Renormalization-group flow in the 3D Georgi-Glashow model

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The entropy formula for the Ricci flow and its geometric applications

a rX iv:mat h /211159v1[ma t h.DG]11Nov22The entropy formula for the Ricci ﬂow and its geometric applications Grisha Perelman ∗November 20,2007Introduction 1.The Ricci ﬂow equation,introduced by Richard Hamilton [H 1],is the evolution equation d ∗St.Petersburg branch of Steklov Mathematical Institute,Fontanka 27,St.Petersburg191011,Russia.Email:perelman@pdmi.ras.ru or perelman@ ;I was partially supported by personal savings accumulated during my visits to the Courant Institute in the Fall of 1992,to the SUNY at Stony Brook in the Spring of 1993,and to the UC at Berkeley as a Miller Fellow in 1993-95.I’d like to thank everyone who worked to make those opportunities available to me.1in dimension four converge,modulo scaling,to metrics of constant positivecurvature.Without assumptions on curvature the long time behavior of the metricevolving by Ricciﬂow may be more complicated.In particular,as t ap-proaches someﬁnite time T,the curvatures may become arbitrarily large in some region while staying bounded in its complement.In such a case,it isuseful to look at the blow up of the solution for t close to T at a point where curvature is large(the time is scaled with the same factor as the metric ten-sor).Hamilton[H9]proved a convergence theorem,which implies that asubsequence of such scalings smoothly converges(modulo diﬀeomorphisms) to a complete solution to the Ricciﬂow whenever the curvatures of the scaledmetrics are uniformly bounded(on some time interval),and their injectivity radii at the origin are bounded away from zero;moreover,if the size of thescaled time interval goes to inﬁnity,then the limit solution is ancient,thatis deﬁned on a time interval of the form(−∞,T).In general it may be hard to analyze an arbitrary ancient solution.However,Ivey[I]and Hamilton[H4]proved that in dimension three,at the points where scalar curvatureis large,the negative part of the curvature tensor is small compared to the scalar curvature,and therefore the blow-up limits have necessarily nonneg-ative sectional curvature.On the other hand,Hamilton[H3]discovered a remarkable property of solutions with nonnegative curvature operator in ar-bitrary dimension,called a diﬀerential Harnack inequality,which allows,inparticular,to compare the curvatures of the solution at diﬀerent points and diﬀerent times.These results lead Hamilton to certain conjectures on thestructure of the blow-up limits in dimension three,see[H4,§26];the presentwork conﬁrms them.The most natural way of forming a singularity inﬁnite time is by pinchingan(almost)round cylindrical neck.In this case it is natural to make a surgery by cutting open the neck and gluing small caps to each of the boundaries,andthen to continue running the Ricciﬂow.The exact procedure was describedby Hamilton[H5]in the case of four-manifolds,satisfying certain curvature assumptions.He also expressed the hope that a similar procedure wouldwork in the three dimensional case,without any a priory assumptions,and that afterﬁnite number of surgeries,the Ricciﬂow would exist for all timet→∞,and be nonsingular,in the sense that the normalized curvatures ˜Rm(x,t)=tRm(x,t)would stay bounded.The topology of such nonsingular solutions was described by Hamilton[H6]to the extent suﬃcient to makesure that no counterexample to the Thurston geometrization conjecture can2occur among them.Thus,the implementation of Hamilton program would imply the geometrization conjecture for closed three-manifolds.In this paper we carry out some details of Hamilton program.The more technically complicated arguments,related to the surgery,will be discussed elsewhere.We have not been able to conﬁrm Hamilton’s hope that the so-lution that exists for all time t→∞necessarily has bounded normalized curvature;still we are able to show that the region where this does not hold is locally collapsed with curvature bounded below;by our earlier(partly unpublished)work this is enough for topological conclusions.Our present work has also some applications to the Hamilton-Tian con-jecture concerning K¨a hler-Ricciﬂow on K¨a hler manifolds with positiveﬁrst Chern class;these will be discussed in a separate paper.2.The Ricciﬂow has also been discussed in quantumﬁeld theory,as an ap-proximation to the renormalization group(RG)ﬂow for the two-dimensional nonlinearσ-model,see[Gaw,§3]and references therein.While my back-ground in quantum physics is insuﬃcient to discuss this on a technical level, I would like to speculate on the Wilsonian picture of the RGﬂow.In this picture,t corresponds to the scale parameter;the larger is t,the larger is the distance scale and the smaller is the energy scale;to compute something on a lower energy scale one has to average the contributions of the degrees of freedom,corresponding to the higher energy scale.In other words,decreasing of t should correspond to looking at our Space through a microscope with higher resolution,where Space is now described not by some(riemannian or any other)metric,but by an hierarchy of riemannian metrics,connected by the Ricciﬂow equation.Note that we have a paradox here:the regions that appear to be far from each other at larger distance scale may become close at smaller distance scale;moreover,if we allow Ricci ﬂow through singularities,the regions that are in diﬀerent connected compo-nents at larger distance scale may become neighboring when viewed through microscope.Anyway,this connection between the Ricciﬂow and the RGﬂow sug-gests that Ricciﬂow must be gradient-like;the present work conﬁrms this expectation.3.The paper is organized as follows.In§1we explain why Ricciﬂow can be regarded as a gradientﬂow.In§2,3we prove that Ricciﬂow,considered as a dynamical system on the space of riemannian metrics modulo diﬀeomor-phisms and scaling,has no nontrivial periodic orbits.The easy(and known)3case of metrics with negative minimum of scalar curvature is treated in§2; the other case is dealt with in§3,using our main monotonicity formula(3.4) and the Gaussian logarithmic Sobolev inequality,due to L.Gross.In§4we apply our monotonicity formula to prove that for a smooth solution on a ﬁnite time interval,the injectivity radius at each point is controlled by the curvatures at nearby points.This result removes the major stumbling block in Hamilton’s approach to geometrization.In§5we give an interpretation of our monotonicity formula in terms of the entropy for certain canonical ensemble.In§6we try to interpret the formal expressions,arising in the study of the Ricciﬂow,as the natural geometric quantities for a certain Riemannian manifold of potentially inﬁnite dimension.The Bishop-Gromov relative volume comparison theorem for this particular manifold can in turn be interpreted as another monotonicity formula for the Ricciﬂow.This for-mula is rigorously proved in§7;it may be more useful than theﬁrst one in local considerations.In§8it is applied to obtain the injectivity radius control under somewhat diﬀerent assumptions than in§4.In§9we consider one more way to localize the original monotonicity formula,this time using the diﬀerential Harnack inequality for the solutions of the conjugate heat equation,in the spirit of Li-Yau and Hamilton.The technique of§9and the logarithmic Sobolev inequality are then used in§10to show that Ricciﬂow can not quickly turn an almost euclidean region into a very curved one,no matter what happens far away.The results of sections1through10require no dimensional or curvature restrictions,and are not immediately related to Hamilton program for geometrization of three manifolds.The work on details of this program starts in§11,where we describe the ancient solutions with nonnegative curvature that may occur as blow-up limits ofﬁnite time singularities(they must satisfy a certain noncollaps-ing assumption,which,in the interpretation of§5,corresponds to having bounded entropy).Then in§12we describe the regions of high curvature under the assumption of almost nonnegative curvature,which is guaranteed to hold by the Hamilton and Ivey result,mentioned above.We also prove, under the same assumption,some results on the control of the curvatures forward and backward in time in terms of the curvature and volume at a given time in a given ball.Finally,in§13we give a brief sketch of the proof of geometrization conjecture.The subsections marked by*contain historical remarks and references. See also[Cao-C]for a relatively recent survey on the Ricciﬂow.41Ricciﬂow as a gradientﬂow1.1.Consider the functional F= M(R+|∇f|2)e−f dV for a riemannian metric g ij and a function f on a closed manifold M.Itsﬁrst variation can be expressed as follows:δF(v ij,h)= M e−f[−△v+∇i∇j v ij−R ij v ij−v ij∇i f∇j f+2<∇f,∇h>+(R+|∇f|2)(v/2−h)]= M e−f[−v ij(R ij+∇i∇j f)+(v/2−h)(2△f−|∇f|2+R)], whereδg ij=v ij,δf=h,v=g ij v ij.Notice that v/2−h vanishes identically iﬀthe measure dm=e−f dV is keptﬁxed.Therefore,the symmetric tensor −(R ij+∇i∇j f)is the L2gradient of the functional F m= M(R+|∇f|2)dm, where now f denotes log(dV/dm).Thus given a measure m,we may consider the gradientﬂow(g ij)t=−2(R ij+∇i∇j f)for F m.For general m thisﬂow may not exist even for short time;however,when it exists,it is just the Ricciﬂow,modiﬁed by a diﬀeomorphism.The remarkable fact here is that diﬀerent choices of m lead to the sameﬂow,up to a diﬀeomorphism;that is, the choice of m is analogous to the choice of gauge.1.2Proposition.Suppose that the gradientﬂow for F m exists for t∈[0,T]. Then at t=0we have F m≤nNow we computeF t≥2n( (R+△f)e−f dV)2=2t1and t2,are called Ricci solitons.