numerator(i.e.,dividing by1/m2is the same as multiplying by m2),and that taking the square root of m2yields m,so our answer has the desired units.(2)Star B has twice the intrinsic luminosity of star A,but it is twice as far away.What is its apparentﬂux relative to that of star A?You may be able to answer this question in your head:the apparentﬂux is proportional to the luminosity and to the inverse square of the distance,so star B appears2/22=1/2as bright as star A.To do it more formally with the equation,note that we are being asked for the ratio f B/f A. We know thatf B=L B/4πd2B and f A=L A/4πd2A,because ourﬂux-luminosity-distance equation should apply to both stars individually.To get our desired ratio,divide the left side of theﬁrst equation by the left side of the second equation and the right side of theﬁrst equation by the right side of the second equation:f BL A/4πd2A =L B4πd2B=2×12.This strategy—write an equation twice and take ratios—is often useful,and it often allows you to cancel out numerical constants(like4π)or physical constants(like c or G)before you ever have to plug them into your calculator.。

材料科学与工程基础(英文)_南京航空航天大学中国大学mooc课后章节答案期末考试题库2023年

材料科学与工程基础（英文）_南京航空航天大学中国大学mooc课后章节答案期末考试题库2023年1.The driving force for steady-state diffusion is the __________.答案:concentration gradient2.Diffusion coefficient is with the increasing diffusion temperature.答案:exponentially increased;3.Due to , alloys are usually than pure metals of the solvent.答案:solid solution strengthening, stronger;4.The finer the grains, the larger the , and .答案:strength, hardness, toughness;5.With plastic deformation,the increase of dislocationdensity will result in .答案:higher strength;6.In general, Brinell Hardness test is to measure thematerial’s hardness.答案:relatively softer7.Yield strength is corresponding to the occurrenceof deformation.答案:noticeable plastic8.Strain Hardening is also named as .答案:work hardening9.Vacancy diffusion is usually interstitial one.答案:slower than10.Edge and screw dislocations differ in what way?答案:angle between Burgers vector and line direction.11. A ____ may form when impurity atoms are added to a solid, in which case theoriginal crystal structure is retained and no new phases are formed.答案:solid solution12.One explanation for why graphite powder acts so well as a “solid lubricant”is .答案:carbon atoms in graphite are covalently bonded within planar layers but have weaker secondary bonds between layers13.Substitutional atom (impurity) is an example of ______.答案:point defect14.Interstitial solid solution belongs to .答案:finite solid solution;15.The atomic packing factor for FCC is .答案:0.7416.The coordination number of BCC crystal structure is .答案:817.The crystal structure of Cu is ?答案:FCC18.How many atoms does the face centered cubic unit cell contain?答案:Four19.If the electron configuration of Fe is 1s2 2s2 2p6 3s2 3p6 3d6 4s2, then theelectron configurations for the Fe3+ is 1s2 2s2 2p6 3s2 _____.答案:3p6 3d520.Bonds in most metals are referred to as ______.答案:Non-directional21.Covalent bonding occurs as a result of _________ sharing.答案:electron22.Which of the following is NOT an example of primary bonding?答案:Van der Waals23.Atomic weight (A) of an element corresponds to the weighted average of theatomic masses of the atom’s naturally occurring ___________.答案:isotopes24.The point on a phase diagram where the maximum number of allowablephases are in equilibrium is .答案:eutectic point25.Sterling silver (92.5%Ag/7.5%Cu) is an example of ___________.答案:Solid solution26.Engineering stress-strain curve and true stress-strain curve are equal up to .答案:Yeild point27.Among thefollowingtypical transformations of austenite in steels,____________transformation is diffusionless.答案:martensitic28.The heat-treatable aluminum alloy can be strengthened by .答案:Both of above29.In the as-quenched state, martensite is very hard and so brittle that a heattreatment known as must be accomplished sequently.答案:tempering30.During heat treatment of steel, austenite transforms into martensite by .答案:quenching31.Which of the following plane has the highest planar density for fcc.答案:(111)32.Which of the following describes recrystallization?答案:Diffusion dependent with no change in phase composition33.Heating the cold-worked metal progresses in three stages: .答案:recovery, recrystallization, grain growth;34.Strength is increased by making dislocation motion .答案:difficult35.The boundary above which only liquid phase exist is called _________.答案:liquidus36.We have an annealed carbon steel which has hardness of 150HBS. Supposewe know the hardness of Pearlite is 200HBS and the hardness of Ferrite is 80HBS, determine the carbon amount of this steel.答案:0.45%37.The maximum solubility of C in γ-austenite - solid solution is .答案:2.1438.In a plain steel that contains 0.2 percentage carbon, we should expect: .答案:a 25% pearlite and 75% pro-eutectoid ferrite39. A copper-nickel alloy is high-temperature heat treated; the diffusion of Cuinto Ni and Ni into Cu regions is referred to as _____________________.答案:Inter-diffusion40.The phase diagram of Sn-Pb alloy is called .答案:Eutectic phase diagram。

integral equation methods in scattering theory

Integral equation methods in scattering theory are a set of mathematical techniques used to analyze the interaction of waves with obstacles. These methods are essential in understanding the behavior of waves in complex media and in particular, in determining the scattering properties of objects.In scattering theory, the interaction of a wave with an obstacle is typically described using integral equations. These equations express the relationship between the scattered field and the incident field, as well as the properties of the obstacle itself. The most common integral equation method in scattering theory is the Lippmann-Schwinger equation.The Lippmann-Schwinger equation is a Fredholm integral equation that relates the scattered field to the incident field and the obstacle's scattering operator. It is derived from the conservation of energy and momentum in the scattering process. The equation provides a means to calculate the scattered field efficiently, given a known incident field and obstacle's scattering operator.Another important integral equation method in scattering theory is the Born approximation. The Born approximation is a perturbative method that approximates the exact solution of the Lippmann-Schwinger equation using a series expansion. It is useful when the obstacle's scattering operator is small compared to the incident field, allowing for an analytical solution of the scattering problem.In addition to these two methods, there are other integral equation techniques that can be used in scattering theory, such as the Rayleigh-Sommerfeld diffraction formula and the Kirchhoff integral formula. These methods are derived from different physical assumptions and are suitable for different types of scattering problems.Integral equation methods in scattering theory have found applications in various fields, including acoustics, electromagnetics, and quantum mechanics. Inacoustics, for example, these methods are used to study the scattering of sound waves by obstacles such as buildings or mountains. In electromagnetics, they are used to analyze the interaction of electromagnetic waves with conducting objects or dielectrics. In quantum mechanics, integral equation methods are used to study the scattering of particles by potentials or potentials.Integral equation methods in scattering theory provide a powerful tool for understanding wave interactions with obstacles. They allow for efficient calculations of scattered fields and provide insights into the physical properties of scattering systems. As such, these methods continue to play a crucial role in various fields of applied mathematics and physics.。

A New Algebraization of the Lame Equation

uction
ψ ′′ (x) + E − m ℓ(ℓ + 1) sn2 x ψ (x) = 0 , (1.1)
The Lam´ e equation,
where ℓ is a real parameter1 , and sn x ≡ sn(x|m) is the usual Jacobian elliptic function of modulus m, occupies a central position in the theory of diﬀerential equations with periodic coeﬃcients. The study of its properties has attracted the attention of many illustrious mathematicians over the last century; classical references are [1–4]. Basic properties of the Lam´ e equation are as follows. First, it arises by separation of variables in the Laplace equation in ellipsoidal coordinates. Secondly, it possesses two linearly independent 2K (k ) or 4K (k )-periodic solutions (for characteristic √ values of E ) if and only if ℓ is a nonnegative integer. Here k = m, and K (k ) (denoted by K from now on) is the complete elliptic integral of the ﬁrst kind with parameter k :

A New Equation of State for Carbon Dioxide Covering the Fluid Region

1514
1514 1516 1516 1516 1516 1517 1517 1518 1518
3.3 3.4 3.5 3.6 3.7 3.8 4.
Melting Pressure. . . . . . . . . . . . . . . . . . . . . . .. Sublimation Pressure. . . . . . . . . . . . . . . . . . . .. Vapor Pressure ....................... " Saturated Liquid Density. . . . . . . . . . . . . . . .. Saturated Vapor Density. . . . . . . . . . . . . . . . .. Caloric Data on the Liquid-Vapor Phase Boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Experimental Basis of the New Equation of State. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.1 Thermal Properties .................... " 4.2 Specific Isobaric Heat Capacity. . . . . . . . . . .. 4.2.1 Experimental Results for the Specific Isobaric Heat Capacity. . . . . . . . . . . . .. 4.2.2 Results for the Specific Isobaric Heat