(Thus,if one considers Ricciﬂow as a dy-namical system on the space of riemannian metrics modulo diﬀeomorphism and scaling,then breathers and solitons correspond to periodic orbits and ﬁxed points respectively).At each time the Ricci soliton metric satisﬁes an equation of the form R ij+cg ij+∇i b j+∇j b i=0,where c is a number and b i is a one-form;in particular,when b i=1log V=1dtV(2−n)/nλ −RdV≥n2V2/n[ |R ij+∇i∇j f−1( (R+△f)2e−f dV−( (R+△f)e−f dV)2)]≥0,nwhere f is the minimizer for F.72.4.The arguments above also show that there are no nontrivial(that is with non-constant Ricci curvature)steady or expanding Ricci solitons(on closed M).Indeed,the equality case in the chain of inequalities above requires that R+△f be constant on M;on the other hand,the Euler-Lagrange equation for the minimizer f is2△f−|∇f|2+R=const.Thus,△f−|∇f|2=const=0, because (△f−|∇f|2)e−f dV=0.Therefore,f is constant by the maximum principle.2.5*.A similar,but simpler proof of the results in this section,follows im-mediately from[H6,§2],where Hamilton checks that the minimum of RV22e−f dV,(3.1)restricted to f satisfying(4πτ)−nM,τt=−1(3.3)2τThe evolution equation for f can also be written as follows:2∗u=0,where u=(4πτ)−ng ij|2(4πτ)−n2τalong the Ricciﬂow.It is not hard to show that in the deﬁnition ofµthere always exists a smooth minimizer f(on a closed M).It is also clear that limτ→∞µ(g ij,τ)=+∞whenever theﬁrst eigenvalue of−4△+R is positive. Thus,our statement that there is no shrinking breathers other than gradient solitons,is implied by the followingClaim For an arbitrary metric g ij on a closed manifold M,the function µ(g ij,τ)is negative for smallτ>0and tends to zero asτtends to zero.Proof of the Claim.(sketch)Assume that¯τ>0is so small that Ricci ﬂow starting from g ij exists on[0,¯τ].Let u=(4πτ)−n2τ−1g ij,fτ,12τ−1g ij”converge”to the euclidean metric,and if we couldextract a converging subsequence from fτ,we would get a function f on R n, such that R n(2π)−n2|∇f|2+f−n](2π)−n2−t)=µ(g ij(0),12)satisﬁesR ij+∇i∇j f−g ij=0.Of course,this argument requires the existence of minimizer,and justiﬁcation of the integration by parts;this is easy if M is closed,but can also be done with more eﬀorts on some complete M,for instance when M is the Gaussian soliton.93.3*The no breathers theorem in dimension three was proved by Ivey[I]; in fact,he also ruled out nontrivial Ricci solitons;his proof uses the almost nonnegative curvature estimate,mentioned in the introduction.Logarithmic Sobolev inequalities is a vast area of research;see[G]for a survey and bibliography up to the year1992;the inﬂuence of the curvature was discussed by Bakry-Emery[B-Em].In the context of geometric evolution equations,the logarithmic Sobolev inequality occurs in Ecker[E1].4No local collapsing theorem IIn this section we present an application of the monotonicity formula(3.4) to the analysis of singularities of the Ricciﬂow.4.1.Let g ij(t)be a smooth solution to the Ricciﬂow(g ij)t=−2R ij on[0,T). We say that g ij(t)is locally collapsing at T,if there is a sequence of times t k→T and a sequence of metric balls B k=B(p k,r k)at times t k,such that r2k/t k is bounded,|Rm|(g ij(t k))≤r−2k in B k and r−n k V ol(B k)→0.Theorem.If M is closed and T<∞,then g ij(t)is not locally collapsing at T.Proof.Assume that there is a sequence of collapsing balls B k=B(p k,r k) at times t k→T.Then we claim thatµ(g ij(t k),r2k)→−∞.Indeed one(x,p k)r−1k)+c k,whereφis a function of one can take f k(x)=−logφ(dist tkvariable,equal1on[0,1/2],decreasing on[1/2,1],and very close to0on [1,∞),and c k is a constant;clearly c k→−∞as r−n k V ol(B k)→0.Therefore, applying the monotonicity formula(3.4),we getµ(g ij(0),t k+r2k)→−∞. However this is impossible,since t k+r2k is bounded.4.2.Deﬁnition We say that a metric g ij isκ-noncollapsed on the scaleρ,if every metric ball B of radius r<ρ,which satisﬁes|Rm|(x)≤r−2for every x∈B,has volume at leastκr n.It is clear that a limit ofκ-noncollapsed metrics on the scaleρis also κ-noncollapsed on the scaleρ;it is also clear thatα2g ij isκ-noncollapsed on the scaleαρwhenever g ij isκ-noncollapsed on the scaleρ.The theorem above essentially says that given a metric g ij on a closed manifold M and T<∞,one canﬁndκ=κ(g ij,T)>0,such that the solution g ij(t)to the Ricciﬂow starting at g ij isκ-noncollapsed on the scale T1/2for all t∈[0,T), provided it exists on this interval.Therefore,using the convergence theorem of Hamilton,we obtain the following10Corollary.Let g ij (t ),t ∈[0,T )be a solution to the Ricci ﬂow on a closed manifold M,T <∞.Assume that for some sequences t k →T,p k ∈M and some constant C we have Q k =|Rm |(p k ,t k )→∞and |Rm |(x,t )≤CQ k ,whenever t <t k .Then (a subsequence of)the scalings of g ij (t k )at p k with factors Q k converges to a complete ancient solution to the Ricci ﬂow,which is κ-noncollapsed on all scales for some κ>0.5A statistical analogyIn this section we show that the functional W ,introduced in section 3,is in a sense analogous to minus entropy.5.1Recall that the partition function for the canonical ensemble at tem-perature β−1is given by Z = exp (−βE )dω(E ),where ω(E )is a ”density of states”measure,which does not depend on β.Then one computes the average energy <E >=−∂(∂β)2log Z.Now ﬁx a closed manifold M with a probability measure m ,and suppose that our system is described by a metric g ij (τ),which depends on the temper-ature τaccording to equation (g ij )τ=2(R ij +∇i ∇j f ),where dm =udV,u =(4πτ)−n 2)dm.(We do not discuss here what assumptions on g ij guarantee that the corre-sponding ”density of states”measure can be found)Then we compute<E >=−τ2 M(R +|∇f |2−n 2τg ij |2dmAlternatively,we could prescribe the evolution equations by replacing the t -derivatives by minus τ-derivatives in (3.3),and get the same formulas for Z,<E >,S,σ,with dm replaced by udV.Clearly,σis nonnegative;it vanishes only on a gradient shrinking soliton.<E >is nonnegative as well,whenever the ﬂow exists for all suﬃciently small τ>0(by proposition 1.2).Furthermore,if (a)u tends to a δ-function as τ→0,or (b)u is a limit of a sequence of functions u i ,such that each u i11tends to aδ-function asτ→τi>0,andτi→0,then S is also nonnegative. In case(a)all the quantities<E>,S,σtend to zero asτ→0,while in case (b),which may be interesting if g ij(τ)goes singular atτ=0,the entropy S may tend to a positive limit.If theﬂow is deﬁned for all suﬃciently largeτ(that is,we have an ancient solution to the Ricciﬂow,in Hamilton’s terminology),we may be interested in the behavior of the entropy S asτ→∞.A natural question is whether we have a gradient shrinking soliton whenever S stays bounded.5.2Remark.Heuristically,this statistical analogy is related to the de-scription of the renormalization groupﬂow,mentioned in the introduction: in the latter one obtains various quantities by averaging over higher energy states,whereas in the former those states are suppressed by the exponential factor.5.3*An entropy formula for the Ricciﬂow in dimension two was found by Chow[C];there seems to be no relation between his formula and ours.The interplay of statistical physics and(pseudo)-riemannian geometry occurs in the subject of Black Hole Thermodynamics,developed by Hawking et al.Unfortunately,this subject is beyond my understanding at the moment.6Riemannian formalism in potentially inﬁ-nite dimensionsWhen one is talking of the canonical ensemble,one is usually considering an embedding of the system of interest into a much larger standard system of ﬁxed temperature(thermostat).In this section we attempt to describe such an embedding using the formalism of Rimannian geometry.6.1Consider the manifold˜M=M×S N×R+with the following metric:˜g ij=g ij,˜gαβ=τgαβ,˜g00=N2N .It turns out that the components of the curvaturetensor of this metric coincide(modulo N−1)with the components of the matrix Harnack expression(and its traces),discovered by Hamilton[H3]. One can also compute that all the components of the Ricci tensor are equal12to zero(mod N−1).The heat equation and the conjugate heat equation on M can be interpreted via Laplace equation on˜M for functions and volume forms respectively:u satisﬁes the heat equation on M iﬀ˜u(the extension of u to˜M constant along the S Nﬁbres)satisﬁes˜△˜u=0mod N−1;similarly,u satisﬁes the conjugate heat equation on M iﬀ˜u∗=τ−N−12e−f dV).To achieve this,ﬁrst apply to˜g a(small)diﬀeomor-phism,mapping each point(x i,yα,τ)into(x i,yα,τ(1−2fN)˜gαβ,˜g m00=˜g00−2fτ−fN)˜gαβ,g m00=˜g m00−|∇f|2=12−[τ(2△f−|∇f|2+R)+f−n]),g m i0=g mα0=g m iα=0Note that the hypersurfaceτ=const in the metric g m has the volume form τN/2e−f times the canonical form on M and S N,and the scalar curvatureof this hypersurface is12+τ(2△f−|∇f|2+R)+f)mod N−1.Thus theentropy S multiplied by the inverse temperatureβis essentially minus the total scalar curvature of this hypersurface.6.3Now we return to the metric˜g and try to use its Ricci-ﬂatness by interpreting the Bishop-Gromov relative volume comparison theorem.Con-sider a metric ball in(˜M,˜g)centered at some point p whereτ=0.Then clearly the shortest geodesic between p and an arbitrary point q is always orthogonal to the S Nﬁbre.The length of such curveγ(τ)can be computedas τ(q)2τ+R+|˙γM(τ)|2dτ= √τ(R+|˙γM(τ)|2)dτ+O(N−3Thus a shortest geodesic should minimize L(γ)= τ(q)0√2Nτ(q) centered at p is O(N−1)-close to the hypersurfaceτ=τ(q),and its volume can be computed as V(S N) M( 2N L(x)+O(N−2))N dx,so the ratio of this volume to 2timesτ(q)−nMτ(R(γ(τ))+|˙γ(τ)|2)dτ(of course,R(γ(τ))and|˙γ(τ)|2are computed using g ij(τ))Let X(τ)=˙γ(τ),and let Y(τ)be any vectorﬁeld alongγ(τ).Then the ﬁrst variation formula can be derived as follows:δY(L)=14τ2τ1√τ(<Y,∇R >+2<∇X Y,X >)dτ= τ2τ1√dτ<Y,X >−2<Y,∇X X >−4Ric(Y,X ))dτ=2√τ<Y,∇R −2∇X X −4Ric(X,·)−12∇R +1τX (τ)has a limit as τ→0.From now on we ﬁx p and τ1=0and denote by L (q,¯τ)the L -length of the L -shortest curve γ(τ),0≤τ≤¯τ,connecting p and q.In the computations below we pretend that shortest L -geodesics between p and q are unique for all pairs (q,¯τ);if this is not the case,the inequalities that we obtain are still valid when understood in the barrier sense,or in the sense of distributions.The ﬁrst variation formula (7.1)implies that ∇L (q,¯τ)=2√¯τ(R +|X |2)−<X,∇L >=2√¯τ(R +|X |2)To evaluate R +|X |2we compute (using (7.2))dτR +2<∇R,X >−2Ric(X,X )−1τ(R +|X |2),(7.3)where H (X )is the Hamilton’s expression for the trace Harnack inequality (with t =−τ).Hence,¯τ32L (q,¯τ),(7.4)15where K =K (γ,¯τ)denotes the integral ¯τ0τ3¯τR −1¯τK (7.5)|∇L |2=−4¯τR +2¯τL −4¯τK (7.6)Finally we need to estimate the second variation of L.We computeδ2Y (L )=¯τ0√τ(Y ·Y ·R +2<∇X ∇Y Y,X >+2<R (Y,X ),Y,X >+2|∇X Y |2)dτNowd¯τ+¯τ0√2τY (7.8)We computed τ<Y,Y >,16so |Y (τ)|2=ττ(∇Y ∇Y R +2<R (Y,X ),Y,X >+2∇X Ric(Y,Y )−4∇Y Ric(Y,X )+2|Ric(Y,·)|2−22τ¯τ)dτTo put this in a more convenient form,observe thatdτRic(Y,Y )−2|Ric(Y,·)|2,so Hess L (Y,Y )≤1¯τ−2√τH (X,Y )dτ,(7.9)whereH (X,Y )=−∇Y ∇Y R −2<R (Y,X )Y,X >−4(∇X Ric(Y,Y )−∇Y Ric(Y,X ))−2Ric τ(Y,Y )+2|Ric(Y,·)|2−1τR +n τ−1dτ|Y |2=2Ric(Y,Y )+2<∇X Y,Y >=2Ric(Y,Y )+2<∇Y X,Y >=2Ric(Y,Y )+1¯τHess L (Y,Y )≤1√2H (X,˜Y )dτ,(7.11)where ˜Y is obtained by solving ODE (7.8)with initial data ˜Y (¯τ)=Y (¯τ).Moreover,the equality in (7.11)holds only if ˜Y is L -Jacobi and hence d √¯τ.17Now we can deduce an estimate for the jacobian J of the L-exponential map,given by L exp X(¯τ)=γ(¯τ),whereγ(τ)is the L-geodesic,starting at p and having X as the limit of√dτlog J(τ)≤n2¯τ−3√¯τg.Let l(q,τ)=1τL(q,τ)be thereduced distance.Then along an L-geodesicγ(τ)we have(by(7.4))d2¯τl+12¯τ−32exp(−l(τ))J(τ)is nonincreasing inτalongγ, and monotonicity is strict unless we are on a gradient shrinking soliton. Integrating over M,we get monotonicity of the reduced volume function ˜V(τ)= Mτ−n2¯τ≥0,(7.13) which follows immediately from(7.5),(7.6)and(7.10).Note also a useful inequality2△l−|∇l|2+R+l−nτL(q,τ),then from(7.5), (7.10)we obtain¯L¯τ+△¯L≤2n(7.15) Therefore,the minimum of¯L(·,¯τ)−2n¯τis nonincreasing,so in particular, the minimum of l(·,¯τ)does not exceed n2(τ0−τ),whenever theﬂow exists forτ∈[0,τ0].)7.2If the metrics g ij(τ)have nonnegative curvature operator,then Hamil-ton’s diﬀerential Harnack inequalities hold,and one can say more about the behavior of l.Indeed,in this case,if the solution is deﬁned forτ∈[0,τ0],then H(X,Y)≥−Ric(Y,Y)(1τ0−τ)≥−R(1τ0−τ)|Y|2and18H(X)≥−R(1τ0−τ).Therefore,wheneverτis bounded away fromτ0(say,τ≤(1−c)τ0,c>0),we get(using(7.6),(7.11))|∇l|2+R≤Cldτlog|Y|2≤1n.We claim that˜V k(ǫk r2k)<3ǫn2ǫ−12k;on the otherhand,the contribution of the longer vectors does not exceed exp(−12k)by the jacobian comparison theorem.However,˜V k(t k)(that is,at t=0)stays bounded away from zero.Indeed,since min l k(·,t k−12,we can pick a point q k,where it is attained,and obtain a universal upper bound on l k(·,t k)by considering only curvesγwithγ(t k−12T].Sincethe monotonicity of the reduced volume requires˜V k(t k)≤˜V k(ǫk r2k),this is a contradiction.A similar argument shows that the statement of the corollary in4.2can be strengthened by adding another property of the ancient solution,obtained as a blow-up ly,we may claim that if,say,this solution is deﬁned for t∈(−∞,0),then for any point p and any t0>0,the reduced volume function˜V(τ),constructed using p andτ(t)=t0−t,is bounded below byκ.7.4*The computations in this section are just natural modiﬁcations of those in the classical variational theory of geodesics that can be found in any textbook on Riemannian geometry;an even closer reference is[L-Y],where they use”length”,associated to a linear parabolic equation,which is pretty much the same as in our case.198No local collapsing theorem II8.1Let usﬁrst formalize the notion of local collapsing,that was used in7.3.Deﬁnition.A solution to the Ricciﬂow(g ij)t=−2R ij is said to be κ-collapsed at(x0,t0)on the scale r>0if|Rm|(x,t)≤r−2for all(x,t) satisfying dist t(x,x0)<r and t0−r2≤t≤t0,and the volume of the metric ball B(x0,r2)at time t0is less thanκr n.8.2Theorem.For any A>0there existsκ=κ(A)>0with the fol-lowing property.If g ij(t)is a smooth solution to the Ricciﬂow(g ij)t=−2R ij,0≤t≤r20,which has|Rm|(x,t)≤r−20for all(x,t),satisfying dist0(x,x0)<r0,and the volume of the metric ball B(x0,r0)at time zero is at least A−1r n0,then g ij(t)can not beκ-collapsed on the scales less than r0at a point(x,r20)with dist r20(x,x0)≤Ar0.Proof.By scaling we may assume r0=1;we may also assume dist1(x,x0)= A.Let us apply the constructions of7.1choosing p=x,τ(t)=1−t.Arguing as in7.3,we see that if our solution is collapsed at x on the scale r≤1,then the reduced volume˜V(r2)must be very small;on the other hand,˜V(1)can not be small unless min l(x,12(x,x0)≤13Kr0+r−10)(the inequality must be understood in the barrier sense,when necessary)(b)(cf.[H4,§17])Suppose Ric(x,t0)≤(n−1)K when dist t(x,x0)<r0, or dist t(x,x1)<r0.Thend3Kr0+r−10)at t=t0 Proof of Lemma.(a)Clearly,d t(x)= γ−Ric(X,X),whereγis the shortest geodesic between x and x0and X is its unit tangent vector,On the other hand,△d≤ n−1k=1s′′Y k(γ),where Y k are vectorﬁelds alongγ,vanishing at20x0and forming an orthonormal basis at x when complemented by X,ands′′Yk (γ)denotes the second variation along Y k of the length ofγ.Take Y k to beparallel between x and x1,and linear between x1and x0,where d(x1,t0)=r0. Then△d≤n−1k=1s′′Y k(γ)= d(x,t0)r0−Ric(X,X)ds+ r00(s2r20)ds= γ−Ric(X,X)+ r00(Ric(X,X)(1−s2r20)ds≤d t+(n−1)(220),andrapidly increasing to inﬁnity on(110),in such a way that2(φ′)2/φ−φ′′≥(2A+100n)φ′−C(A)φ,(8.1) for some constant C(A)<∞.Note that¯L+2n+1≥1for t≥12)is achieved for some y satisfying d(y,110.Now we compute2h=(¯L+2n+1)(−φ′′+(d t−△d−2A)φ′)−2<∇φ∇¯L>+(¯L t−△¯L)φ(8.2)∇h=(¯L+2n+1)∇φ+φ∇¯L(8.3) At a minimum point of h we have∇h=0,so(8.2)becomes2h=(¯L+2n+1)(−φ′′+(d t−△d−2A)φ′+2(φ′)2/φ)+(¯L t−△¯L)φ(8.4)Now since d(y,t)≥120),we can apply our lemma(a)to get d t−△d≥−100(n−1)on the set where φ′=0.Thus,using(8.1)and(7.15),we get2h≥−(¯L+2n+1)C(A)φ−2nφ≥−(2n+C(A))hThis implies that min h can not decrease too fast,and we get the required estimate.219Diﬀerential Harnack inequality for solutions of the conjugate heat equation9.1Proposition.Let g ij(t)be a solution to the Ricciﬂow(g ij)t=−2R ij,0≤t≤T,and let u=(4π(T−t))−ng ij|2(9.1)2(T−t)Proof.Routine computation.Clearly,this proposition immediately implies the monotonicity formula (3.4);its advantage over(3.4)shows up when one has to work locally.9.2Corollary.Under the same assumptions,on a closed manifold M,or whenever the application of the maximum principle can be justiﬁed,min v/u is nondecreasing in t.9.3Corollary.Under the same assumptions,if u tends to aδ-function as t→T,then v≤0for all t<T.Proof.If h satisﬁes the ordinary heat equation h t=△h with respect to the evolving metric g ij(t),then we have ddt hv≥0.Thus we only need to check that for everywhere positive h the limit of hv as t→T is nonpositive.But it is easy to see,that this limit is in fact zero.9.4Corollary.Under assumptions of the previous corollary,for any smooth curveγ(t)in M holds−d2(R(γ(t),t)+|˙γ(t)|2)−1and2(T−t)v≤0we get f t+12|∇f|2−f dt f(γ(t),t)=−f t−<∇f,˙γ(t)>≤−f t+12|˙γ|2.Summing these two inequalities, we get(9.2).9.5Corollary.If under assumptions of the previous corollary,p is the point where the limitδ-function is concentrated,then f(q,t)≤l(q,T−t),where l is the reduced distance,deﬁned in7.1,using p andτ(t)=T−t.22。