Stability of peakons for the Degasperis-Procesi equation

a rXiv:0712.27v1[mat h.A P]12D ec27Stability of peakons for the Degasperis-Procesi equation Zhiwu Lin ∗and Yue Liu †Abstract The Degasperis-Procesi equation can be derived as a member of a one-parameter family of asymptotic shallow water approximations to the Eu-ler equations with the same asymptotic accuracy as that of the Camassa-Holm equation.In this paper,we study the orbital stability problem of the peaked solitons to the Degasperis-Procesi equation on the line.By con-structing a Liapunov function,we prove that the shapes of these peakon solitons are stable under small perturbations.Keywords:Stabiltiy;Degasperis-Procesi equation;Peakons Mathematics Subject Classiﬁcation (2000):35G35,35Q51,35G25,35L051Introduction The Degasperis-Procesi (DP)equation y t +y x u +3yu x =0,x ∈R ,t >0,(1.1)with y =u −u xx ,was originally derived by Degasperis-Procesi [14]using the method of asymptotic integrability up to third order as one of three equations in the family of third order dispersive PDE conservation laws of the form u t −α2u xxt +γu xxx +c 0u x =(c 1u 2+c 2u 2x +c 3uu xx )x .(1.2)The other two integrable equations in the family,after rescaling and applying a Galilean transformation,are the Korteweg-de Vries (KdV)equationu t +u xxx +uu x =0and the Camassa-Holm (CH)shallow water equation [2,15,18,22],y t +y x u +2yu x =0,y =u −u xx .(1.3)These three cases exhaust in the completely integrable candidates for(1.2)by Painlev´e analysis.Degasperis,Holm and Hone[13]showed the formal integra-bility of the DP equation as Hamiltonian systems by constructing a Lax pair and a bi-Hamiltonian structure.The Camassa-Holm equation wasﬁrst derived by Fokas and Fuchassteiner [18]as a bi-Hamiltonian system,and then as a model for shallow water waves by Camassa and Holm[2].The DP equation is also in dimensionless space-time variables(x,t)an approximation to the incompressible Euler equations for shallow water under the Kodama transformation[20,21]and its asymptotic accuracy is the same as that of the Camassa-Holm(CH)shallow water equation, where u(t,x)is considered as theﬂuid velocity at time t in the spatial x-direction with momentum density y.Recently,Liu and Yin[24]proved that theﬁrst blow-up inﬁnite time to equation(1.1)must occur as wave breaking and shock waves possibly appear afterwards.It is shown in[24]that the lifespan of solutions of the DP equation (1.1)is not aﬀected by the smoothness and size of the initial proﬁles,but aﬀected by the shape of the initial proﬁles(for the CH equation,see[1,8]).This can be viewed as a signiﬁcant diﬀerence between the DP equation(or the CH equation )and the KdV.It is also noted that the KdV equation,unlike the CH equation or DP equation,does not have wave breaking phenomena[30].Under wave breaking we understand that development of singularities inﬁnite time by which the wave remains bounded but its slope becomes unbounded[31].It is well known that the KdV equation is an integrable Hamiltonian equation that possesses smooth solitons as traveling waves.In the KdV equation,the leading order asymptotic balance that conﬁnes the traveling wave solitons occurs between nonlinear steepening and linear dispersion.However,the nonlinear dispersion and nonlocal balance in the CH equation and the DP equation,even in the absence of linear dispersion,can still produce a conﬁned solitary traveling wavesu(t,x)=cϕ(x−ct)(1.4) traveling at constant speed c>0,whereϕ(x)=e−|x|.Because of their shape (they are smooth except for a peak at their crest),these solutions are called the peakons[2,13].Peakons of both equations are true solitons that interact via elastic collisions under the CH dynamics,or the DP dynamics,respectively.The peakons of the CH equation are orbitally stable[12].For waves that approximate the peakons in a special way,a stability result was proved by a variation method [11].Note that we can rewrite the DP equation asu t−u txx+4uu x=3u x u xx+uu xxx,t>0,x∈R.(1.5) The peaked solitons are not classical solutions of(1.5).They satisfy the Degasperis-Procesi equation in the conservation law formu t+∂x 12ϕ∗ 3where∗stands for convolution with respect to the spatial variable x∈R.This is the exact meaning in which the peakons are solutions.Recently,Lundmark and Szmigielski[26]presented an inverse scattering ap-proach for computing n-peakon solutions to equation(1.5).Holm and Staley[20] studied stability of solitons and peakons numerically to equation(1.5).Anal-ogous to the case of Camassa-Holm equation[6],Henry[19]and Mustafa[29] showed that smooth solutions to equation(1.5)have inﬁnite speed of propaga-tion.The following are three useful conservation laws of the Degasperis-Procesi equation.E1(u)= R y dx,E2(u)= R yv dx,E3(u)= R u3dx,where y=(1−∂2x)u and v=(4−∂2x)−1u,while the corresponding three useful conservation laws of the Camassa-Holm equation are the following:F1(u)= R y dx,F2(u)= R(u2+u2x)dx,F3(u)= R(u3+uu2x)dx.(1.7)The stability of solitary waves is one of the fundamental qualitative proper-ties of the solutions of nonlinear wave equations.Numerical simulations[13,25] suggest that the sizes and velocities of the peakons do not change as a result of collision so these patterns are expected to be stable.Furthermore,it is observed that the shape of the peakons remains approximately the same as time evolves. As far as we know,the case of stability of the peakons for the Camassa-Holm equation is well understood by now[11,12],while the Degasperis-Procesi equa-tion case is the subject of this paper.The goal of this paper is to establish a stability result of peaked solitons for equation(1.5).It is found that the corresponding conservation laws of the Degasperis-Procesi equation are much weaker than those of the Camassa-Holm equation. In particular,one can see that the conservation law E2(u)for the DP equation.In fact,by the Fourier transform,we haveis equivalent to u 2L2E2(u)= R yvdx= R1+ξ2Let us denoteE2(u)= u 2X.The following stability theorem is the principal result of the present paper. Theorem1(Stability)Let cϕbe the peaked soliton deﬁned in(1.4)traveling with speed c>0.Then cϕis orbitally stable in the following sense.If u0∈H sfor some s>3/2,y0=u0−∂2x u0is a nonnegative Radon measure ofﬁnite total mass,andu0−cϕ X<cε,|E3(u0)−E3(cϕ)|<c3ε,0<ε<16 ≤c√ε.(1.10)Remark1The state of aﬀairs about these maxima/minima implied by the pre-vious theorem is a consequence of the assumption on y0,as shown in Lemma 3.1.For an initial proﬁle u0∈H s,s>3/2,there exists a local solution u∈C([0,T),H s)of(1.5)with initial data u(0)=u0[32].Under the assump-tion y0=u0−∂2x u0≥0in Theorem1,the existence is global in time[24],that is T=+∞.For peakons cϕwith c>0,we have 1−∂2x (cϕ)=2cδ(hereδis the Dirac distribution).Hence the assumption on y0that it is a nonnegative measure is quite natural for a small perturbation of the peakons.Existence of global weak solution in H1of the DP equation is also proved in[16].Note that peakons cϕare not strong solutions,sinceϕ∈H s,only for s<3/2.The above theorem of orbital stability states that any solution starting close to peakons cϕremains close to some translate of cϕin the norm X,at any later time.More information about this stability is contained in(1.9)and (1.10).Notice that for peakons cϕ,the function v cϕis single-humped with the height1There are two standard methods to study stability issues of dispersive waveequations.One is the variational approach which constructs the solitary waves as energy minimizers under appropriate constraints,and the stability automat-ically follows.However,without uniqueness of the minimizer,one can onlyobtain the stability of the set of minima.The variational approach is used in [11]for the CH equation.It is shown in[11]that the each peakon cϕis theunique minimum(ground state)of constrained energy,from which its orbital stability is proved for initial data u0∈H3with y0=(1−∂2x)u0≥0.Their proof strongly relies on the fact that the conserved energy F2in(1.7)of the CHequation is the H1−norm of the solution.However,for the DP equation the energy E2in(1.8)is only the L2norm of the solution.Consequently,it is morediﬃcult to use such a variational approach for the DP equation.Another approach to study stability is to linearize the equation around the solitary waves,and it is commonly believed that nonlinear stability is governed by the linearized equation.However,for the CH and DP equations,the non-linearity plays the dominant role rather than being a higher-order correction to linear terms.Thus it is unclear how one can get nonlinear stability of peakons by studying the linearized problem.Morover,the peaked solitons cϕare not diﬀerentiable,which makes it diﬃcult to analyze the spectrum of the linearized operator around cϕ.To establish the stability result for the DP equation,we extend the approach in[12]for the CH equation.The idea in[12]is to directly use the energy F2as the Liapunov functional.By expanding F2in(1.7)around the peakon cϕ,the error term is in the form of the diﬀerence of the maxima of cϕand the perturbed solution u.To estimate this diﬀerence,they establish two integral relations g2=F2(u)−2(max u)2and ug2=F3(u)−4M3,M=max u(x)3and the error estimate|M−maxϕ|then follows from the structure of the above polynomial inequality.To extend the above approach to nonlinear stability of the DP peakons,we have to overcome several diﬃculties.By expanding the energy E2(u)around the peakon cϕ,the error term turns out to be max v cϕ−max v u,with v u= (4−∂2x)−1u.We can derive the following two integral relations for M1=max v u, E2(u)and E3(u)byg2=E2(u)−12M21and hg2=E3(u)−144M31with some functions g and h related to v u.To get the required polynomial inequality from the above two identities,we need to show h≤18max v u.How-ever,since h is of the form−∂2x v u±6∂x v u+16v u,generally it can not be5bounded by v u.This new diﬃculty is due to the more complicated nonlinearstructure and weaker conservation laws of the DP equation.To overcome it,we introduce a new idea.By constructing g and h piecewise according to mono-tonicity of the function v u,we then establish two new integral identities(3.7)and(3.9)for E2,E3and all local maxima and minima of v u.The crucial es-timate h≤18max v u can now be shown by using this monotonicity structureand properties of the DP solutions.This results in inequality(3.13)related to E2,E3and all local maxima and minima of v u.By analyzing the structure ofequality(3.13),we can obtain not only the error estimate|M1−max v cϕ|but more precise stability information from(1.10).We note that the same approach can also be used for the CH equation to gain more stability information(seeRemark2).Although the DP equation is similar to the CH equation in several aspects, we would like to point out that these two equations are truly diﬀerent.One of the novel features of the DP equation is it has not only peaked solitons[13], u(t,x)=ce−|x−ct|,c>0but also shock peakons[4,25]of the form1u(t,x)=−for t≥T.k+(t−T)On the other hand,the isospectral problem in the Lax pair for equation(1.5)is the third-order equationψx−ψxxx−λyψ=0cf.[13],while the isospectral problem for the Camassa-Holm equation is thesecond order equationψxx−1(in both cases y=u−u xx)cf.[2].Another indication of the fact that there is no simple transformation of equation(1.5)into the Camassa-Holm equation is the entirely diﬀerent form of conservation laws for these two equations[2,13]. Furthermore,the Camassa-Holm equation is a re-expression of geodesicﬂow on the diﬀeomorphism group[5,10]and on the Bott-Virasoro group[9,28],while no such geometric derivation of the Degasperis-Procesi equation is available.The remainder of the paper is organized as follows.In Section2,we recall the local well-posedness of the Cauchy problem of equation(1.5),the precise blow-up scenario of strong solutions,and several useful results which are crucial in the proof of stability theorem for equation(1.5)from[32,33].Section3is devoted to the proof of the stability result(Theorem1).Notation.As above and henceforth,we denote by∗convolution with respect to the spatial variable x∈R.We use · L p to denote the norm in the Lebesgue space L p(R)(1≤p≤∞),and · H s,s≥0for the norm in the Sobolev spaces H s(R).2PreliminariesIn the present section,we discuss the issue of well-posedness.The local exis-tence theory of the initial-value problem is necessary for our study of nonlinear stability.We brieﬂy collect the needed results from[24,32,33].Denote p(x):=12u2 =0,t>0,x∈R.