Gravitational Stability and Renormalization-Group Flow

a rXiv:h ep-th/9997v22O ct1999DAMTP-1999-111SPIN-1999/20hep-th/9909070Gravitational Stability and Renormalization-Group Flow Kostas Skenderis ∗Spinoza Institute,University of Utrecht Leuvenlaan 4,3584CE Utrecht The Netherlands and Paul K.Townsend †DAMTP,University of Cambridge Silver Street,Cambridge CB39EW,UK AbstractFirst-order ‘Bogomol’nyi’equations are found for dilaton domain walls of D-dimensional gravity with the general dilaton potential admitting a stable anti-de Sitter vacuum.Im-plications for renormalization group ﬂow in the holographically dual ﬁeld theory are discussed.1IntroductionThe strong t’Hooft-coupling limit of certain non-conformal supersymmetric quantum ﬁeld theories associated with coincident non-conformal branes has a description in terms of supergravity theory[1].This description involves gauged supergravities admitting domain-wall vacua[2].The Minkowski vacuum of the gauge theory at a given scale is a ‘horosphere’of a supergravity dilaton domain wall,i.e.a hypersurface in the‘holographic frame’anti-de Sitter(adS)metric on which the dilaton is constant[2].The position of the horosphere and the value of the dilaton is directly related to the energy scale of the gauge theory.The domain wall solution itself therefore corresponds in the gauge theory to renormalization-group(RG)ﬂow from one scale to another.The cases considered in [2],and similar lower-dimensional cases[3],are all ones for which the dilaton potential is a simple exponential.In such cases there is no maximally-supersymmetric adS vacuum but there is a1/2supersymmetric linear-dilaton vacuum which can be interpreted as a domain wall.Another type of domain wall,interpolating between adS vacua with diﬀerent radii of curvature,has been extensively studied in the context of D=4supergravity[4,5],and similar solutions have recently been found for D=5supergravity theories[6,7,8,9].These domain walls correspond to RGﬂow from one superconformalﬁeld theory to another. Other examples of RGﬂows of d=4,N=4SYM theory that have a description in terms of D=5supergravity can be found in[10,11,12,13,14,15].More recently,the RGﬂow associated with domain walls has been used in the context of‘Brane World’scenarios to explain the origin of mass hierarchies and as a possible explanation for the smallness of the cosmological constant[16,17,18].Given these new applications of domain wall spacetimes,it would be helpful to have a model-independent analysis of the possibilities in which basic physical requirements are the only input.Since matterﬁelds other than scalars play no role in domain wall solutions,the general framework is gravity coupled to a scalarﬁeld theory in D spacetime dimensions.The scalarﬁelds will take values in some target space M and the model is characterized by the metric on M,which determines the scalar kinetic terms,and a function V on M,which determines the scalar potential.The target space metric must be positive deﬁnite for vacuum stability.Intuition from non-gravitationalﬁeld theory mightlead one to suppose that vacuum stability also requires that V be positive but in gauged supergravity theories V is typically unbounded from below,and the supersymmetric adS vacua are either maxima or saddle points of V[19].The perturbative stability of these adS vacua is guaranteed by the fact that the eigenvalues of the scalar mass matrix satisfy the Breitenlohner-Freedman bound[20]or its D-dimensional generalization[21]. Non-perturbative stability has also been established in many cases by an extension of the spinorial proof of the positive energy theorem[22]to asymptotically adS spacetimes [23,24].This method was used in[25,26]to determine the restrictions on V that arise from the requirement that there exist a stable adS vacuum,whether supersymmetric or not.The results imply the perturbative stability bounds of[20,21]but go well beyond them by providing information about the potential away from its critical points.This information is particularly useful if one supposes that there is only a single scalar ﬁeldφ,which we shall call the‘dilaton’.In this case M=R so the target space metric is diﬀeomorphic to a constant and V becomes a function of a single real variable.The general model discussed above reduces to one with Lagrangian densityL= 2R−1restrictive than those used in its derivation.Notice also that a potential of the form(2) for the multi-scalar case still guarantees gravitational stability[26]although the converse is not necessarily true,i.e.in the multi-scalar case there may be more general potentials than(2)compatible with gravitational stability.In particular,the potential of a subset of the scalars of the D=5supergravity used in recent studies[8,9,13]is of the form(2). The potential(2)has a form that is typical in supergravity theories,hence the choice of terminology‘superpotential’for w,even though supersymmetry is not an ingredient in its derivation.In this paper we will investigate general properties of domain wall solutions in the theory with Lagrangian(1)with V given by(2).Our interest in domain wall spacetimes stems from their connection to the RGﬂow of the dualﬁeld theories.Such models are characterized by their superpotential w.Let usﬁrst note thatV′=4(D−2)[(D−2)w′′−(D−1)w]w′(3) so that V has critical points at critical points of w,and at points for which w′′=D−1but we shallﬁrst consider the general case.2Domain walls and the c-functionWe begin by making the domain-wall ansatzds2=e2A(r)ds2 E(1,D−2) +dr2(4) with dilatonﬁeldφ(r).Let us introduce a new radial coordinate U=e A.The domain-wall spacetime then takes the formds2=U2ds2 E(1,D−2) +1U2.(5) At critical points of V the dilaton is constant,as is(∂r A),and the geometry becomes anti-de Sitter with a cosmological constantΛequal to the value of V at the critical point;Λ=−1(∂rφ)2.(9)D−2In [6,8]thefunctionC(U )=C 0/[∂r A (r )]D −2(10)was proposed as a c-function,where C 0is a constant related to the universal coeﬃcient appearing in the “holographic”Weyl anomaly [27](for odd D ).By deﬁnition,a c-function is a positive function of the coupling constant(s)that is non-increasing along the RG ﬂow from the UV to the IR.We can easily establish monotonicity:U ∂(∂r A )2∂2r A =C ∂r φ2 ∞−∞dr e (D −1)A (∂r φ)2−(D −1)(D −2)(∂r A )2+2V .(12)The integrand is minus the eﬀective Lagrangian obtained by substitution of the domain wall ansatz into the Lagrangian (1),and the functional E is simply related to expressions for the energy obtained by other means 1.The functional (12)can be rewritten,`a la Bogomol’nyi,asE =11For example,use of the ﬁeld equations allows E to be expressed as a surface integral of the second fundamental form,which was shown in [28]to be proportional to the energy.It is straightforward to verify that solutions of these equations indeed solve the second-order equations(6)-(8).We shall call solutions of these equations‘BPS domain walls’.Another way to arrive at theﬁrst-order equations(14)is to note that the energy bound established in[26]is saturated byﬁeld conﬁgurations for which the equations(D m+wΓm)ǫ=0,[Γm∂mφ−2(D−2)w′]ǫ=0,(15)admit solutions for a non-zero spinorǫ,which we shall call a Killing spinor.Suchﬁeld con-ﬁgurations automatically solve the second-order Einstein-dilaton equations.Substitution of the domain wall ansatz into the equations(15)leads immediately to the equations(14). The Killing spinor isǫ=e A/2ǫ0withǫ0a constant spinor satisfyingΓrǫ0=±ǫ0.In the context of supergravity,the domain walls admitting Killing spinors are supersymmetric.4ExamplesWe now consider two examples involving the N=1,D=7supergravity[29]and the N=2,D=6F(4)supergravity[30].The D=7theory is obtained by compactiﬁcation of D=11supergravity on S4[31],and it is associated with the near-horizon limit of the M5 brane with an orbifold projection on the transverse sphere[32,33,34].The D=6theory is obtained by a warped S4-compactiﬁcation of massive IIA supergravity[35],and it is associated with the near-horizon limit of the D4-D8system[36,37].In both cases there is a single scalarﬁeldφ,and we may discuss them simultaneously.The potential is given by(2)withw(φ)=−12(D−2)ge1D−2φ+mD−5e−D−3D−2φ (16)Here g is the coupling constant of the gauge group and m is a‘topological’mass parame-ter.The potential has two critical points:at w′=0and at w′′=D−1D−2φ=m√D−2φ=m D−5,(17) Only theﬁrst(w′=0)critical point is supersymmetric.Domain-wall solutions of these supergravity theories,preserving1/2supersymmetry,were found in[38].In terms of a new radial parameterρ,they take the formD−2ds2=e2BD−2w′(φ).(19) There is an apparent singularity at the critical point w′=0but this is only a coordinate singularity,as one can verify by choosing U=e B2dU2∂ρe−Bw w′=0=0(21)D−2is required in order for the domain wall solution to become the supersymmetric adS solution as the critical point is approached.This relation turns out to be satisﬁed.In the new radial coordinate the critical point is at U=∞,so it corresponds to a UVﬁxed point of the dualﬁeld theory.We conclude with an example of a superpotential admitting a BPS domain wall but which is not the superpotential of any known supergravity theory(at least not for general D).A class of solutions of equations(14)is obtained byﬁrst considering these equations for complexφ,w(φ),A(r),and then imposing reality conditions2.As an example we consider the case the superpotential w(φ)is equal to the Weierstrass elliptic function, w(φ)=℘(φ;g2,g3).The dilatonφis then the uniformizing variable of the elliptic curve associated to the Weierstrass function.Let us recall some standard facts about the Weiersrass function,℘(φ;g2,g3).It satisﬁes the diﬀerential equation,℘′2=4℘3−g2℘−g3.(22) It follows that the superpotential has three critical points.The value of the superpotential at the critical points is given by the three roots,e1,e2,e3,of the cubic polynomial in theright hand side of(22).The critical points occur atφ=ω1,φ=ω1+ω2,φ=ω2,where ω1andω2are the half periods of℘.One can easily integrate equations(14).The result isr−r0=±1(e1−e2)(e1−e3)log(℘−e1)+1(e3−e1)(e3−e2)log(℘−e3)A(r(φ))−A0=−1(e1−e2)(e1−e3)log(℘−e1)+e2(e3−e1)(e3−e2)log(℘−e3)(23)where r0and A0are integration constants,which we set to zero so that the critical points occur for r=±∞,and U=0or U=∞.When g2,g3are real one can impose reality conditions on the solution.There are two cases to consider.When the discriminant∆=g32−27g23is positive,one may chooseprimitive periods such thatω1is real andω2is imaginary.In this case all three roots e i (e1>e2>e3,e1>0,e3<0)are real.The(real)superpotential has one critical point atφ=ω1(the other two critical points occur for complex values of the dilaton).When ∆<0,one may chooseω1,ω2to be complex conjugate of each other.The roots e1and e3are complex conjugates and e2is real.The(real)superpotential has one critical point atφ=ω1+ω2.When two of the roots coincide,or what is the same,one of the periods becomes inﬁnite,the Weierstrass function reduces to an elementary function.Acknowledgments:We thank Harm Jan Boonstra for collaboration at early stages of this work.KS is supported by the Netherlands Organization for Scientiﬁc Research (NWO).References[1]N.Itzhaki,J.M.Maldacena,J.Sonnenschein and S.Yankielowicz,Supergravity andthe large N limit of theories with sixteen supercharges,Phys.Rev.D58(1998) 046004.[2]H.J.Boonstra,K.Skenderis and P.K.Townsend,The domain-wall/QFT correspon-dence,JHEP9901(1999)003,hep-th/9807137;K.Skenderis,Field theory limit of branes and gauged supergravities,hep-th/9903003.[3]K.Behrndt, E.Bergshoeﬀ,R.Halbersma and J.P.van de Schaar,On domainwall/QFT dualities in various dimensions,hep-th/9907006.[4]M.Cvetiˇc,S.Griﬃes and S-J Rey,Static domain walls in N=1supergravity,Nucl.Phys.B381(1997)301.[5]M.Cvetiˇc and H.H.Soleng,Supergravity domain walls,Phys.Rept.282(1997)159.[6]L.Girardello,M.Petrini,M.Porrati and A.Zaﬀaroni,Novel Local CFT and ExactResults on Perturbations of N=4Super Yang Mills from AdS Dynamics,JHEP9812 (1998)022,hep-th/9810126.[7]J.Distler and F.Zamora,Non-supersymmetric Conformal Field Theories from StableAnti-de Sitter Spaces,Adv.Theor.Math.Phys.2(1998)1405-1439,hep-th/9810206.[8]D.Z.Freedman,S.S.Gubser,K.Pilch and N.P.Warner,Renormalization groupﬂowsfrom holography–supersymmetry and a c-theorem,hep-th/9904017.[9]K.Berndt,Domain walls of D=5supergravity andﬁxpoints of N=1super Yang-Mills,hep-th/9907070.[10]A.Kehagias and K.Sfetsos,On the runnings in gauge theories from type IIB super-gravity,Phys.Lett.B454(1999)270-276,hep-th/9902125.[11]S.Gubser,Dilaton driven conﬁnement,hep-th/9902155.[12]L.Girardello,M.Petrini,M.Porrati and A.Zaﬀaroni,Conﬁnement and CondensatesWithout Fine Tuning in Supergravity Duals of Gauge Theories,JHEP9905(1999) 026,hep-th/9903026.[13]D.Z.Freedman,S.S.Gubser,K.Pilch and N.P.Warner,Continuous distributions ofD3-branes and gauged supergravity,hep-th/9906194.[14]A.Brandhuber and K.Sfetsos,Wilson loops from multicentre and rotating branes,mass gaps and phase structure in gauge theories,hep-th/9906201.[15]L.Girardello,M.Petrini,M.Porrati and A.Zaﬀaroni,The Supergravity Dual ofN=1Super Yang-Mills Theory,hep-th/9909047.[16]L.Randall and R.Sundrum,A large mass hierarchy from a small extra dimension,hep-ph/9905221;An alternative to compactiﬁcation,hep-th/9906064.[17]H.Verlinde,Holography and Compactiﬁcation,hep-th/9906182.[18]E.Verlinde,Holography,Compactiﬁcation and the Cosmological Constant,talk inString’99,http://strings99.aei-potsdam.mpg.de.[19]N.P.Warner,Some properties of the scalar potential in gauged supergravity theories,Nucl.Phys.B231(1984)250.[20]P.Breitenlohner and D.Z.Freedman,Positive energy in anti-de Sitter backgroundsand gauged extended supergravity,Phys.Lett.115B(1982)197;Stability in gauged extended supergravity,Ann.Phys.144(1982)249.[21]L.Mezincescu and P.K.Townsend,Stability at a local maximum in higher dimen-sional anti-deSitter space,and applications to supergravity,Ann.Phys.(N.Y.)160 (1985)406.[22]E.Witten,A simple proof of the positive energy theorem,Commun.Math.Phys.80(1981)381.[23]G.W.Gibbons,C.M.Hull and N.Warner,The stability of gauged supergravity,Nucl.Phys.B218(1983)173.[24]L.Mezincescu,P.K.Townsend and P.van Nieuwenhuizen,Stability of gauged d=7supergravity and the deﬁnition of masslessness in(adS)7,Phys.Lett.143B(1984) 384.[25]W.Boucher,Positive energy without supersymmetry,Nucl.Phys.B242(1984)282.[26]P.K.Townsend,Positive energy and the scalar potential in higher dimensional(su-per)gravity theories,Phys.Lett.148B(1984)55.[27]M.Henningson and K.Skenderis,The Holographic Weyl anomaly,JHEP9807(1998)023,hep-th/9806087;Holography and the Weyl anomaly,hep-th/9812032.[28]S.W.Hawking and G.T.Horowitz,The gravitational hamiltonian,action,entropyand surface terms,Class.Quantum Grav.13(1996)1487-1498,gr-qc/9501014.[29]P.K.Townsend and P.van Nieuwenhuizen,Gauged seven-dimensional supergravity,Phys.Lett.B125(1983)41.[30]L.J.Romans,The F(4)gauged supergravity in six dimensions,Nucl.Phys.B269(1986)691.[31]H.L¨u and C.N.Pope,Exact embedding of N=1D=7gauged supergravity in D=11,hep-th/9906168.[32]M.Berkooz,A Supergravity Dual of a(1,0)Field Theory in Six Dimensions,Phys.Lett.B437(1998)315-317,hep-th/9802195.[33]S.Ferrara,A.Kehagias,H.Partouche and A.Zaﬀaroni,Membranes and Fivebraneswith Lower Supersymmetry and their AdS Supergravity Duals,Phys.Lett.B431 (1998)42-48,hep-th/9803109.[34]C.Ahn,K.Oh and R.Tatar,Orbifolds AdS7×S4and Six Dimensional(0,1)SCFT,Phys.Lett.B442(1998)109-116,hep-th/9804093.[35]M.Cvetiˇc,H.L¨u and C.N.Pope,Gauged six-dimensional supergravity from massiveType IIA,hep-th/9906221.[36]S.Ferrara,A.Kehagias,H.Partouche and A.Zaﬀaroni,AdS6Interpretation of5dSuperconformal Field Theories,Phys.Lett.B431(1998)57-62,hep-th/9804006. 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金兹堡朗道理论

Ginzburg–Landau theoryFrom Wikipedia, the free encyclopediaIn physics, Ginzburg–Landau theory, named after Vitaly Lazarevich Ginzburg and Lev Landau, is a mathematical physical theory used to describe superconductivity. In its initial form, it was postulated as a phenomenological model which could describe type-I superconductors without examining their microscopic properties. Later, a version of Ginzburg–Landau theory was derived from the Bardeen-Cooper-Schrieffer microscopic theory by Lev Gor'kov, thus showing that it also appears in some limit of microscopic theory and giving microscopic interpretation of all its parameters.Contents•1Introduction•2Simple interpretation•3Coherence length and penetration depth•4Fluctuations in the Ginzburg–Landau model•5Classification of superconductors based on Ginzburg–Landau theory•6Landau–Ginzburg theories in string theory•7See also•8References•8.1PapersIntroduction[edit]Based on Landau's previously-established theory of second-order phase transitions, Ginzburg and Landau argued that the free energy, F, of a superconductor near the superconducting transition can be expressed in terms ofa complex order parameter field, ψ, which is nonzero below a phase transition into a superconducting state and isrelated to the density of the superconducting component, although no direct interpretation of this parameter was given in the original paper. Assuming smallness of |ψ| and smallness of its gradients, the free energy has the form ofa field theory.where F n is the free energy in the normal phase, α and β in the initial argument were treated as phenomenologicalparameters, m is an effective mass, e is the charge of an electron, A is the magnetic vector potential, and is the magnetic field. By minimizing the free energy with respect to variations in the order parameter and the vector potential, one arrives at the Ginzburg–Landau equationswhere j denotes the dissipation-less electric current density and Re the real part. The first equation — which bears some similarities to the time-independent Schrödinger equation, but is principally different due to a nonlinear term —determines the order parameter, ψ. The second equation then provides the superconducting current.Simple interpretation[edit]Consider a homogeneous superconductor where there is no superconducting current and the equation for ψ simplifies to:This equation has a trivial solution: ψ = 0. This corresponds to the normal state of the superconductor, that is for temperatures above the superconducting transition temperature, T>T c.Below the superconducting transition temperature, the above equation is expected to have a non-trivial solution (that is ψ ≠ 0). Under this assumption the equation above can be rearranged into:When the right hand side of this equation is positive, there is a nonzero solution for ψ (remember that the magnitude of a complex number can be positive or zero). This can be achieved by assuming the following temperature dependence of α: α(T) = α0 (T - T c) with α0/ β > 0:•Above the superconducting transition temperature, T > T c, the expression α(T) / β is positive and the right hand side of the equation above is negative. The magnitude of a complex number must be a non-negative number, so only ψ = 0 solves the Ginzburg–Landau equation.•Below the superconducting transition temperature, T < T c, the right hand side of the equation above is positive and there is a non-trivial solution for ψ. Furthermorethat is ψ approaches zero as T gets closer to T c from below. Such a behaviour is typical for a second order phase transition.In Ginzburg–Landau theory the electrons that contribute to superconductivity were proposed to forma superfluid.[1] In this interpretation, |ψ|2 indicates the fraction of electrons that have condensed into a superfluid.[1] Coherence length and penetration depth[edit]The Ginzburg–Landau equations predicted two new characteristic lengths in a superconductor which wastermed coherence length, ξ. For T > T c (normal phase), it is given bywhile for T < T c (superconducting phase), where it is more relevant, it is given byIt sets the exponential law according to which small perturbations of density of superconducting electrons recover their equilibrium value ψ0. Thus this theory characterized all superconductors by two length scales. The second one is the penetration depth, λ. It was previously introduced by the London brothers in their London theory. Expressed in terms of the parameters of Ginzburg-Landau model it iswhere ψ0 is the equilibrium value of the order parameter in the absence of an electromagnetic field. The penetration depth sets the exponential law according to which an external magnetic field decays inside the superconductor. The original idea on the parameter "k" belongs to Landau. The ratio κ = λ/ξ is presently known asthe Ginzburg–Landau parameter. It has been proposed by Landau that Type I superconductors are those with 0 < κ< 1/√2, and Type II superconductors those with κ> 1/√2.The exponential decay of the magnetic field is equivalent with the Higgs mechanism in high-energy physics. Fluctuations in the Ginzburg–Landau model[edit]Taking into account fluctuations. For Type II superconductors, the phase transition from the normal state is of second order, as demonstrated by Dasgupta and Halperin. While for Type I superconductors it is of first order as demonstrated by Halperin, Lubensky and Ma.Classification of superconductors based on Ginzburg–Landau theory[edit]In the original paper Ginzburg and Landau observed the existence of two types of superconductors depending on the energy of the interface between the normal and superconducting states.The Meissner state breaks down when the applied magnetic field is too large. Superconductors can be divided into two classes according to how this breakdown occurs. In Type I superconductors, superconductivity is abruptly destroyed when the strength of the applied field rises above a critical value H c. Depending on the geometry of the sample, one may obtain an intermediate state[2] consisting of a baroque pattern[3] of regions of normal material carrying a magnetic field mixed with regions of superconducting material containing no field. In Type II superconductors, raising the applied field past a critical value H c1 leads to a mixed state (also known as the vortex state) in which an increasing amount of magnetic flux penetrates the material, but there remains no resistance to the flow of electric current as long as the current is not too large. At a second critical field strength H c2, superconductivity is destroyed. The mixed state is actually caused by vortices in the electronic superfluid, sometimes called fluxons because the flux carried by these vortices is quantized. Most pure elemental superconductors, except niobium and carbon nanotubes, are Type I, while almost all impure and compound superconductors are Type II.The most important finding from Ginzburg–Landau theory was made by Alexei Abrikosov in 1957. He used Ginzburg–Landau theory to explain experiments on superconducting alloys and thin films. He found that in a type-II superconductor in a high magnetic field, the field penetrates in a triangular lattice of quantized tubes offlux vortices.[citation needed]Landau–Ginzburg theories in string theory[edit]In particle physics, any quantum field theory with a unique classical vacuum state and a potential energy witha degenerate critical point is called a Landau–Ginzburg theory. The generalization to N=(2,2) supersymmetric theories in 2 spacetime dimensions was proposed by Cumrun Vafa and Nicholas Warner in the November 1988 article Catastrophes and the Classification of Conformal Theories, in this generalization one imposes thatthe superpotential possess a degenerate critical point. The same month, together with Brian Greene they argued that these theories are related by a renormalization group flow to sigma models on Calabi–Yau manifolds in thepaper Calabi–Yau Manifolds and Renormalization Group Flows. In his 1993 paper Phases of N=2 theories intwo-dimensions, Edward Witten argued that Landau–Ginzburg theories and sigma models on Calabi–Yau manifolds are different phases of the same theory. A construction of such a duality was given by relating the Gromov-Witten theory of Calabi-Yau orbifolds to FJRW theory an analogous Landau-Ginzburg "FJRW" theory in The Witten Equation, Mirror Symmetry and Quantum Singularity Theory. Witten's sigma models were later used to describe the low energy dynamics of 4-dimensional gauge theories with monopoles as well as brane constructions. Gaiotto, Gukov & Seiberg (2013)See also[edit]•Domain wall (magnetism)•Flux pinning•Gross–Pitaevskii equation•Husimi Q representation•Landau theory•Magnetic domain•Magnetic flux quantum•Reaction–diffusion systems•Quantum vortex•Topological defectReferences[edit]1.^ Jump up to:a b Ginzburg VL (July 2004). "On superconductivity and superfluidity (what I have and havenot managed to do), as well as on the 'physical minimum' at the beginning of the 21 st century". Chemphyschem.5 (7): 930–945. doi:10.1002/cphc.200400182. PMID15298379.2.Jump up^ Lev D. Landau; Evgeny M. Lifschitz (1984). Electrodynamics of Continuous Media. Course ofTheoretical Physics8. Oxford: Butterworth-Heinemann. ISBN0-7506-2634-8.3.Jump up^ David J. E. Callaway (1990). "On the remarkable structure of the superconductingintermediate state". Nuclear Physics B344 (3): 627–645. Bibcode:1990NuPhB.344..627C.doi:10.1016/0550-3213(90)90672-Z.Papers[edit]•V.L. Ginzburg and L.D. Landau, Zh. Eksp. Teor. Fiz.20, 1064 (1950). English translation in: L. D. Landau, Collected papers (Oxford: Pergamon Press, 1965) p. 546• A.A. Abrikosov, Zh. Eksp. Teor. Fiz.32, 1442 (1957) (English translation: Sov. Phys. JETP5 1174 (1957)].) Abrikosov's original paper on vortex structure of Type-II superconductors derived as a solution of G–L equations for κ > 1/√2•L.P. Gor'kov, Sov. Phys. JETP36, 1364 (1959)• A.A. Abrikosov's 2003 Nobel lecture: pdf file or video•V.L. Ginzburg's 2003 Nobel Lecture: pdf file or video•Gaiotto, David; Gukov, Sergei; Seiberg, Nathan (2013), "Surface Defects and Resolvents" (PDF), Journal of High Energy Physics。