(2.1) The local well-posedness of the Cauchy problem of equation(1.5)with initial data u0∈H s(R),s>32,there exist a maximal T=T(u0)>0and a unique solution u to equation(1.5)(or equation(2.1)),such thatu=u(·,u0)∈C([0,T);H s(R))∩C1([0,T);H s−1(R)). Moreover,the solution depends continuously on the initial data,i.e.the mapping u0→u(·,u0):H s(R)→C([0,T);H s(R))∩C1([0,T);H s−1(R))is continuous and the maximal time of existence T>0can be chosen to be independent of s.The following two lemmas show that the only way that a classical solution to(1.5)may fail to exist for all time is that the wave may break. Lemma2.2[32]Given u0∈H s(R),s>3Lemma2.3[24]Assume u0∈H s(R),s>32.Assume there exists x0∈R such thaty0(x)=u0(x)−u0,xx(x)≥0if x≤x0,y0(x)=u0(x)−u0,xx(x)≤0if x≥x0,and y0changes sign.Then,the corresponding solution to equation(1.5)blows up in aﬁnite time.Lemma2.7[24]Assume u0∈H s(R),s>3The following lemma is a special case of Lemma 2.7.Lemma2.8[24]Assumeu0∈H s (R ),s >32x −∞e ηy (t,η)dη+e x 2 x −∞e ηy (t,η)dη+e x2(k 1∓1)e −x x−∞e ηydη+14 ∞−∞e −2|x −ξ|w (t,ξ)dξ=14e 2x ∞xe −2ξw (t,ξ)dξand ∂x (4−∂2x )−1w =−12e 2x ∞xe −2ξw (t,ξ)dξ.Combining above two identities,we get(k 2±∂x )(4−∂2x )−1w =14(k 2±2)e 2x ∞xe −2ξw (t,ξ)dξ≥0. 93Proof of stabilityIn this primary section of the paper,we prove the stability theorem(Theorem 1)stated in the introduction.Note that the assumptions on the initial proﬁles guarantee the existence of an unique global solution of equation(1.5).The stability theorem provides a quantitative estimate of how closely the wave must approximate the peakon initially in order to be close enough to some translate of the peakon at any later time.That translate must be located at a point where the wave is tallest.The proof of Theorem1is based on a series of lemmas including some in the previous section.We take the wave speed c=1and the case of general c follows by scaling the estimates.Note thatϕ(x)=e−|x|∈H1(R)has the peak at x=0andE3(ϕ)= ∞−∞e−3|x|dx=24e−2|x|∗u.Thenvϕ(x)=13e−|x|−16.(3.3)Noteϕ−∂2xϕ=2δ.Here,δdenotes the Dirac distribution.For simplicity, we abuse notation by writing integrals instead of the H−1/H1duality pairing. Hence we haveE2(ϕ)= ϕ 2X= R(1−∂2x)ϕ(4−∂2x)−1ϕdx=2 Rδ(x)(4−∂2x)−1ϕ(x)dx=2vϕ(0)=1where use has been made of integration by parts and the fact that E2(ϕ)= ϕ 2X=2vϕ(0).This completes the proof of the lemma.In the next two lemmas,we establish two formulas related the critical values of v u to the two invariants E2(u)and E3(u).Consider a function0=u∈H s,s>3/2and u≥0.Then0<v u= 4−∂2x −1u∈H s+2⊂C2.Since v u is positive and decays at inﬁnity,it must have n points{ξi}n i=1with local maximal values and n−1points{ηi}n−1i=1with local minimal values for some integer n≥1.We arrange these critical points in their order by−∞<ξ1<η1<ξ2<η2<...<ξi−1<ηi−1<ξi<ηi<...<ηn−1<ξn<+∞. Letv u(ξi)=M i,1≤i≤n and v u(ηi)=m i,1≤i≤n−1.(3.5) Here,we assume n<+∞,that is,there are aﬁnite number of minima and maxima of v u.In the case when there are inﬁnitely many maxima and minima, the proofs below can be modiﬁed simply by changing theﬁnite sums to inﬁnite sums.Lemma3.2Let0=u∈H s,s>3/2and u≥0.By the above notations, deﬁne the function g by1≤i≤n.(3.6) g(x)= 2v u+∂2x v u−3∂x v u,ηi−1<x<ξi,2v u+∂2x v u+3∂x v u,ξi<x<ηi,withη0=−∞andηn=+∞.Then we haveR g2(x)dx=E2(u)−12 n i=1M2i−n−1 i=1m2i .(3.7)Proof.To simplify notations,we use v for v u below.Then u=4v−∂2x v.First, we note thatE2(u)= R (1−∂2x)u vdx= R(uv+∂x u∂x v)dx= 4v−∂2x v v+ 4−∂2x ∂x v ∂x v dx= 4v2+5(∂x v)2+ ∂2x v 2 dx.To show(3.7),we evaluate the integral of g2on each interval[ηi−1,ηi],1≤i≤n. We haveηiηi−1g2(x)dx= ξiηi−1(2v+∂xx v−3∂x v)2dx+ ηiξi(2v+∂xx v+3∂x v)2dx =I+II.11To estimate theﬁrst term,by integration by parts,we haveI= ξiηi−1 4v2+(∂xx v)2+9(∂x v)2+4v∂xx v−12v∂x v−6∂xx v∂x v dx= ξiηi−1 4v2+(∂xx v)2+5(∂x v)2 dx−6 v(ξi)2−v(ηi−1)2 ,where use has been made of the fact that∂x v(ξi)=∂x v(ηi−1)=0.Similarly,II= ηiξi 4v2+ ∂2x v 2+5(∂x v)2 dx+6 v(ηi)2−v(ξi)2 .Soηiηi−1g2(x)dx= ηiηi−1 4v2+ ∂2x v 2+5(∂x v)2 dx−12v(ξi)2+6v(ηi−1)2+6v(ηi)2andR g2(x)dx= R 4v2+ ∂2x v 2+5(∂x v)2 dx−n i=1 12v(ξi)2−6v(ηi−1)2−6v(ηi)2 =E2(u)−12 n i=1M2i−n−1 i=1m2i ,where use has been made of the fact that v(η0)=v(ηn)=0and the notationsin(3.5).Lemma3.3With the same assumptions and notations in Lemma3.2.Deﬁnethe function h byh(x)= −∂2x v u−6∂x v u+16v u,ηi−1<x<ξi,−∂2x v u+6∂x v u+16v u,ξi<x<ηi,1≤i≤n.(3.8)withη0=−∞andηn=+∞.Then we haveR h(x)g2(x)dx=E3(u)−144 n i=1M3i−n−1 i=1m3i .(3.9) Proof.We still use v for v u.First,note thatE3(u)= R(4v−∂xx v)3dx= R −(∂xx v)3+12(∂xx v)2v−48v2∂xx v+64v3 dx.To show(3.9),we evaluate the integral of h(x)g2(x)on each interval[ηi−1,ηi],1≤i≤n.We haveηiηi−1h(x)g2(x)dx= ξiηi−1(−∂xx v−6∂x v+16v)(2v+∂xx v−3∂x v)2dx+ ηiξi(−∂xx v+6∂x v+16v)(2v+∂xx v+3∂x v)2dx=I+II.12It is found that theﬁrst termI= ξiηi−1{−(∂xx v)3+12(∂xx v)2v+27∂xx v(∂x v)2−108v∂xx v∂x v+60v2∂xx v −54(∂x v)3+216(∂x v)2v−216v2∂x v+64v3}dx= ξiηi−1{−(∂xx v)3+12(∂xx v)2v+54(∂x v)3+60v2∂xx v−54(∂x v)3−108v2∂xx v+64v3}dx−72 v(ξi)3−v(ηi−1)3= ξiηi−1 −(∂xx v)3+12(∂xx v)2v−48v2∂xx v+64v3 dx−72 v(ξi)3−v(ηi−1)3 ,where use has been made of the following integral identities due to integrationby parts and∂x v(ξi)=∂x v(ηi−1)=0,ξiηi−1∂xx v(∂x v)2dx=1(∂x v)2 dx=−12v2 dx=−12∂x v3 dx=13Without changing the integral identities(3.7)and(3.9),we can rearrange M i and m i in the order:M1≥M2···≥M n≥0,m1≥···≥m n−1≥0.Moreover,since each local minimum is less than the neighboring local maximum, we have M i≥m i−1(2≤i≤n).The following two elementary inequalities are needed in the later proofs.Lemma3.4For any n≥2,assume{M i}n i=1and{m i}n−1i=1are2n−1numbers satisfyM1≥M2···≥M n≥0,m1≥···≥m n−1≥0and M i≥m i−1(2≤i≤n).Then(i)ni=2 M3i−m3i−1 ≤32≥ M31+n 2 M3i−m3i−1 1M1 M2i−m2i−121=−2and B n= M31+n 2 M3i−m3i−1 1if M1≥M2≥m1≥0.We haveM21+M22−m21 3− M31+M32−m31 2=3M41M22−3M41m21−2M31M32+2M31m31+3M21M42−6M21M22m21+3M21m41−3M42m21+2M32m31+3M22m41−2m61=(M2−m1)(M1−m1){3M1M32+3M31M2−2M1m31+3M31m1−2M2m31+3M32m1−2m41−2M21M22+M21m21+M22m21−2M1M2m21+M1M22m1+M21M2m1},which is obviously nonnegative by the assumption M1≥M2≥m1≥0.Assume the inequality A n≥B n is true for n≤k(k≥2).Our goal is to deduce A k+1≥B k+1.Since A k≥M1≥M k+1≥m k,we haveA6k+1= M21+k+1 2 M2i−m2i−1 3= A2k+M2k+1−m2k 3≥ A3k+M3k+1−m3k 2(by the n=2inequality)≥ B3k+M3k+1−m3k 2(since A k≥B k by the induction assumption)=B6k+1.Thus A k+1≥B k+1and A n≥B n is true for any n≥2.The following lemma is crucial in the proof of stability of the peakons. Lemma3.5Assume u0∈H s,s>3/2and y0≥0.Let M1=v u(t,ξ1)= max x∈R{v(t,x)}Then for t≥0,E3(u)−144B3n≤18M1 E2(u)−12A2n (3.13) where u is the global solution of equation(1.5)with initial value u0,v u=(4−∂2x)−1u,and A n and B n are deﬁned in(3.12).Proof.First,by Lemma2.8the global solution u of equation(1.5)satisﬁes u(t,x)≥0and y(t,x)=u−∂2x u≥0for all(t,x)∈R+×R.We now claim that h≤18v for(t,x)∈R+×R.To see this,we rewrite the expression of h ash(x)= − ∂2x+3∂x+12 v−3∂x v+18v,ηi−1<x<ξi,− ∂2x−3∂x+2 v+3∂x v+18v,ξi<x<ηi,1≤i≤n. Ifηi−1<x<ξi,1≤i≤n,then v x>0.On the other hand,it follows from Lemma2.10that− ∂2x+3∂x+2 v=−(2+∂x)(4−∂2x)−1(1+∂x)u≤0.Hence− ∂2x+3∂x+2 v−3∂x v+18v≤18v.(3.14)15A similar argument also shows that forξi<x<ηi,1≤i≤n,∂x v<0and− ∂2x−3∂x+2 v+3∂x v+18v(3.15)=−(2−∂x)(4−∂2x)−1(1−∂x)u+3∂x v+18v≤18v.The combination of(3.14)and(3.15)yieldsh≤18v≤18max v=18M1,∀(t,x)∈R+×R.By the notation in(3.12),the integral identities(3.7)and(3.9)becomeR g2(x)dx=E2(u)−12A2nandRh(x)g2(x)dx=E3(u)−144B3n.Note that when n=1,A1=B1=M1.Relating the above integrals,we getE3(u)−144B3n≤18M1(E2(u)−12A2n).Lemma3.6Assume u∈H s,s>3/2and y≥0.LetM1=v u(t,ξ1)=maxx∈R{v(t,x)}andA n=M21+ni=2 M2i−m2i−1where M i and m i−1(2≤i≤n)are local maxima and minima of v u.If|E2(u)−E2(ϕ)|≤δand|E3(u)−E3(ϕ)|≤δwith0<δ<1,then(i)M1−1δ,(3.16) recalling that vϕ(0)=16 <√3√Proof.To obtain (i),we ﬁrst claim thatM 31−172E 3(u )≤0.(3.19)In the case when M 1isthe onlylocal maximum of v u ,we have n =1,A 1=B 1=M 1and (3.19)follows directly from (3.13).When n ≥2,in view of (3.13)and inequality (3.10)in Lemma 3.4(i),there appears the relationM 31−172E 3(u )≤144 n i =2 M 3i −m 3i −1 −34E 2(u )y +13and E 3(ϕ)=212y +16 2 y +14(E 2(u )−E 2(ϕ))M 1−16 2≤324(E 3(u )−E 3(ϕ)).(3.22)On the other hand,observing E 2(u )−12A 2n ≥0,we have0<M 1≤A n ≤ (1/3+δ)/12<16 ≤ 4|E 2(u )−E 2(ϕ)|+1δ.We now prove claim (ii).When n =1,A 1=M 1and it is reduced to (i).When n ≥2,it is thereby inferred from (3.13)thatA 3n −172E 3(u )≤0,(3.24)due to 0≤M 1≤A n and 0≤B n ≤A n by Lemma 3.4(ii).In consequence,(3.17)follows from the same argument as in part (i).17(iii)can be obtained from (i)and (ii).In fact,combining (i)and (ii),we have 2√A n +M 1,which implies (3.18)by using (3.23).ProofofTheorem1.Letu∈C ([0,∞),H s ),s >3/2be the solution of (1.5)with initial data u (0)=u 0.Since E 2and E 3are both conserved by the evolution equation (1.5),we haveE 2(u (t,·))=E 2(u 0)and E 3(u (t,·)=E 3(u 0),∀t ≥0.(3.25)Since u 0−ϕ X <ε,we obtain|E 2(u 0)−E 2(ϕ)|=|( u 0 X − ϕ X )( u 0 X + ϕ X )|≤ε(2 ϕ X +ε)=ε 23+ε <2ε,under the assumption ε<16≤√2ε+4√E 2(u )/12.For peakons cϕ,we havemax v cϕ=6c .So among all waves of a ﬁxed energy E 2,the peakon is tallest in terms of v u .(2)In our proof,we use inequality (3.22)to get estimates (3.16)and (3.18)more directly,compared with the argument in [12]by analyzing the root struc-ture of the polynomial P (y ).Moreover,it implies that the peakons are energy minimizers with a ﬁxed invariant E 3,which explains their stability.Indeed,if E 3(u )=E 3(ϕ),it follows from (3.22)that E 2(u )≥E 2(ϕ).The same re-mark also applies to the CH equation and shows that the CH-peakons are energy minima with ﬁxed F 3.(3)Compared with [12],our construction of the integral relations (3.7)and (3.9)is more delicate.It not only is required in our current case,but also 18provides us more information about stability via(1.10).For the CH equation, even if the orbital stability is proved by a simpler construction[12],our approach can also give the additional stability information.More speciﬁcally,for the CH equation(1.3)with y0≥0,by reﬁning the integrals of[12,Lemma2]to each monotonic interval of u,one can obtain4F3(u)−[10]A.Constantin and B.Kolev,Geodesicﬂow on the diﬀeomorphism group ofthe circle,Comment.Math.Helv.,78(2003),787–804.[11]A.Constantin and L.Molinet,Obtital stability of solitary waves for a shallowwater equation,Physica D,157(2001),75–89.[12]A.Constantin and W.A.Strauss,Stability of peakons,Comm.Pure Appl.Math.,53(2000),603–610.[13]A.Degasperis,D.D.Holm,and A.N.W.Hone,A New Integral Equationwith Peakon Solutions,Theoretical and Mathematical Physics,133(2002),1463–1474.[14]A.Degasperis and M.Procesi,Asymptotic integrability,in Symmetry andPerturbation Theory,edited by A.Degasperis and G.Gaeta,World Scientiﬁc (1999),23–37.[15]H.R.Dullin,G.A.Gottwald,and D.D.Holm,An integrable shallow waterequation with linear and nonlinear dispersion,Phys.Rev.Letters,87(2001), 4501–4504.[16]J.Escher,Y.Liu,and Z.Yin,Global weak solutions and blow-up structurefor the Degasperis-Procesi equation,J.Funct.Anal.,241(2006),457–485. [17]J.Escher,Y.Liu,and Z.Yin,Shock waves and blow-up phenomena for theperiodic Degasperis-Procesi equation,Indiana Univ.Math.J.,56(2007),87–117.[18]A.Fokas and B.Fuchssteiner,Symplectic structures,their B¨a cklund trans-formation and hereditary symmetries,Physica D,4(1981),47–66.[19]D.Henry,Inﬁnite propagation speed for the Degasperis-Procesi equation.J.Math.Anal.Appl.,311(2005),755–759.[20]D.D.Holm and M.F.Staley,Wave structure and nonlinear balances in afamily of evolutionary PDEs,SIAM J.Appl.Dyn.Syst.(electronic),2(2003), 323–380.[21]D.D.Holm and M.F.Staley,Nonlinear balance and exchange of stabil-ity in dynamics of solitons,peakons,ramps/cliﬀs and leftons in a1-1nonlinear evolutionary PDE,Phys.Lett.A,308(2003),437–444.[22]R.S.Johnson,Camassa-Holm,Korteweg-de Vries and related models for waterwaves,J.Fluid Mech.,455(2002),63–82.[23]T.Kato,Quasi-linear equations of evolution,with applications to partial diﬀer-ential equations,in:Spectral Theory and Diﬀerential Equations,Lecture Notes in Math.,Springer Verlag,Berlin,448(1975),25–70.[24]Y.Liu and Z.Yin,Global existence and blow-up phenomena for the Degasperis-Procesi equation,Comm.Math.Phys.,267(2006),801–820.20[25]H.Lundmark,Formation and dynamics of shock waves in the Degasperis-Procesiequation,J.Nonlinear Science,17(2007),169–198.[26]H.Lundmark and J.Szmigielski,Multi-peakon solutions of the Degasperis-Procesi equation,Inverse Problems,19(2003),1241–1245.[27]Y.Matsuno,Multisoliton solutions of the Degasperis-Procesi equation and theirpeakon limit,Inverse Problems,21(2005),1553–1570.[28]G.Misiolek,A shallow water equation as a geodesicﬂow on the Bott-Virasorogroup,J.Geom.Phys.,24(1998),203–208.[29]O.G.Mustafa,A note on the Degasperis-Procesi equation,J.Nonlinear Math.Phys.,12(2005),10–14.[30]T.Tao,Low-regularity global solutions to nonlinear dispersive equations.Sur-veys in analysis and operator theory(Canberra,2001),Proc.Centre Math.Appl.Austral.Nat.Univ.,Canberra,40(2002),19–48.[31]G.B.Whitham,Linear and Nonlinear Waves,J.Wiley&Sons,New York,1980.[32]Z.Yin,On the Cauchy problem for an integrable equation with peakon solutions,Illinois J.Math.,47(2003),649–666.[33]Z.Yin,Global solutions to a new integrable equation with peakons,IndianaUniv.Math.J.,53(2004),1189–1210.21。

A New Supersymmetric Index

∞
Lyman Laboratory of Physics
Harvard University, Cambridge, MA 02138, USA
dimensions, ependent of almost all deformations of the theory. This index is related to the geometry of the vacua (Berry’s curvature) and satisﬁes an exact diﬀerential equation as a function of β . For integrable theories we can also compute the index thermodynamically, using the exact S -matrix. The equivalence of these two results implies a highly non-trivial equivalence of a set of coupled integral equations with these diﬀerential equations, among them Painleve III and the aﬃne Toda equations.
HUTP-92/A021 SISSA 68/92/EP BUHEP-92-14
arXiv:hep-th/9204102v1 30 Apr 1992
A New Supersymmetric Index
Sergio Cecotti† , Paul Fendley⋆ , Ken Intriligator∞ and Cumrun Vafa∞
partition function Tr e−βH . This powerful method is known as the thermodynamic Bethe ansatz (TBA)[10]. In particular, the TBA analysis for a large class of N =2 integrable theories in two dimensions was carried out in [11,12], conﬁrming the conjectured S -matrices as in particular reproducing the correct central charges in the UV limit. One can extend the usual TBA analysis by allowing arbitrary chemical potentials, and in particular one can compute objects such as TreiαF e−βH . This allows us, as a special case, to compute Tr(−1)F F e−βH in these theories in terms of integral equations. Thus for integrable theories we seem to have two inequivalent methods to compute the