k-ε双方程湍流模型对制退机内流场计算的适用性分析

k-ε双方程湍流模型对制退机内流场计算的适用性分析张晓东;张培林;傅建平;王成;杨玉栋【摘要】A three-dimensional computation model with actual sizes was developed by choosing the recoil brake of a certain gun as the research object. Dynamic meshes and sliding meshes techniques were used to numerically calculate the three-dimensional flow field inside the recoil brake at the actual recoiling velocity of the gun. The pressures of the different chambers in the recoil brake were calculated by using the standard k-ε model, the RNG k-ε model and t he realizable k-ε model, respectively. And the calculated pressures were compared with the experimental results. Comparisons show that the pressures calculated by the standard k-ε model are in the best agreement with the experimental results.%为研究不同双方程湍流模型对制退机内复杂流场计算的适用性,以某火炮制堪机为研究对象,建立了实际结构下的三维计算模型,利用动网格与滑移网格技术.实现了火炮实际后坐速度下的制退机内部三维运动流场的数值计算.分别采用标准k-ε模型、RNGk-ε模型和Realizablek-ε模型计算制退机内部各腔室压力,与实验曲线对比,结果表明,应用标准k-ε模型对后坐冲击过程的制退机内部压力计算的误差最小,与实验结果吻合最好.【期刊名称】《爆炸与冲击》【年(卷),期】2011(031)005【总页数】5页(P516-520)【关键词】流体力学;流场;湍流模型;制退机【作者】张晓东;张培林;傅建平;王成;杨玉栋【作者单位】军械工程学院一系,河北石家庄050003;军械工程学院一系,河北石家庄050003;军械工程学院一系,河北石家庄050003;军械工程学院一系,河北石家庄050003;军械工程学院一系,河北石家庄050003【正文语种】中文【中图分类】O357.5火炮会在射击瞬间产生巨大的后坐冲击力，制退机是消耗后坐能量、控制平稳后坐的关键部件。

apm闪存日志分析

闪存日志目录[显示]日志类型(闪存VS 数传日志)有两种方法可以记录你飞行时的数据。

尽管两种记录方法十分类似，但也有些区别：∙闪存日志(本页的主题)飞行完成后，可以从APM或PX4的板载闪存上下载。

飞机和地面车辆，只要一开机就会自动创建闪存日志。

四轴上则是解锁后才创建闪存日志。

∙数传日志(也称为―tlogs‖)用3DR或XBee数传模块连接APM到电脑上，会被Mission Planner(或其他地面站)记录下来。

你可以在这里找到详细信息。

设置你想要记录的数据LOG_BITMASK参数控制闪存内记录什么类型的数据。

如果你想指定某个消息类型附加到默认的方法上，在Mission Planner标准参数列表页面，可以在Log Bitmask的下拉列表设置。

使用终端控制会更加方便（命令行界面），如下所示：∙进入Mission Planner的终端界面∙单击―连接APM‖或―连接PX‖∙键入logs∙启用或禁用某个消息类型，在enable或disable后输入信息的类型即可。

如：enable IMU用于Arducopter的所有可能的闪存信息：ATT：roll，pitch和yaw（启用ATTITUDE_FAST记录频率是50Hz，启用ATTITUDE_MED记录频率是10HZ）。

ATUN：自动调参概览（从开头记录每一次―晃动‖测试）ATDE：自动调参详情（以100HZ记录飞行器的晃动情况）CAMERA（相机）：快门按下后，记录当时的GPS时间、roll、pitch、yaw、纬度、精度、高度。