麦克斯韦方程组英文

麦克斯韦方程组英文English: The Maxwell's equations are a set of fundamental equations in classical electromagnetism that describe how electric and magnetic fields interact with each other and with electric charges and currents. They were formulated by the physicist James Clerk Maxwell in the 19th century and played a crucial role in the development of electromagnetic theory. The equations consist of four equations: Gauss's law for electric fields, Gauss's law for magnetic fields, Faraday's law of electromagnetic induction, and Ampère's law with Maxwell's addition. These equations are mainly concerned with the spatial and temporal changes of electric and magnetic fields, and they are usually written in differential form or integral form. In the differential form, the equations express how the fields change at each point in space, while in the integral form, they describe the flux of the fields through closed surfaces or the circulation of the fields along closed paths. The Maxwell's equations have important implications in many areas of physics and engineering, as they govern the behavior of electromagnetic waves, the propagation of signals through transmission lines, the behavior of antennas, the operation of electric motors and generators, andvarious other electromagnetic phenomena. In addition, the Maxwell's equations also played a crucial role in the development of the theory of relativity, as they led to the realization that electric and magnetic fields are two different manifestations of a single electromagnetic field, and they can transform into each other under certain conditions. Overall, the Maxwell's equations are of fundamental importance in understanding the behavior of electric and magnetic fields and their interactions with matter, and they have paved the way for numerous technological advancements and scientific discoveries.中文翻译: 麦克斯韦方程组是经典电磁学中描述电场、磁场与电荷电流相互作用的一组基本方程。

Analyzing the properties of ionic liquids

Analyzing the properties of ionicliquidsIonic liquids are an emerging class of solvents that have been gaining popularity in various scientific fields. They are essentially salts that exist as liquids at room temperature, owing to the fact that they have low melting points. One of the key properties of ionic liquids is their extremely low volatility, making them highly useful in applications requiring high thermal stability and the prevention of vaporization. In this article, we will delve into the properties of ionic liquids and their applications.Firstly, let us examine the unique properties of ionic liquids that make them so attractive to researchers. One of the most significant factors is their low melting point, which typically lies in the range of -100 to 100 degrees celsius. This leads to a low vapor pressure and high thermal stability, making them a much safer and more environmentally friendly alternative to traditional solvents. Additionally, ionic liquids can dissolve a wide variety of compounds, including polar and non-polar substances, inorganic salts, and gases. This wide range of solvency makes them useful in a diverse range of applications.Another fascinating property of ionic liquids is their ability to form a variety of unique structures. Ionic liquids consist of positively and negatively charged ions. Depending on the size and shape of these ions, different forces of attraction can occur between them. This can lead to the formation of different types of aggregates, such as micelles, nanodroplets, and even crystals. Such structures can be utilized in fields such as nanochemistry and materials science.The chemical and physical properties of ionic liquids can also be tuned by altering the types of ions and their composition. This can be achieved by designing new ions or modifying existing ones. For instance, the addition of a hydroxyl group to an anion can make it more hydrophilic, while the use of fluorinated anions can increase their solvency for organic compounds. Such modifications have led to the development of new types of ionic liquids for use in various fields.Now, let us explore some of the applications of ionic liquids. One area where ionic liquids are beginning to find increasing use is in the field of electrochemistry. Their high thermal stability and ability to facilitate charge transfer make them excellent electrolytes for batteries and other energy storage devices. Additionally, the large electrochemical window of some ionic liquids allows for the development of high voltage batteries.Ionic liquids also find use in various chemical processes. They have been utilized as solvents for organic reactions, especially those involving transition metal catalysis. Their low volatility and ability to facilitate chemical reactions have led to their adoption in greener and more sustainable chemical processes. Furthermore, the ability of ionic liquids to dissolve a wide range of compounds has made them useful in extraction and separation processes.Finally, ionic liquids have shown great potential in biological applications. Their low toxicity and biocompatibility make them attractive for use in drug delivery and biomaterials. They have been used as solvents for the production of pharmaceuticals, and in the synthesis of biocompatible polymers.In conclusion, ionic liquids are a fascinating and versatile class of solvents with unique properties that make them attractive to researchers in various fields. Their low volatility, ability to form diverse structures, and tunable properties make them attractive for use in numerous applications ranging from energy storage to biomedicine. Researchers are continuously exploring new ways to design and utilize ionic liquids for various applications, and it is undoubtedly an exciting area for future research.。

量子信息——连续变量篇

Quantum information with continuous variablesSamuel L.BraunsteinComputer Science,University of York,York YO105DD,United KingdomPeter van LoockNational Institute of Informatics(NII),Tokyo101-8430,Japan and Institute of TheoreticalPhysics,Institute of Optics,Information and Photonics(Max-Planck Forschungsgruppe),Universität Erlangen-Nürnberg,D-91058Erlangen,Germany͑Published29June2005͒Quantum information is a rapidly advancing area of interdisciplinary research.It may lead to real-world applications for communication and computation unavailable without the exploitation of quantum properties such as nonorthogonality or entanglement.This article reviews the progress in quantum information based on continuous quantum variables,with emphasis on quantum optical implementations in terms of the quadrature amplitudes of the electromagneticﬁeld.CONTENTSI.Introduction513II.Continuous Variables in Quantum Optics516A.The quadratures of the quantizedﬁeld516B.Phase-space representations518C.Gaussian states519D.Linear optics519E.Nonlinear optics520F.Polarization and spin representations522G.Necessity of phase reference523 III.Continuous-Variable Entanglement523A.Bipartite entanglement5251.Pure states5252.Mixed states and inseparability criteria526B.Multipartite entanglement5291.Discrete variables5292.Genuine multipartite entanglement5303.Separability properties of Gaussian states5304.Generating entanglement5315.Measuring entanglement533C.Bound entanglement534D.Nonlocality5341.Traditional EPR-type approach5352.Phase-space approach5363.Pseudospin approach536E.Verifying entanglement experimentally537 IV.Quantum Communication with Continuous Variables538A.Quantum teleportation5401.Teleportation protocol5412.Teleportation criteria5433.Entanglement swapping546B.Dense coding546rmation:A measure5472.Mutual information5473.Classical communication5474.Classical communication via quantum states5475.Dense coding548C.Quantum error correction550D.Quantum cryptography5501.Entanglement-based versus prepare andmeasure5502.Early ideas and recent progress5513.Absolute theoretical security5524.Verifying experimental security5535.Quantum secret sharing553E.Entanglement distillation554F.Quantum memory555V.Quantum Cloning with Continuous Variables555A.Local universal cloning5551.Beyond no-cloning5552.Universal cloners556B.Local cloning of Gaussian states5571.Fidelity bounds for Gaussian cloners5572.An optical cloning circuit for coherentstates558C.Telecloning559 VI.Quantum Computation with Continuous Variables560A.Universal quantum computation560B.Extension of the Gottesman-Knill theorem563 VII.Experiments with Continuous Quantum Variables565A.Generation of squeezed-state EPR entanglement5651.Broadband entanglement via opticalparametric ampliﬁcation5652.Kerr effect and linear interference567B.Generation of long-lived atomic entanglement568C.Generation of genuine multipartite entanglement569D.Quantum teleportation of coherent states569E.Experimental dense coding570F.Experimental quantum key distribution571G.Demonstration of a quantum memory effect572 VIII.Concluding Remarks572 Acknowledgments573 References573I.INTRODUCTIONQuantum information is a relatively young branch of physics.One of its goals is to interpret the concepts of quantum physics from an information-theoretic point of view.This may lead to a deeper understanding of quan-REVIEWS OF MODERN PHYSICS,VOLUME77,APRIL20050034-6861/2005/77͑2͒/513͑65͒/$50.00©2005The American Physical Society513tum theory.Conversely,information and computation are intrinsically physical concepts,since they rely on physical systems in which information is stored and by means of which information is processed or transmitted. Hence physical concepts,and at a more fundamental level quantum physical concepts,must be incorporated in a theory of information and computation.Further-more,the exploitation of quantum effects may even prove beneﬁcial for various kinds of information pro-cessing and communication.The most prominent ex-amples of this are quantum computation and quantum key distribution.Quantum computation means in par-ticular cases,in principle,computation faster than any known classical computation.Quantum key distribution makes possible,in principle,unconditionally secure communication as opposed to communication based on classical key distribution.From a conceptual point of view,it is illuminating to consider continuous quantum variables in quantum in-formation theory.This includes the extension of quan-tum communication protocols from discrete to continu-ous variables and hence fromﬁnite to inﬁnite dimensions.For instance,the original discrete-variable quantum teleportation protocol for qubits and other ﬁnite-dimensional systems͑Bennett et al.,1993͒was soon after its publication translated into the continuous-variable setting͑Vaidman,1994͒.The main motivation for dealing with continuous variables in quantum infor-mation,however,originated in a more practical observa-tion:efﬁcient implementation of the essential steps in quantum communication protocols,namely,preparing, unitarily manipulating,and measuring͑entangled͒quan-tum states,is achievable in quantum optics utilizing con-tinuous quadrature amplitudes of the quantized electro-magneticﬁeld.For example,the tools for measuring a quadrature with near-unit efﬁciency or for displacing an optical mode in phase space are provided by homodyne-detection and feedforward techniques,respectively. Continuous-variable entanglement can be efﬁciently produced using squeezed light͓in which the squeezing of a quadrature’s quantumﬂuctuations is due to a non-linear optical interaction͑Walls and Milburn,1994͔͒and linear optics.A valuable feature of quantum optical implementa-tions based upon continuous variables,related to their high efﬁciency,is their unconditionalness.Quantum re-sources such as entangled states emerge from the non-linear optical interaction of a laser with a crystal͑supple-mented if necessary by some linear optics͒in an unconditional fashion,i.e.,every inverse bandwidth time.This unconditionalness is hard to obtain in discrete-variable qubit-based implementations using single-photon states.In that case,the desired prepara-tion due to the nonlinear optical interaction depends on particular͑coincidence͒measurement results ruling out the unwanted͑in particular,vacuum͒contributions in the outgoing state vector.However,the unconditional-ness of the continuous-variable implementations has its price:it is at the expense of the quality of the entangle-ment of the prepared states.This entanglement and hence any entanglement-based quantum protocol is al-ways imperfect,the degree of imperfection depending on the amount of squeezing of the laser light involved. Good quality and performance require large squeezing which is technologically demanding,but to a certain ex-tent͓about10dB͑Wu et al.,1986͔͒already state of the art.Of course,in continuous-variable protocols that do not rely on entanglement,for instance,coherent-state-based quantum key distribution,these imperfections do not occur.To summarize,in the most commonly used optical ap-proaches,the continuous-variable implementations al-ways work pretty well͑and hence efﬁciently and uncon-ditionally͒,but never perfectly.Their discrete-variable counterparts only work sometimes͑conditioned upon rare successful events͒,but they succeed,in principle, perfectly.A similar tradeoff occurs when optical quan-tum states are sent through noisy channels͑opticalﬁ-bers͒,for example,in a realistic quantum key distribu-tion scenario.Subject to losses,the continuous-variable states accumulate noise and emerge at the receiver as contaminated versions of the sender’s input states.The discrete-variable quantum information encoded in single-photon states is reliably conveyed for each photon that is not absorbed during transmission.Due to the recent results of Knill,Laﬂamme,and Mil-burn͑Knill et al.,2001͒,it is now known that efﬁcient quantum information processing is possible,in principle, solely by means of linear optics.Their scheme is formu-lated in a discrete-variable setting in which the quantum information is encoded in single-photon states.Apart from entangled auxiliary photon states,generated off-line without restriction to linear optics,conditional dy-namics͑feedforward͒is the essential ingredient in mak-ing this approach work.Universal quantum gates such as a controlled-NOT gate can,in principle,be built using this scheme without need of any Kerr-type nonlinear op-tical interaction͑corresponding to an interaction Hamil-tonian quartic in the optical modes’annihilation and creation operators͒.This Kerr-type interaction would be hard to obtain on the level of single photons.However, the off-line generation of the complicated auxiliary states needed in the Knill-Laﬂamme-Milburn scheme seems impractical too.Similarly,in the continuous-variable setting,when it comes to more advanced quantum information proto-cols,such as universal quantum computation or,in a communication scenario,entanglement distillation,it turns out that tools more sophisticated than mere Gaussian operations are needed.In fact,the Gaussian operations are effectively those described by interaction Hamiltonians at most quadratic in the optical modes’annihilation and creation operators,thus leading to lin-ear input-output relations as in beam-splitter or squeez-ing transformations.Gaussian operations,mapping Gaussian states onto Gaussian states,also include ho-modyne detections and phase-space displacements.In contrast,the non-Gaussian operations required for ad-vanced continuous-variable quantum communication͑in particular,long-distance communication based on en-514S.L.Braunstein and P.van Loock:Quantum information with continuous variables Rev.Mod.Phys.,Vol.77,No.2,April2005tanglement distillation and swapping,quantum memory,and teleportation͒are due either to at least cubic non-linear optical interactions or to conditional transforma-tions depending on non-Gaussian measurements such asphoton counting.It seems that,at this very sophisticatedlevel,the difﬁculties and requirements of the discrete-and continuous-variable implementations are analogous.In this review,our aim is to highlight the strengths ofthe continuous-variable approaches to quantum infor-mation processing.Therefore we focus on those proto-cols that are based on Gaussian states and their feasiblemanipulation through Gaussian operations.This leads tocontinuous-variable proposals for the implementation ofthe simplest quantum communication protocols,such asquantum teleportation and quantum key distribution,and includes the efﬁcient generation and detection ofcontinuous-variable entanglement.Before dealing with quantum communication andcomputation,in Sec.II,weﬁrst introduce continuousquantum variables within the framework of quantumoptics.The discussions about the quadratures of quan-tized electromagnetic modes,about phase-space repre-sentations,and about Gaussian states include the nota-tions and conventions that we use throughout thisarticle.We conclude Sec.II with a few remarks on linearand nonlinear optics,on alternative polarization andspin representations,and on the necessity of a phasereference in continuous-variable implementations.Thenotion of entanglement,indispensable in many quantumprotocols,is described in Sec.III in the context of con-tinuous variables.We discuss pure and mixed entangledstates,entanglement between two͑bipartite͒and be-tween many͑multipartite͒parties,and so-called bound ͑undistillable͒entanglement.The generation,measure-ment,and veriﬁcation͑both theoretical and experimen-tal͒of continuous-variable entanglement are here of par-ticular interest.As for the properties of the continuous-variable entangled states related with theirinseparability,we explain how the nonlocal character ofthese states is revealed.This involves,for instance,vio-lations of Bell-type inequalities imposed by local real-ism.Such violations,however,cannot occur when themeasurements considered are exclusively of continuous-variable type.This is due to the strict positivity of theWigner function of the Gaussian continuous-variable en-tangled states,which allows for a hidden-variable de-scription in terms of the quadrature observables.In Sec.IV,we describe the conceptually and practi-cally most important quantum communication protocols formulated in terms of continuous variables and thus utilizing the continuous-variable͑entangled͒states. These schemes include quantum teleportation and en-tanglement swapping͑teleportation of entanglement͒, quantum͑super͒dense coding,quantum error correc-tion,quantum cryptography,and entanglement distilla-tion.Since quantum teleportation based on nonmaxi-mum continuous-variable entanglement,usingﬁnitely squeezed two-mode squeezed states,is always imperfect, teleportation criteria are needed both for the theoretical and for the experimental veriﬁcation.As is known from classical communication,light,propagating at high speed and offering a broad range of different frequen-cies,is an ideal carrier for the transmission of informa-tion.This applies to quantum communication as well. However,light is less suited for the storage of informa-tion.In order to store quantum information,for in-stance,at the intermediate stations in a quantum re-peater,atoms are more appropriate media than light. Signiﬁcantly,as another motivation to deal with continu-ous variables,a feasible light-atom interface can be built via free-space interaction of light with an atomic en-semble based on the alternative polarization and spin-type variables.No strong cavity QED coupling is needed as with single photons.The concepts of this transfer of quantum information from light to atoms and vice versa, as the essential ingredients of a quantum memory,are discussed in Sec.IV.FSection V is devoted to quantum cloning with con-tinuous variables.One of the most fundamental͑and historically one of theﬁrst͒“laws”of quantum informa-tion theory is the so-called no-cloning theorem͑Dieks, 1982;Wootters and Zurek,1982͒.It forbids the exact copying of arbitrary quantum states.However,arbitrary quantum states can be copied approximately,and the resemblance͑in mathematical terms,the overlap orﬁ-delity͒between the clones may attain an optimal value independent of the original states.Such optimal cloning can be accomplished locally by sending the original states͑together with some auxiliary system͒through a local unitary quantum circuit.Optimal cloning of Gauss-ian continuous-variable states appears to be more inter-esting than that of general continuous-variable states, because the latter can be mimicked by a simple coin toss.We describe a non-entanglement-based implemen-tation for the optimal local cloning of Gaussian continuous-variable states.In addition,for Gaussian continuous-variable states,an optical implementation exists of optimal cloning at a distance͑telecloning͒.In this case,the optimality requires entanglement.The cor-responding multiparty entanglement is again producible with nonlinear optics͑squeezed light͒and linear optics ͑beam splitters͒.Quantum computation over continuous variables,dis-cussed in Sec.VI,is a more subtle issue than the in some sense straightforward continuous-variable extensions of quantum communication protocols.Atﬁrst sight,con-tinuous variables do not appear well suited for the pro-cessing of digital information in a computation.On the other hand,a continuous-variable quantum state having an inﬁnite-dimensional spectrum of eigenstates contains a vast amount of quantum information.Hence it might be promising to adjust the continuous-variable states theoretically to the task of computation͑for instance,by discretization͒and yet to exploit their continuous-variable character experimentally in efﬁcient͑optical͒implementations.We explain in Sec.VI why universal quantum computation over continuous variables re-quires Hamiltonians at least cubic in the position and momentum͑quadrature͒operators.Similarly,any quan-tum circuit that consists exclusively of unitary gates from515S.L.Braunstein and P.van Loock:Quantum information with continuous variables Rev.Mod.Phys.,Vol.77,No.2,April2005the continuous-variable Clifford group can be efﬁcientlysimulated by purely classical means.This is acontinuous-variable extension of the discrete-variableGottesman-Knill theorem in which the Clifford groupelements include gates such as the Hadamard͑in thecontinuous-variable case,Fourier͒transform or the con-trolled NOT͑CNOT͒.The theorem applies,for example,to quantum teleportation which is fully describable by CNOT’s and Hadamard͑or Fourier͒transforms of some eigenstates supplemented by measurements in thateigenbasis and spin or phaseﬂip operations͑or phase-space displacements͒.Before some concluding remarks in Sec.VIII,wepresent some of the experimental approaches to squeez-ing of light and squeezed-state entanglement generationin Sec.VII.A.Both quadratic and quartic optical nonlin-earities are suitable for this,namely,parametric downconversion and the Kerr effect,respectively.Quantumteleportation experiments that have been performed al-ready based on continuous-variable squeezed-state en-tanglement are described in Sec.VII.D.In Sec.VII,wefurther discuss experiments with long-lived atomic en-tanglement,with genuine multipartite entanglement ofoptical modes,experimental dense coding,experimentalquantum key distribution,and the demonstration of aquantum memory effect.II.CONTINUOUS VARIABLES IN QUANTUM OPTICSFor the transition from classical to quantum mechan-ics,the position and momentum observables of the par-ticles turn into noncommuting Hermitian operators inthe Hamiltonian.In quantum optics,the quantized elec-tromagnetic modes correspond to quantum harmonicoscillators.The modes’quadratures play the roles of theoscillators’position and momentum operators obeyingan analogous Heisenberg uncertainty relation.A.The quadratures of the quantizedﬁeldFrom the Hamiltonian of a quantum harmonic oscil-lator expressed in terms of͑dimensionless͒creation and annihilation operators and representing a single mode k, Hˆk=ប␻k͑aˆk†aˆk+12͒,we obtain the well-known form writ-ten in terms of“position”and“momentum”operators ͑unit mass͒,Hˆk=12͑pˆk2+␻k2xˆk2͒,͑1͒withaˆk=1ͱ2ប␻k͑␻k xˆk+ipˆk͒,͑2͒aˆk†=1ͱ2ប␻k͑␻k xˆk−ipˆk͒,͑3͒or,conversely,xˆk=ͱប2␻k͑aˆk+aˆk†͒,͑4͒pˆk=−iͱប␻k2͑aˆk−aˆk†͒.͑5͒Here,we have used the well-known commutation rela-tion for position and momentum,͓xˆk,pˆkЈ͔=iប␦kkЈ,͑6͒which is consistent with the bosonic commutation rela-tions͓aˆk,aˆkЈ†͔=␦kkЈ,͓aˆk,aˆkЈ͔=0.In Eq.͑2͒,we see that up to normalization factors the position and the momentum are the real and imaginary parts of the annihilation op-erator.Let us now deﬁne the dimensionless pair of con-jugate variables,Xˆkϵͱ␻k2បxˆk=Re aˆk,Pˆkϵ1ͱ2ប␻k pˆk=Im aˆk.͑7͒Their commutation relation is then͓Xˆk,PˆkЈ͔=i2␦kkЈ.͑8͒In other words,the dimensionless position and momen-tum operators,Xˆk and Pˆk,are deﬁned as if we setប=1/2.These operators represent the quadratures of a single mode k,in classical terms corresponding to the real and imaginary parts of the oscillator’s complex am-plitude.In the following,by using͑Xˆ,Pˆ͒or equivalently ͑xˆ,pˆ͒,we shall always refer to these dimensionless quadratures as playing the roles of position and momen-tum.Hence͑xˆ,pˆ͒will also stand for a conjugate pair of dimensionless quadratures.The Heisenberg uncertainty relation,expressed in terms of the variances of two arbitrary noncommuting observables Aˆand Bˆfor an arbitrary given quantum state,͗͑⌬Aˆ͒2͘ϵŠ͑Aˆ−͗Aˆ͒͘2‹=͗Aˆ2͘−͗Aˆ͘2,͗͑⌬Bˆ͒2͘ϵŠ͑Bˆ−͗Bˆ͒͘2‹=͗Bˆ2͘−͗Bˆ͘2,͑9͒becomes͗͑⌬Aˆ͒2͗͑͘⌬Bˆ͒2͘ജ14͉͓͗Aˆ,Bˆ͔͉͘2.͑10͒Inserting Eq.͑8͒into Eq.͑10͒yields the uncertainty re-lation for a pair of conjugate quadrature observables of a single mode k,xˆk=͑aˆk+aˆk†͒/2,pˆk=͑aˆk−aˆk†͒/2i,͑11͒namely,͗͑⌬xˆk͒2͗͑͘⌬pˆk͒2͘ജ14͉͓͗xˆk,pˆk͔͉͘2=116.͑12͒Thus,in our units,the quadrature variance for a vacuum or coherent state of a single mode is1/4.Let us further516S.L.Braunstein and P.van Loock:Quantum information with continuous variables Rev.Mod.Phys.,Vol.77,No.2,April2005illuminate the meaning of the quadratures by looking at a single frequency mode of the electric ﬁeld ͑for a single polarization ͒,E ˆk ͑r ,t ͒=E 0͓a ˆk ei ͑k ·r −␻k t ͒+a ˆk †e −i ͑k ·r −␻k t ͔͒.͑13͒The constant E 0contains all the dimensional prefactors.By using Eq.͑11͒,we can rewrite the mode asE ˆk ͑r ,t ͒=2E 0͓x ˆk cos ͑␻k t −k ·r ͒+pˆk sin ͑␻k t −k ·r ͔͒.͑14͒Clearly,the position and momentum operators xˆk and p ˆk represent the in-phase and out-of-phase components of the electric-ﬁeld amplitude of the single mode k with respect to a ͑classical ͒reference wave ϰcos ͑␻k t −k ·r ͒.The choice of the phase of this wave is arbitrary,of course,and a more general reference wave would lead us to the single-mode descriptionE ˆk ͑r ,t ͒=2E 0͓x ˆk ͑⌰͒cos ͑␻k t −k ·r −⌰͒+pˆk ͑⌰͒sin ͑␻k t −k ·r −⌰͔͒,͑15͒with the more general quadraturesxˆk ͑⌰͒=͑a ˆk e −i ⌰+a ˆk †e +i ⌰͒/2,͑16͒p ˆk ͑⌰͒=͑a ˆk e −i ⌰−a ˆk †e +i ⌰͒/2i .͑17͒These new quadratures can be obtained from x ˆk and p ˆk via the rotationͩx ˆk ͑⌰͒pˆk ͑⌰͒ͪ=ͩcos ⌰sin ⌰−sin ⌰cos ⌰ͪͩxˆk pˆk ͪ.͑18͒Since this is a unitary transformation,we again end upwith a pair of conjugate observables fulﬁlling the com-mutation relation ͑8͒.Furthermore,because pˆk ͑⌰͒=x ˆk ͑⌰+␲/2͒,the whole continuum of quadratures is cov-ered by x ˆk ͑⌰͒with ⌰෈͓0,␲͒.This continuum of observ-ables is indeed measurable by relatively simple means.Such a so-called homodyne detection works as follows.A photodetector measuring an electromagnetic mode converts the photons into electrons and hence into an electric current,called the photocurrent i ˆ.It is therefore sensible to assume i ˆϰn ˆ=a ˆ†a ˆor i ˆ=qaˆ†a ˆwhere q is a con-stant ͑Paul,1995͒.In order to detect a quadrature of themode aˆ,the mode must be combined with an intense local oscillator at a 50:50beam splitter.The local oscil-lator is assumed to be in a coherent state with large photon number,͉␣LO ͘.It is therefore reasonable to de-scribe this oscillator by a classical complex amplitude␣LO rather than by an annihilation operator aˆLO .The two output modes of the beam splitter,͑aˆLO +a ˆ͒/ͱ2and ͑a ˆLO −a ˆ͒/ͱ2͑see Sec.II.D ͒,may then be approximated byaˆ1=͑␣LO +a ˆ͒/ͱ2,aˆ2=͑␣LO −a ˆ͒/ͱ2.͑19͒This yields the photocurrentsi ˆ1=qa ˆ1†aˆ1=q ͑␣LO *+a ˆ†͒͑␣LO +a ˆ͒/2,i ˆ2=qa ˆ2†aˆ2=q ͑␣LO *−a ˆ†͒͑␣LO −a ˆ͒/2.͑20͒The actual quantity to be measured will be the differ-ence photocurrent␦i ˆϵi ˆ1−i ˆ2=q ͑␣LO *aˆ+␣LO a ˆ†͒.͑21͒By introducing the phase ⌰of the local oscillator,␣LO=͉␣LO ͉exp ͑i ⌰͒,we recognize that the quadrature observ-able xˆ͑⌰͒from Eq.͑16͒is measured ͑without mode index k ͒.Now adjustment of the local oscillator’s phase ⌰෈͓0,␲͔enables us to detect any quadrature from thewhole continuum of quadratures xˆ͑⌰͒.A possible way to realize quantum tomography ͑Leonhardt,1997͒,i.e.,the reconstruction of the mode’s quantum state given by its Wigner function,relies on this measurement method,called ͑balanced ͒homodyne detection .A broadband rather than a single-mode description of homodyne de-tection can be found in the work of Braunstein and Crouch ͑1991͒,who also investigate the inﬂuence of a quantized local oscillator.We have now seen that it is not too hard to measure the quadratures of an electromagnetic mode.Unitary transformations such as quadrature displacements ͑phase-space displacements ͒can also be relatively easily performed via the so-called feedforward technique,as opposed to,for example,photon number displacements.This simplicity and the high efﬁciency when measuring and manipulating continuous quadratures are the main reasons why continuous-variable schemes appear more attractive than those based on discrete variables such as the photon number.In the following,we shall refer mainly to the conju-gate pair of quadratures xˆk and p ˆk ͑position and momen-tum,i.e.,⌰=0and ⌰=␲/2͒.In terms of these quadra-tures,the number operator becomesn ˆk =a ˆk †a ˆk =x ˆk 2+p ˆk 2−12,͑22͒using Eq.͑8͒.Let us ﬁnally review some useful formulas for the single-mode quadrature eigenstates,xˆ͉x ͘=x ͉x ͘,pˆ͉p ͘=p ͉p ͘,͑23͒where we have now dropped the mode index k .They are orthogonal,͗x ͉x Ј͘=␦͑x −x Ј͒,͗p ͉p Ј͘=␦͑p −p Ј͒,͑24͒and complete,͵−ϱϱ͉x ͗͘x ͉dx =1,͵−ϱϱ͉p ͗͘p ͉dp =1.͑25͒Just as for position and momentum eigenstates,the quadrature eigenstates are mutually related to each other by a Fourier transformation,͉x ͘=1ͱ␲͵−ϱϱe −2ixp ͉p ͘dp ,͑26͒517S.L.Braunstein and P .van Loock:Quantum information with continuous variablesRev.Mod.Phys.,Vol.77,No.2,April 2005͉p͘=1ͱ͵−ϱϱe+2ixp͉x͘dx.͑27͒Despite being unphysical and not square integrable,the quadrature eigenstates can be very useful in calculations involving the wave functions␺͑x͒=͗x͉␺͘,etc.,and inidealized quantum communication protocols based on continuous variables.For instance,a vacuum state inﬁ-nitely squeezed in position may be expressed by a zero-position eigenstate͉x=0͘=͉͐p͘dp/ͱ␲.The physical,ﬁ-nitely squeezed states are characterized by the quadrature probability distributions͉␺͑x͉͒2,etc.,ofwhich the widths correspond to the quadrature uncer-tainties.B.Phase-space representationsThe Wigner function is particularly suitable as a “quantum phase-space distribution”for describing the effects on the quadrature observables that may arise from quantum theory and classical statistics.It behaves partly as a classical probability distribution,thus en-abling us to calculate measurable quantities such as mean values and variances of the quadratures in a classical-like fashion.On the other hand,in contrast to a classical probability distribution,the Wigner function can become negative.The Wigner function was originally proposed by Wigner in his1932paper“On the quantum correction for thermodynamic equilibrium”͑Wigner,1932͒.There, he gave an expression for the Wigner function in terms of the position basis which reads͑with x and p being a dimensionless pair of quadratures in our units withប=1/2as introduced in the previous section;Wigner, 1932͒W͑x,p͒=2␲͵dye+4iyp͗x−y͉␳ˆ͉x+y͘.͑28͒Here and throughout,unless otherwise speciﬁed,the in-tegration will be over the entire space of the integration variable͑i.e.,here the integration goes from−ϱtoϱ͒. We gave Wigner’s original formula for only one mode or one particle͓Wigner’s͑1932͒original equation was in N-particle form͔because it simpliﬁes the understanding of the concept behind the Wigner function approach. The extension to N modes is straightforward.Why does W͑x,p͒resemble a classical-like probability distribution?The most important attributes that explain this are the proper normalization,͵W͑␣͒d2␣=1,͑29͒the property of yielding the correct marginal distribu-tions,͵W͑x,p͒dx=͗p͉␳ˆ͉p͘,͵W͑x,p͒dp=͗x͉␳ˆ͉x͘,͑30͒and the equivalence to a probability distribution in clas-sical averaging when mean values of a certain class of operators Aˆin a quantum state␳ˆare to be calculated,͗Aˆ͘=Tr͑␳ˆAˆ͒=͵W͑␣͒A͑␣͒d2␣,͑31͒with a function A͑␣͒related to the operator Aˆ.The measure of integration is in our case d2␣=d͑Re␣͒d͑Im␣͒=dxdp with W͑␣=x+ip͒ϵW͑x,p͒,and we shall use d2␣and dxdp interchangeably.The opera-tor Aˆrepresents a particular class of functions of aˆand aˆ†or xˆand pˆ.The marginal distribution for p,͗p͉␳ˆ͉p͘,is obtained by changing the integration variables͑x−y =u,x+y=v͒and using Eq.͑26͒,that for x,͗x͉␳ˆ͉x͘,by using͐exp͑+4iyp͒dp=͑␲/2͒␦͑y͒.The normalization of the Wigner function then follows from Tr͑␳ˆ͒=1.For any symmetrized operator͑Leonhardt,1997͒,the so-called Weyl correspondence͑Weyl,1950͒,Tr͓␳ˆS͑xˆn pˆm͔͒=͵W͑x,p͒x n p m dxdp,͑32͒provides a rule for calculating quantum-mechanical ex-pectation values in a classical-like fashion according to Eq.͑31͒.Here,S͑xˆn pˆm͒indicates symmetrization.For example,S͑xˆ2pˆ͒=͑xˆ2pˆ+xˆpˆxˆ+pˆxˆ2͒/3corresponds to x2p ͑Leonhardt,1997͒.Such a classical-like formulation of quantum optics in terms of quasiprobability distributions is not unique.In fact,there is a whole family of distributions P͑␣,s͒of which each member corresponds to a particular value of a real parameter s,P͑␣,s͒=1␲2͵␹͑␤,s͒exp͑i␤␣*+i␤*␣͒d2␤,͑33͒with the s-parametrized characteristic functions ␹͑␤,s͒=Tr͓␳ˆexp͑−i␤aˆ†−i␤*aˆ͔͒exp͑s͉␤͉2/2͒.͑34͒The mean values of operators normally and antinor-mally ordered in aˆand aˆ†may be calculated via the so-called P function͑s=1͒and Q function͑s=−1͒,re-spectively.The Wigner function͑s=0͒and its character-istic function␹͑␤,0͒are perfectly suited to provide ex-pectation values of quantities symmetric in aˆand aˆ†such as the quadratures.Hence the Wigner function,though not always positive deﬁnite,appears to be a good com-promise in describing quantum states in terms of quan-tum phase-space variables such as single-mode quadra-tures.We may formulate various quantum states relevant to continuous-variable quantum communica-tion by means of the Wigner representation.These par-ticular quantum states exhibit extremely nonclassical features such as entanglement and nonlocality.Yet their Wigner functions are positive deﬁnite,and thus belong to the class of Gaussian states.518S.L.Braunstein and P.van Loock:Quantum information with continuous variables Rev.Mod.Phys.,Vol.77,No.2,April2005。