CMD（命令）：从地面站接收命令，或者作为执行任务之一。

COMPASS（罗盘）：罗盘原始数据和compassmot补偿值。

CURRENT（电流）：以10HZ的频率，记录电流和主板电压信息。

CUTN：油门和高度信息，包括油门输入大小、超声波测得的高度（sonar alt）、气压测得的高度（baro alt），以10HZ频率记录。

RNG+k-ε模型数值模拟油雾燃烧流场

收稿日期:2004-10-24;修订日期:2005-01-29作者简介:郭尚群(1978-) 女贵州人南京航空航天大学能源与动力学院硕士生主要从事航空发动机燃烧方面的研究.第20卷第5期2005年10月航空动力学报Journal of Aerospace PowerVol.20No.5:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ct.2005文章编号:1000-8055(2005)05-0807-06RNG k -E 模型数值模拟油雾燃烧流场郭尚群赵坚行(南京航空航天大学能源与动力学院江苏南京210016)摘要:采用RNG (Renormalization Group )k -E 紊流模型对某环形燃烧室火焰筒内三维两相燃烧流场进行数值模拟0运用偏微分方程法和区域法生成贴体网格 EBU -Arrhenius 紊流燃烧模型估算化学反应速率六通量热辐射模型估算辐射通量0在非交错网格下求解气相采用SIMPLE 算法液相采用颗粒轨道模型和PSIC 算法0近壁区处理分别采用两层和三层壁面函数0计算数据与试验值比较表明采用三层壁面函数的RNG k -E 模型更适用于模拟三维两相燃烧流场0关键词:航空,航天推进系统;RNG k -E 模型;三维两相燃烧;数值模拟;贴体网格;三层壁面函数中图分类号:V 231.2文献标识码:ANumerical Simulation of Three -DimensionalSpray combustion Flow Field by RNG k -E Turbulent modelGU Shang -gun ZHA Jian -xing(College o f Energ y an d Po w er EngineeringNan j ing Uni V ersit y o f Aeronautics an d Astronautics Nan j ing 210016 China )Abstract :RNG (Renormalization Group )k -E tur b ulent mo d el w as applie d to the numerical simulation o f the three -d imensional t w o -phase com b ustion f lo w f iel d in an annular com b ustor .A b o dy -f itte d gri d w as generate d by an elliptical gri d generation proce d ure an d the zone metho d .EBU -Arrhenius com b ustion mo d el w as use d to d etermine the rate o f reaction an d six -f lux ra d iation mo d el w as use d to pre d ict the d istri b ution o f the heat f lux .SIMPLE algorithm w ith the 3-D non -staggere d gri d s y stem w as emplo y e d to sol V e the go V erning eguations .T he ligui d phase eguations w ere sol V e d w ith the particle tra j ector y mo d el an d the PSIC algorithm .T he three -la y er an d t w o -la y er w all f unction w ere applie d to pre d iction o f f lo w in the near w all region .Computations are in f airl y goo d agreement w ith experiments .T he comparison in d icates that RNG k -E tur b ulent mo d el w ith three -la y er w all f unction is more relia b le f or mo d eling 3-D spra y com b ustion f lo w f iel d .K ey words :aerospace propulsion s y stem ;RNG k -E tur b ulence mo d el ;three -d imensional t w o -phase com b ustion f lo w f iel d ;numerical simulation ;b o dy -f itte d gri d ;three -la y er w all f unction多年来k-E紊流模型以其形式简单~使用方便等优点被广泛应用于科学和工程领域中的紊流问题O但是许多计算值与实验数据比较表明k-E模型适用于射流~管流~自由剪切流以及弱旋流等简单的紊流流动而不太适用于强旋流~回流及曲壁边界层等复杂紊流流动[1]O其原因:一是它的模型系数是从简单紊流流动中得到的对于一些复杂紊流流动不太适合;二是该模型是根据Boussinesg的各向同性涡旋粘性假设建立的实际上紊流粘性不是一种流体性质而是随流动变化是各向异性的O为了扩大k-E模型的使用范围许多学者提出了各种改进形式O其中Yakhot和Orszag[2]在紊流问题中引入重整化群理论(Renormalization Group缩写RNG)将非稳态Navier-Stokes方程对一个平衡态作Gauss统计展开通过频谱分析消去其中的小尺度涡并将其影响归并到涡粘性中从而改善了对耗散率E的模拟O该模型在形式上与标准k-E模型完全一样不同之处在于E方程中5个模型系数的取值O由于RNG k-E模型考虑了非平衡流对紊流的影响改进了对复杂紊流问题的预测效果因而受到越来越广泛的重视O1数学模型1.1RNG k-6紊流模型RNG k-E模型是基于重整化群理论把紊流视为受随机力驱动的输运过程通过频谱分析消去其中的小尺度涡并将其影响归并到涡粘性中从而得到所需尺度上的输运方程O模型中各模型系数是利用RNG理论推导出来的具有一定的通用性O在高雷诺数时RNG模型的紊流动能k及其耗散率E的输运方程形式为:H(0~z k) H x z =G k-0E+HH x zu T6kH kH x z(1)H(0~z E) H x z =C E1EkG k-C E10E2k+HH x zu T6EH EH x z(2)式中紊流涡旋粘性系数uT:u T=C u0E2 k紊流动能产生项Gk:G k=2u T S zj S zj时均应变率Szj:S zj=H~z/H x j+H~j/H~z25个模型系数为[3]:C u=0.085C E1=1.42-7(1-7/70)/(1+B73)C E2=1.686k=0.71796E=0.7179其中:7=Sk/E S=(2S z j S z j)1/270=4.38B=0.015耗散率E方程源项系数CE1中引入了一个附加产生项该项主要是考虑流动中的不平衡应变率能及时反映主流流动情况对具有大应变率的流动~具有强曲率影响和壁面约束的紊流分离流动都起着重要的作用[4]O由于该模型在一定程度上考虑了紊流的各向异性效应间接改善了对E模拟增强了对较复杂紊流(旋流等)流动的预测能力研究表明同时采用RNG k-E模型与3层壁面函数可得到更为合理的计算结果[3]O1.2紊流燃烧模型~两相燃烧模型~辐射模型(1)为了同时考虑紊流和化学动力因素对化学反应速率的影响本文采用EBU-Arrhenius紊流燃烧模型O按该模型化学反应速率可由下式表示:R fu=-min(R fu1R fu2)(3)其中:R fu1=-C R g1/20E/kR fu2=-A002m fu m ox exp(-E/RT)(2)本文采用颗粒群轨道模型求解油珠运动轨迹以及其沿轨道的质量~速度和温度的变化历程O对气液两相之间的耦合采用SIC法O在圆柱坐标系下油滴运动方程为[7]:d~pdt=-1/r d(~p-~g)d z pdt=-1/r d(z p-z g)+z2p/v pd z pdt=-1/r d(z p-z g)-z p z p/v\p(4)式中:~p z p z p和~g z g z g分别为油滴和气相速度分量rd为颗粒驰豫时间O对于蒸发的油滴其直径随时间变化的速率由下列蒸发方程求得:d D p/dt=-C D/2D p(1+0.23R e0.5)(5)808航空动力学报第20卷其中:CD=8/g ln 1+C pg(T g-T c)/Lp p C pg为蒸发率常数,pp,T c,L 分别为液相密度~沸点温度和汽化潜热,Dp 为油滴直径,Re为油滴相对雷诺数,Cpg为气相比热G油滴达到沸点之前被周围高温气体加热,其温度随时间变化率为:dT p/dz=6/g(2+O.6Re1/2P11/3)(T g-T p)/(p p D2p C pp)(6)其中:油滴比热为Cpp=84O.5+4.137T p,T p为油滴温度G(3)本文采用六通量法热辐射模型[1]来估算辐射通量:d dI1c+SdR IdI=c(R I-E Z)+S3(2R I-R1-R)1 1dd11c+S+1/1dR1d1=c(R I-E Z)+S3(2R1-R I-R)1 1dd11c+SdR1d=c(R I-E Z)+S3(2R-R I-R1>W,)(7)其中:RI,R1和R分别为I,1和方向净辐射通量,c为吸收系数,由下列经验公式确定:c=O.2m fu+O.1m pr式中:S为散射系数,本文取O.O1G2曲线坐标系下控制方程2.1气相控制方程设(I,1,)为圆柱坐标,(E,y,C)为任意曲线坐标,两组坐标的雅可比行列式J=8(I,1,)/8 (E,y,C),经坐标变换后,在任意曲线坐标系(E,y, C)下气相控制方程的通用形式为[1]:8 8E (pU )+88y(pU )+88C(pW )= 88E1J (g11E+g12y+g13C[])+88y1J (g21E+g22y+g23C[])+88C1J (g31E+g32y+g33C[])+1JS+S c (8)变量分别为速度z,U和z,紊流动能k和紊流动能耗散率E,焓h,混合分数f,燃油浓度mfu,燃油浓度脉动均方值g,辐射通量RI,R1和R,为各变量的输运系数,S为气相场自身源项,Sc为油滴蒸发产生的源项G U,V和W为任意曲线坐标系(E,y,C)下的速度,gzj为协变度量张量,其具体含义详见文献[1,6],控制方程中变量k,E,RI,R1和R方程的源项分别为:k方程的源项1JS k为:1(G k-pE)E方程的源项1JS E为:1E(C E1G k-C E2pE)/kR I方程的源项1JS I为:-1c(R1-E Z)+(2R1-R I-R)/3R1方程的源项1JS1为:-1c(R1-E Z)+(2R1-R I-R)/3R方程的源项1JS为:-1c(R-E Z)+(2R-R I-R1)/3其他变量方程的源项见文献[6],k方程的产生项Gk为:G