Complete reducibility of systems of equations with respect to

2000 Mathematics Subject Classiﬁcation: 20M05 (primary); 20M07, 20M35 (secondary). Keywords: Relatively free proﬁnite semigroup, system of equations, implicit signature, complete tameness, complete reducibility, rational constraint, R-trivial semigroup, labeled ordinal.
1
1.1
Introduction
Framework
Since the 1960’s, the theory of ﬁnite semigroups has drawn a strong motivation from applications in computer science, namely as a natural algebraic framework for classifying combinatorial phenomena described through ﬁnite automata, rational languages, or various kinds of logical formalisms [30, 31, 40, 41, 51]. A central question in problems arising from such applications is to determine eﬀectively whether a given ﬁnite semigroup belongs to a ﬁxed pseudovariety (that is a class closed under taking homomorphic images, subsemigroups, and ﬁnite direct products) or to show that there is no algorithm that solves the membership problem for the pseudovariety. Pseudovarieties are often deﬁned by describing a set of generators, which are usually constructed by applying some natural algebraic construction to members of given pseudovarieties. Thus, for two given pseudovarieties U and V, the central problem translates into solving the membership problem for their join U ∨ V, their semidirect or wreath product U ∗ V, their bilateral semidirect or block product U V, their Mal’cev product U m V, or the power P V. While there is no hope for a universal solution of this type of problem, as it has been shown that none of these operators preserves decidability of the membership problem [1, 46, 47, 23], in many particular cases the problem can be solved by exploring special structural features of the semigroups involved.

引力波中的相关英语高考考点

引力波中的相关英语高考考点英语可能会在阅读理解中出关于引力波的题目。

The surface barrier in mesoscopic type I and type II superconductors

a r X i v :c o n d -m a t /0102444v 2 [c o n d -m a t .s u p r -c o n ] 29 J a n 2002The surface barrier in mesoscopic type I and type II superconductorsAlexander D.Hern´a ndez 1,2and Daniel Dom´ınguez 21Laboratorio de Superconductividad,Facultad de F´ısica-IMRE,Universidad de la Habana,10400,Ciudad Habana,Cuba.2Centro At´o mico Bariloche,8400San Carlos de Bariloche,R´ıo Negro,ArgentinaWe study the surface barrier for magnetic ﬁeld penetration in mesoscopic samples of both type I andtype II superconductors.Our results are obtained from numerical simulations of the time-dependent Ginzburg-Landau equations.We calculate the dependence of the ﬁrst ﬁeld for ﬂux penetration (H p )with the Ginzburg-Landau parameter (κ)observing an increase of H p with decreasing κfor a superconductor-insulator boundary condition ((∇−iA )Ψ|n =0)while for a superconductor-normal boundary condition (approximated by the limiting case of Ψ|S =0)H p has a smaller value independent of κand proportional to H c .We study the magnetization curves and penetration ﬁelds at diﬀerent sample sizes and for square and thin ﬁlm geometries.For small mesoscopic samples we study the peaks and discontinuous jumps found in the magnetization as a function of magnetic ﬁeld.To interpret these jumps we consider that vortices located inside the sample induce a reinforcement of the surface barrier at ﬁelds greater than the ﬁrst penetration ﬁeld H p 1.This leads to multiple penetration ﬁelds H pi =H p 1,H p 2,H p 3,...for vortex entrance in mesoscopic samples.We study the dependence with sample size of the penetration ﬁelds H pi .We explain these multiple penetration ﬁelds extending the usual Bean-Livingston analysis by considering the eﬀect of vortices inside the superconductor and the ﬁnite size of the sample.PACS numbers:74.20.De,74.25.Ha,74.60.EcI.INTRODUCTIONIn the last years there has been an important exper-imental and theoretical interest in the study of vortex physics in a mesoscopic scale.1–4The smallness of these systems imply that the sample geometry and the inter-action between the vortices and the sample surface be-come important.The interaction between vortices and the surface manifests itself fundamentally in the exis-tence of a surface barrier,ﬁrst studied by Bean and Livingston,5which delays the vortex penetration and generates metastable states.If the surface eﬀects are ig-nored,the penetration of magnetic ﬁeld is energetically favorable at the ﬁrst critical ﬁeld H c 1.However the en-ergy barrier of the surface prevents the vortex entry until a higher ﬁeld H p at which the barrier vanishes.H p ,also known as the superheating ﬁeld,is associated with the peak in the magnetization curves and is strongly inﬂu-enced by the presence of surface irregularities.The surface barrier has attracted a renewed interest re-cently in the study of mesoscopic superconductors.For example,Enomoto and Okada 6by means of numeri-cal simulations of the time-dependent Ginzburg-Landau equations (TDGL),studied the inﬂuence of temperature and surface irregularities on the surface barrier.Sonin and Traito 7showed that the presence of the surface bar-rier aﬀects the entry and exit of vortices inﬂuencing the surface resistance.They found a surface-induced sup-pression of the ac losses.One important line of research of mesoscopic super-conductors are the superconducting disks.1,2,4,8–14The study of small superconducting disks was started by Buis-son et al.1for disks with radius ∼7µm.Recent ad-vances in the microfabrication technology and measure-ment techniques now allow the fabrication and study of superconducting disks with sizes comparable to the co-herence length ξ,with radius as small as 0.3µm.2Most of the studies were done in Al disks,a material with κ≃0.3,however for small samples the eﬀective penetration depth Λ=λ2/d increases for decreasing disk thickness (d )re-sulting in eﬀective κvalues in the type II region that can be studied theoretically using the equilibrium Ginzburg-Landau equations.In this regime the Al disk can develop Abrikosov multivortex states 9and depending on the ra-dius (R )and thickness of the disk it is possible to observe ﬁrst or second order phase transitions,8by increasing the disk sizes the second order reversible phase transition ob-served for small disk radius is replaced by a ﬁrst order transition.There is also an intermediate regime where jumps in the magnetization appear associated with the vortex entrance.Other interesting phenomena have been studied for mesoscopic Al disks,for example in Ref.14hysteresis in the magnetization curves was observed ex-perimentally and explained in terms of the presence of a “Bean-Livingston”surface barrier and in Ref.12the behavior of the third critical ﬁeld H c 3was investigated for diﬀerent sample sizes and geometries.The time dependent Ginzburg-Landau (TDGL)equa-tions have been proposed 15as a time dependent gener-alization of the mean ﬁeld approach of the Ginzburg-Landau theory.Gorkov and Eliasberg 16obtained the TDGL equations from the microscopic BCS theory in the gapless case.In the last years,numerical simulations of the time dependent Ginzburg-Landau (TDGL)equations have been suscesfully used to study the magnetic proper-ties and ﬂux dynamics in superconductors.17–22Frah et al.17and Liu et al.18simulated the TDGL equations for κ=0.3−20,Kato et al.19and Machida and Kaburaki 20for κ=2,Aranson et al.21studied vortex dynamics in theκ=∞limit ,and Vicente-´Alvarez et al.studied the dy-namics of d -wave superconductors.22Only few studies of the TDGL equations have been done in superconductors with type I behavior,17,18possibly because the theory is better to describe a superconductor near a second order phase transition at a temperature near T c .In this paper,we present a numerical simulation of the TDGL equations to study the surface barrier in meso-scopic samples for κ=0.15−2.We neglect demagneti-zation eﬀects and therefore we assume that the sample is inﬁnite in the direction of the external magnetic ﬁeld (the ˆz direction).We consider square samples that are meso-scopic in the xy plane (perpendicular to the magnetic ﬁeld)with linear sizes of 5−30λ,with λthe penetration depth.The paper is organized as follows.In Sec.II we present the TDGL equations with their discretized form in ﬁ-nite diﬀerences,and we discuss their possible boundary conditions.In the Sec.III of the paper we study the de-pendence of the penetration ﬁeld H p with the Ginzburg-Landau parameter κ(κ=λ/ξ),exploring both the type I (κ<1/√2)for large samples.We study the eﬀects of the surface barrier from a comparison of two types of boundary conditions.(i)The superconductor-insulator (S-I)boundary condition:consisting in the vanishing of the superconducting cur-rent perpendicular to the boundary (J s ·ˆn =0).In this case we ﬁnd an increase of H p with decreasing κ.(ii)The superconductor-normal (S-N)boundary condition:approximated as the vanishing of the superconducting order parameter at the boundary (Ψ|S =0).A diﬀerent behavior is observed for the S-N boundary condition,the ﬁeld H p is independent of κand nearly equal to H c .In the Sec.IV we study the surface barrier in mesoscopic su-perconductors.In particular,in Sec.IVA we study mag-netization curves in type II superconductors at diﬀer-ent sample dimensions in the region where the transition from a macroscopic to a mesoscopic behavior takes place.In Sec.IVB we show that the discontinuities that appear in the magnetization curves of mesoscopic samples can be explained by considering that the vortices that are inside the sample induce a reinforcement of the surface barrier at ﬁelds greater than the ﬁrst penetration ﬁeld.In this way,it is possible to deﬁne a second,third,fourth,etc.penetration ﬁelds which are a consequence of the in-teraction between vortices and the surface currents.We study the sample size dependence of the ﬁrst,second and third penetration ﬁelds and we show that for suﬃciently large sample sizes the known macroscopic behavior is re-covered,i.e.a continuous magnetization curve appears since H p 3→H p 2→H p .Finally in Sec.V we give a summary of our results and conclusions.II.MODEL AND DYNAMICSA.TDGL equationsOur numerical simulations are carried out using the time-dependent Ginzburg-Landau equations comple-mented with the appropriate Maxwell equations.In the zero-electric potential gauge we have:15,23∂Ψη[(∇−iA )2Ψ+(1−T )(1−|Ψ|2)Ψ](1)∂A∂t=1(∆x )2++U y I,J ΨI,J +1−2ΨI,J +U x I,J −1ΨI,J −1T c)(1−|ΨI,J |2)ΨI,J(3)∂A x I,JT c)Im[U x I,J Ψ∗I,J ΨI +1,J ]∆y)(4)∂A y I,JT c)Im[U y I,J Ψ∗I,J ΨI,J +1]∆x)(5)where ∆x and ∆y are the mesh widths of the spatialdiscretization.B.Boundary conditionsThe dynamical equations must be completemented with the appropriate boundary conditions for both the order parameter and the vector potential.The boundary conditions for the vector potentials A µI,J are obtained by makingB =∇×A =H aat the sample surface (where H a is the applied magnetic ﬁeld).The boundary conditions for the order parameter de-pend sensibly onthe physical nature of the boundary.In general is given by:25( ΠΨ)|n =(∇−i A )|n Ψ=ΨN N1ξN,with N the local den-sity of states at the Fermi surface,N N the bulk density of states,T j the transmission coeﬃcient at the bound-ary and ξN is the coherence length in the normal state.25Typically (N N /N )T j ∼1and b is small when compared to ξ(T ).Therefore,the superconducting-normal (S-N)boundary condition can be approximated by b ≈0,giv-ing:Ψ|S =0.(8)The case of b →0is also found for a ferromagnet-superconductor surface.26Moreover,the condition Ψ|S =0is similar to having a high density of defects at the in-terface and therefore a highly defective surface is also represented by (8).It is also interesting to note that for high T c superconductors even the superconductor-insulator boundary is better approximated by (8)since b ∼ξ(0)2/a ≪ξ(T )in a large range of temperatures,due to the smallness of ξ(0)in this case.The bound-ary condition of (8)completely suppresses the currentsat the boundary ( J ⊥s= J s =0)and maximizes the sur-face Helmholtz free energy,which becomes equal to the free energy of a normal metal.In the study of the surface barrier in Sec.III we will compare these two conceptually diﬀerent boundary con-ditions of the order parameter.They represent the two limiting cases of Eq.(6)and we will call them,in short,the “S-I”boundary condition (Eq.(7))and the “S-N”boundary condition (Eq.(8)).A previous discussion of these two types of boundary conditions was done by Buis-son et al.1,where the equilibrium solutions and eigen-values of the linearized Ginzburg-Landau equations were compared to study the behavior near H c 2.However,for the vortex nucleation process at low magnetic ﬁelds the nonlinear terms of the Ginzburg-Landau equations should be considered.III.SURF ACE BARRIER IN MACROSCOPICSAMPLESWhen the magnetic ﬁeld H is increased starting from H =0in a ﬁnite sample,the Meissner state is destroyed at a magnetic ﬁeld H p which is typically higher than H c 1.This is due to the presence of a surface barrier for vortex entrance in ﬁnite samples.The surface bar-rier in macroscopic samples is sometimes known as the “Bean-Livingston barrier”since it was ﬁrst studied by Bean and Livingston (BL)5.In the BL work the sur-face barrier was obtained in the London approximation and for the ideal case of a semi-inﬁnite superconductor,therefore vortex core nucleation eﬀects and geometrical eﬀects were both neglected.With this simpliﬁcation,the BL work was able to identify one of the most relevant causes of the surface barrier:the screening currents near the surface circulate in the opposite direction to the su-perconducting currents around a vortex.This is typically viewed as a competition between the repulsion between the vortex and the surface shielding current and the at-tractive force between a vortex inside the sample and its image.This argument,which is based on the London model,yields the Bean-Livingston result for the pene-tration ﬁeld,H Londonp ≈√2H c ,which is independent of κ.5,25However,in addition to overcoming the barrier in-duced by the shielding Meissner currents,a vortex pene-tration event has also to deplet the superconducting order parameter |Ψ|at the surface,resulting in a higher pene-tration ﬁeld H p (κ)>H London p .30–35This later eﬀect can be sensitive to the boundary conditions of the Ginzburg-Landau equations.In this section we will discuss the eﬀect of boundary conditions on the surface barrier in macroscopic samples as a function of κ.t f −t ot ft =t 0M (t ),(9)FIG.2.Magnetization curves obtained using the S-N boundary condition:(a)κ=0.2,(b)κ=0.7and (c)κ=2.teractions between currents and demagnetization eﬀectsare neglected.28,29Results for a type II superconductor withκ=2are shown in Figure1(c).In this case thesuperconductor is in the Meissner state until a pene-trationﬁeld H p is reached.At H p some vortices enterthe sample and a peak in the magnetization curve is ob-served.Above H p the magnetization increases due tothe penetration of vortices in the sample,until the es-tablishment of surface superconductivity forﬁelds in therange H c3(T)>H>H c2(T).The criterion for surface superconductivity that we use is the existence of super-conductivity in the surface(i.e.|Ψ|=0in a contour around the surface of width equal to the discretizationlength,∆x=0.5ξin this case)and the exact vanishing of superconductivity in the bulk(i.e.|Ψ|=0inside theregion enclosed by the surface contour).The S-N boundary condition leads to a diﬀerent mag-netic behavior as can be observed in Figure2where we have used similar parameters as those reported in Figure1.We observe the following diﬀerences:(i)The magne-tization is smaller for the same external magneticﬁeld (there are less vortices).Since in this case the super-conducting order parameter vanishes at the surface,the Meissner shielding currents are nucleated at a distanceof a fewξinside the sample,instead of being right at the boundary.Therefore less vortices can stay inside thesample for a given magneticﬁeld when compared with the S-I boundary.For example,forκ=2,weﬁnd that theshielding distanceδbetween the vortices and the sample surface isδSN≈5λfor the S-N interface while for the S-I interface we haveδSI≈3λ.In the general case,when the boundary condition is described by Eq.(6)with aﬁniteb,one can expect that the shielding distance will have an intermediate valueδSN>δ(b)>δSI.(ii)There is nosurface superconductivity above H c2.This is an obvious and direct consequence of the S-N boundary condition that enforcesΨ|S=0.(iii)Theﬁrst penetrationﬁeld H p is smaller and therefore the surface barrier is lower, we will discuss this in detail in the following paragraphs. In order to understand the diﬀerence in the surface barrier,let us study the dynamics of theﬁrst vortex pen-etration just at H=H p.Figure3shows the dynamic evolution of the order parameter near a small region close to an S-I interface(Figure3left)and close to an S-N in-terface(Figure3right).If the interface is of the S-I type the order parameter at the boundary is diﬀerent from zero in the Meissner state.When the condition for vor-tex entrance is fulﬁlled the order parameter at a bound-ary point has to decrease until reaching zero.Therefore there is an intermediate time interval whenΨ=0in a point at the boundary.Just in this moment a vortex can enter the sample and afterwards the order parameter at the boundary increases again and returns to a non-zero value.It is interesting to note that this process is always necessary for vortex penetration in S-I interfaces,there is always the need of an intermediateΨ=0state at the boundary.On the contrary,a smooth entrance of vortices is observed for an S-N interface(Figure3right).FIG.parameter in a small region around the boundary were a vor-tex entrance is taking place.For the S-I condition(left)the magneticﬁeld is H=0.199H c2(0),and for the S-N condi-tion(right)H=0.169H c2(0).In both cases H is just above theﬁrst penetrationﬁeld andκ=2.Gray scale from black (|Ψ|=0)to white(|Ψ|=1).For this boundary condition the order parameter at the interface is already zero.A small deformation of the re-gion ofΨ=0allows for the penetration of a vortex.Since there is no need of depressing the order parameter at the boundary,the surface barrier is much smaller in the S-N case.A similar dynamical behavior for vortex entrance would appear in an S-I interface with defects at the sur-face.At the defects the order parameter is depressed and thereforeΨ=0is already established at some boundary points,which are preferred points for vortex entrance. From the magnetization curves we can obtain theﬁrst ﬁeld forﬂux penetration H p as a function ofκfor the diﬀerent boundary conditions,this is shown in Figure4. Theκdependence of the superheatingﬁeld H p has been previosly calculated for the case of a semi-inﬁnite medium with the Ginzburg-Landau equations.30–35Matricon and Saint James31obtained H p(κ)solving the semi-inﬁnite one-dimensional GL equations,in Ref.32–35the stability of the superheated state under smallﬂuctuations of the order parameter and the vector potential was discussed, and very recently Vodolazov36analyzed the eﬀects of sur-face defects on H p.In the case of a one dimensional semi-inﬁnite medium,the Matricon-Saint James31solu-tion can be obtained solving the equations:d2Ψd2A=0at x=0dxA=H=0andΨ=1at x=∞(11) Solving numerically equations(9)with the boundary con-dition(10)it is possible toﬁnd a relationship among H a andΨ(x=0)=Ψo,where H p is the maximum value of H a that allows a physically meaningfulΨo,i.e. 0<Ψo<1.The results obtained in this way are rep-resented by the continuous-line of Figure4.Our simula-tional results,on the other hand,are a numerical solution of the exact problem in a two dimensional square sample (which has geometrical eﬀects)and are represented by closed circles.The H p values reported here are the peaks of the simulated dc-magnetization curves and the error bars correspond to the discreteﬁeld step used in the the magnetization curves.We see in Fig.4that the value ofH p obtained for the S-I interface is always well above H c1√in the type II region(H c1(T)=[(lnκ)/κ.30–32Forκ→∞we obtain that H p→H c in agreement with the result of Ref.30–32.Figure4 also shows that in the type II region the H c2values ob-tained from the S-I simulations(closed squares)are close√to the expected values(H c2(T)=FIG.5.Magnetization curves of square samples at diﬀerent sizes and using the S-I boundary condition for κ=2(a)30λ×30λ,(b)15λ×15λand (c)10λ×10λ.IV.SURF ACE BARRIER IN MESOSCOPICSAMPLES A.Finite-size eﬀects in mesoscopic type IIsuperconductorsThe magnetic behavior of mesoscopic superconductors is diﬀerent from the behavior of bulk samples.In the mesoscopic scale,several maxima appear in the magne-tization curves which are related to the vortex entrance.This result is quite general and appears either in thin ﬁlms at parallel ﬁelds 37or mesoscopic superconducting disks.8,12In this section we study the magnetic behavior of superconducting samples of diﬀerent sizes,in partic-ular we cover the sample size region where a transition from a mesoscopic to a bulk behavior takes place.Figure 5shows the dc-magnetic behavior of supercon-ducting square samples of diﬀerent sizes with S-I bound-ary condition.The behavior of Fig.5(a)is typical of the macroscopic samples that we have studied in the pre-vious section,however if we decrease the sample size to the mesoscopic region the continuous behavior disappears and some magnetization maxima followed by discontinu-ous jumps appear (see Figs.5(b)and (c)).FIG.7.Magnetization curves of thin ﬁlms with the exter-nal magnetic ﬁeld applied parallel to the ﬁlm.The curves are generated for decreasing thicknesses of the ﬁlm and using the S-I boundary condition for κ=2.(a)15λ×∞,(b)10λ×∞and (c)5λ×∞.The changes observed in the magnetization curves when going from the macroscopic to the mesoscopic be-havior are quite general and do not depend on the sample geometry.We also study the magnetic behavior of a thin ﬁlm with the ﬁeld parallel to its faces.In this case there is only one relevant dimension,the thickness of the ﬁlm (d ).We work here in the case d >λ.Figure 7shows the magnetic behavior of thin ﬁlms with diﬀerent thickness obtained using the S-I boundary condition.We observe that in this case the discontinuities in the magnetization curves appear at smaller ﬁlm thickness and that they are less important than in squares samples,possibly because vortices in a mesoscopic square sample are more conﬁned than in thin ﬁlms of the same linear size.We have also done numerical calculations of the magnetic behavior in thin ﬁlms using the S-N boundary condition (see Figure 8).In particular Figure 8(c)shows that for very small ﬁlm thickness there is a continuous change in magnetiza-tion without the jagged structure observed in Fig.8(a-b).In this case vortex lines do not penetrate the sample and the superconducting state disappears gradually.This be-havior is also present when we use the S-I boundary con-dition but it is necessary to explore lower sample sizes than the one shown in the corresponding ﬁgures.FIG.9.Magnetization curves for a mesoscopic 10λ×10λsquare sample using the S-I boundary condition for κ=2.In the right scale the number of vortices N v is shown.To analyze this eﬀect,let us extend the simple Lon-don approximation model of Bean and Livingston for the surface barrier to the case when there are extra vortices inside the sample and near the boundary.In the Lon-don approach,the Gibbs free energy (G )of a vortex line located adistance x from the sample surface can be cal-culated as:5,25G =Φoλ)−12πλ2K o (2xH c 2Φo=[H a exp(−x2k2K o(2x2κ2−H a ](13)where we have used the relation H c 1=(ln κ)/2κ2H c 2.The ﬁrst term of Equations (12)and (13)is related with the repulsive interaction between the vortex line and the external ﬁeld,the second term describes the attraction between the vortex and the surface currents,this term is usually interpreted as the interaction with an image vortex 5,25and the third term is the vortex self energy.The above expressions are valid when there are no vor-tices inside the sample.If there are vortices located inside the sample,some additional terms due to the interaction between the vortices are needed.In the following we will assume that there is only one vortex inside the sample located at position l and that we are analysing the Gibbs free energy in the same line perpendicular to the sample surface.If l is small,the new vortex that is trying to en-ter the sample now feels two additional terms.The ﬁrst term is the repulsive force between the vortices and the second is the attractive interaction with the image of the vortex located inside the sample.The last term is needed in order to take into account the perturbation of the2κ2K o(2x 4κ2−H a ++1λ)−1λ)(14)Equations (13)and (14)are approximate expressions and have the limitation that they are not valid near the sam-ple surface,then we will only use it for x >ξ.It is useful to note that in a more exact treatment we should obtain that G →0when x →0because the Gibbs free energy of a vortex located outside the sample must be zero.Figures 10(a),10(b)and 10(c)were generated using Equation (13)for κ=2.The free energy of the vor-tex depends on both the applied magnetic ﬁeld and the distance to the surface.For H <H c 1the free energy associated with a vortex is positive at all x values,as it is shown in Figure 10(a),vortices are then thermody-namically unstable inside the superconductor and there is an energy cost associated with the vortex entrance.The thermodynamic condition for vortex penetration is not fulﬁlled until H >H c 1,when the free energy of a vortex located well inside the superconductor becomes negative,see Figure 10(b).However,Figure 10(b)alsoshows that because of the attractive interaction between the vortex and the surface currents a barrier to vortex entrance appears nears the surface.This is the “Bean-Livingston”surface barrier originated by the screening currents.Taking into account Equation (13)it is possi-ble to estimate H p as the magnetic ﬁeld for which the expression (∂G/∂x )x =ξbecomes negative,i.e.when the maximum of G (x )moves inside the region x <ξ.This condition is fulﬁlled at H p =√2H c (H p ≈0.24H c 2in Figure 10(c)).At H =H p some vortices penetrate the sample and the free energy associated with the entrance of a new vortex now must be calculated using an analo-gous to Equation (14),the exact expression depends on the number of vortices and their location inside the sam-ple.Considering only one vortex located at x =l =3λand using Equation (14)we have calculated the free en-ergy just after the ﬁrst vortex entrance (H >0.24H c 2),the results are shown in Figure 10(d).Observe that the free energy changes considerably from Figure 10(c)to ﬁg-ure 10(d).Now there are three relevant regions:i)near x =3λ,there is a region where the Gibbs free energy increases with increasing x ,the strong repulsive inter-action with the vortex inside the sample dominates;ii)there is an intermediate region where G decreases at in-creasing x ,this means that a vortex located in this re-gion will be pulled inside the sample,it is possible to allocate vortices in this region and there is also a region iii)near the sample surface that repels vortex entrance,G increases for increasing x .The existence of regions ii)and iii)means a reinforcement of the surface barrier induced by vortex penetration and allows magnetic ﬁeld intervals of metastable states.This energy barrier rein-forcement due to vortex entrance is more important in mesoscopic superconductors because,in small samples,vortices are conﬁned by the potential well generated by the sample surface and even a vortex ﬁxed in the center of the sample is very close to the surface.For example in an 10λ×10λsquare sample,vortices are constrained to be located around the center of the sample at x =5λ,because of its interaction with the surface currents.As a consequence vortices stay at x ≈5λgenerating a new surface barrier and it is necessary an important mag-netic ﬁeld increase to allow new vortex penetration.For macroscopic samples the situation is quite diﬀerent,there is a nearly continuous vortex penetration.In macroscopic samples the vortices that are inside the sample are not conﬁned and they do not have serious restrictions in their movement since they can be allocated very far from the surface.In this case,a small increase of the magnetic ﬁeld is enough to accommodate new vortex lines,as can be observed in Figure 5(a).From the analysis of Figure 9and the previous discus-sion we can conclude that in mesoscopic samples there are preferred values of magnetic ﬁeld for vortex penetration,in this case the process of vortex entrance is discontinu-ous in contrast with the continuous macroscopic regime.This behavior is a consequence of the barrier to vortexentrance that appears after each penetration event.In this way we can deﬁne a second penetration ﬁeld H p 2,a third penetration ﬁeld H p 3and so on.We observe that after increasing the size of the sample the vortex pene-tration becomes continuous (H p 3→H p 2→H p ).The exact delimitation of the macroscopic and meso-scopic regimes depends on the geometry of the sample.We will now study in detail the behavior of a thin ﬁlm because is a simpler case with only one signiﬁcant length scale.The size dependence of the penetration ﬁelds H p ,H p 2,H p 3obtained numerically from the TDGL equa-tions in a thin ﬁlm using the S-I boundary condition are summarized in Figure 11(a).For d >15λa continu-ous vortex penetration is observed,this is the region of a macroscopic behavior.The mesoscopic region is lo-cated at 2λ d 15λin which several penetration ﬁelds H p i can be distinguished.For d <2λthere is a gradual transition to the normal state without vortices.Figure 11(b)shows the sample size dependence of the ﬁrst,sec-ond and third penetration ﬁeld of thin ﬁlms using an S-N boundary condition.For the S-N boundary the scales are shifted to larger sizes since the shielding length is larger (δSN >δSI ),as we discussed before.The macroscopic behavior appears at d >18λ,the mesoscopic region is lo-cated between 5λ d 18λand for d <5λa continuous transition to the normal state appears.It is interesting to note that H p is size dependent in the mesoscopic region.This size dependence can be ex-plained by considering in Equation (13)the eﬀect of two surfaces that are separated by a small distance d .The term generated by the magnetic ﬁeld H exp(−x/λ)now becomes H cosh[(x −d )/λ]/cosh(d/λ)due to the proxim-ity of the other surface,the magnetic force that pulls the vortex inside the sample decreases when the ﬁlm thick-ness (d )decreases.The image term also changes because the vortex lines that are trying to enter the sample are now near both surfaces and new image lines are neces-sary in order to satisfy the boundary condition at both surfaces.The application of the image method to parallel surfaces gives an inﬁnite number of images,but the net eﬀect is the appearance of an attractive interaction to the new surface.Then,in the mesoscopic region,there is also a decrease in the net attractive image force.If we con-sider only three relevant terms in the image forces,the normalized force f =(4π/Φo H c 2)λ(δG/δx )that feels a vortex line that is trying to enter when the sample is in the Meissner state can be estimated by:f =H a sinh(x −d/2cosh(d/2λ)+1λ)−1λ)+1λ)(15)where d is the thickness of the ﬁlm and we have chosen f positive when it repels the vortex entrance.As we analyzed before,H p is usually obtained from f |x =ξ=0.。