k=e/J2{2[(A11z E+A12z y+A13z C)2+(A21U E+A22U y+A23U C)2+(A31z E+A32z y+A33z C+JU)2/12]+(A21z E+A22z y+A23z C+A11U E+A12U y+A13U C)2+[A11z E+A12z y+A13z C+(A31z E+A32z y+A33z C)/1]2+[(A31U E+A32U y+A33U C)/1+A21z E+A22z y+A23z C-Jz/1]2 2.2液相基本方程在曲线坐标系(E,y,C)下,油滴在各方向上的运动速度为:U c=dEdz,V c=dydz,W c=dCdz(9)经过坐标变换,在曲线坐标系下油滴运动方程可写为:9O8第5期郭尚群等:RNG k-E模型数值模拟油雾燃烧流场dU c/dt=-(U c-U g)/Z c1w2c/1c-G z c w c/11cdV c/dt=-(U c-U g)/Z c71w2c/1c-7G z c w c/11cdW c/dt=-(W c-W g)/Z c Z1w2c/1c-Z G z c w c/11c(10)运用4阶Runge-Kutta方程求解方程(10)得到油滴速度U V和W按式(9)可求得计算区域内油滴运动轨迹再由逆变换确定油滴在物理平面上的位置(1G),3边界条件处理进口边界条件2本文进口流量分配是给定的主流速度由流量\进口面积和气流密度确定9主燃孔\掺混孔及气膜冷却孔的进气速度均按要求给定9紊流动能及其耗散率均是按经验公式给定2in=0.03u2in E in=3/2in 0.005其中2为计算区域的特征尺寸,出口边界条件2令各变量法向梯度为零9利用流量连续对速度和压力进行修正,周期边界条件2由于圆周边界条件周期性重复出现所以令周期位置上变量值相即2G1=G n G n 1=G24壁面函数处理由于近壁区速度梯度较大为了避免由于采用细网格分布而增加计算工作量的问题通常采用壁面函数作为固壁的边界条件,目前大都采用2层壁面函数来计算近壁区域流动,按照壁面函数方法壁面的切应力可以写为2Z w=UV tn11.63Z w=V t1/2C1/2U Xln(E )11.63(11)式中2为靠近壁面的节点n为点到壁面的垂直距离Vt为切向速度分量,=1/2C1/2U n/UX=0.4187E=9.793但Amano[5]认为采用3层壁面函数比2层壁面函数更为合理,所谓3层壁面函数其基本原理是把近壁第一个内节点与壁面之间的距离分为粘性底层和缓冲层2部分规定粘性底层的范围是0<<5 缓冲层为5<<30,对每一个区域内的E和Z的分布均给出假设于是可积分求出控制容积中和E方程的产生项和耗散项并用这些数据来求解控制容积的和E方程[4],u=5-3.055ln5<305.5 2.5ln30(12)其中2u=u/uZ u Z是近壁面处由于壁面剪切应力Z w作用所产生的摩擦速度[3],P点的紊流动能和紊流动能耗散率为2=Z wC1/2UE=C3/4U3/2K n本文在固体壁面上给定速度\紊流参数\组分等为0 在近壁区域处理分别采用3层壁面函数和两层壁面函数,通过流场计算研究不同壁面函数对近壁区流动的影响,5计算结果与分析本文研究对象为环形燃烧室火焰筒内部流场该火焰筒为环形周向均布28个头部每个头部装有一个涡流器涡流器通过转接段与后端火焰筒相连接,火焰筒内外环上分别开有一排主燃孔和两排掺混孔火焰筒内外环壁面上分别开有17和15排气膜冷却孔,考虑燃烧室结构周向变化具有周期性因此本文只研究一个头部的扇形区域它是由一个头部\转接段和火焰筒体3部分构成,为了保证网格分布的合理性网格生成采用采用偏微分方程法和区域法生成三维贴体网格[1]即分别生成火焰筒头部和火焰筒体网格然图1计算区域三维贴体网格Fig.1Three-dimensional body-fitted grid ofcalculated domain018航空动力学报第20卷后将其合并为一个整体计算区域网格(见图1)总的网格数为Z O7>4Z>6Z其两相燃烧流场计算的部分结果如图Z~图9所示其中图Z为通过旋流器中心轴线的纵剖面(k=3O)上的速度分布图由图可知在火焰筒头部有明显的回流区它起始于头部进气段结束于主燃孔位置附近另外还可看出火焰筒后半段上下壁面掺混孔进气后流场情况图3为通过主燃孔和掺混孔的纵剖面(k=3Z)的温度分布图由图可知主燃孔进气为主燃区提供了氧气因此主燃区气流温度最高;掺混孔进气主要与高温燃气掺混降低燃气温度使得出口温度分布符合设计要求图Z燃烧室k=3O截面速度矢量分布图Fig.Z Velocity vector distribution at k=3O section图3燃烧室k=5Z截面温度分布图Fig.3Temperature distribution at k=5Z section图4燃烧室k=31截面燃油分布图Fig.4Fuel distribution at k=31section图4为通过主燃孔和掺混孔的纵剖面(k=31)的燃油浓度分布由图可知在火焰筒主燃区存在未烧完的燃油但随着轴向距离增加燃烧时间延长燃油逐渐烧完图5和图6分别为轴向位置通过掺混孔(1=1Z8)时横剖面速度和温度分布由图5可知从掺混孔进入射流大部分顺流而下与高温燃气掺混而少部分在柱状射流后面形成旋涡图5燃烧室1=1Z8截面速度矢量分布图Fig.5Velocity profiles at1=1Z8section图6燃烧室1=1Z8截面温度分布图Fig.6Temperature profiles at1=1Z8section图7和图8分别出口截面上速度和温度周向不均匀系数TPF[8]径向分布情况可以看出不同壁面函数对出口温度周向不均匀系数TPF径向分布的影响要比对出口速度径向分布影响要大些计算结果与实验数据比较表明:3层壁面函数得到的温度TPF径向分布要比两层壁面函数更为合理Z层与和3层壁面函数对外环壁面温度轴向118第5期郭尚群等:RNG k-模型数值模拟油雾燃烧流场图7出口速度径向分布Fig.7Velocity radial profiles at rhe exit图8出口温度TPF系数TPF分布Fig.8Temperature radial profiles at the exit图9外环壁面温度轴向分布图Fig.9Axial profiles of outer annular Wall temperature in the flame tube 分布的影响如图9所示由图可明显看出由于3层壁面函数加强了RNG k E模型对近壁面上紊流流动的处理使所得的壁面温度分布沿轴向幅值变化均匀6结论本文在三维非交错网格贴体坐标系统下采用RNG k E模型计算环形燃烧室两相反应流流场同时数值分析Z层和3层壁面函数对近壁流动的影响计算结果表明RNG k E紊流模型对复杂区域紊流流动的预测效果更为合理提高了数值模拟的精度由于3层壁面函数加强对近壁面流体流动的模拟使计算值得到进一步改进本计算方法预测的结果可为燃烧室优化设计和研制提供了有用的参考依据参考文献:[l]赵坚行.燃烧的数值模拟[M].北京:科学出版社Z OOZ. [Z]Yakhot V Orzag s A.Renormalization Group Analysis of Turbulence:Basic Theory[J]put.l986 l:3~5.[3]speziale C G Thangam s.Analysis of an RNG BasedTurbulence Model for separated FloWs[J].Int.J.Engng.Sci.l99Z3O(l O):l379~l388.[4]陈庆光徐忠张永建.RNG k E模式在工程紊流数值计算中的应用[J].力学季刊Z OO3 Z4(l):88~95.Chen G G Xu Z Zhang Y J.Application of RNG k EModels in Numerical simulations of engineering TurbulentFloWs[J].Chinese@uc1te1l}Mechcncs Z OO3 Z4(l):88~95.[5]Amano R s.Development of a Turbulent Near wall Modeland its Application to separated and Reattached FloWs[J].Nume1. ect1cns e1l984 7:59~75.[6]雷雨冰.数值模拟环形燃烧室整体流场[D].江苏南京:南京航空航天大学Z OOO.[7]雷雨冰.数值模拟环形燃烧室两相反应流场[J].燃烧科学与技术Z OOO 6(3):Z Z Z~Z Z5.L ei Y B Tan H P Zhao J X.Numerical simulation ofThree Dimensional spray Combustion in a Gas TurbineCombustor[J].Jou1ncl o Com乙ustion Science cn dechnolog}Z OOO 6(3):Z Z Z~Z Z5.[8]金如山.航空燃气轮机燃烧室[M].北京:宇航出版社l985.Z l8航空动力学报第Z O一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一卷RNG k-ε模型数值模拟油雾燃烧流场作者：郭尚群，赵坚行， GUO Shang-qun， ZHAO Jian-xing作者单位：南京航空航天大学,能源与动力学院,江苏,南京,210016刊名：航空动力学报英文刊名：JOURNAL OF AEROSPACE POWER年，卷(期)：2005,20(5)被引用次数：7次参考文献(8条)1.赵坚行燃烧的数值模拟 20022.Yakhot V;Orzag S A Renormalization Group Analysis of Turbulence:Basic Theory 19863.Speziale C G;Thangam S Analysis of an RNG Based Turbulence Model for Separated Flows[外文期刊] 1992(10)4.陈庆光;徐忠;张永建RNG k-ε模式在工程紊流数值计算中的应用[期刊论文]-力学季刊 2003(01)5.Amano R S Development of a Turbulent Near Wall Model and its Application to Separated and Reattached Flows 19846.雷雨冰数值模拟环形燃烧室整体流场 20007.雷雨冰数值模拟环形燃烧室两相反应流场[期刊论文]-燃烧科学与技术 2000(03)8.金如山航空燃气轮机燃烧室 1985本文读者也读过(5条)1.马贵阳.解茂昭用RNG k-ε模型计算内燃机缸内湍流流动[期刊论文]-燃烧科学与技术2002,8(2)2.徐英.郑建生.杨会峰.吴经纬.李刚.李巧真.XU Ying.ZHENG Jian-sheng.YANG Hui-feng.WU Jing-wei. LI Gang.LI Qiao-zhen基于RNG k-ε模型的内锥流量计气体流出系数预测[期刊论文]-天津大学学报2007,40(10)3.党新宪.赵坚行.吉洪湖.DANG Xin-xian.ZHAO Jian-xing.JI Hong-hu双旋流器燃烧室NOx生成研究[期刊论文]-航空动力学报2008,23(3)4.颜应文.赵坚行.张靖周.刘勇.YAN Ying-wen.ZHAO Jian-xing.ZHANG Jing-zhou.LIU Yong环形燃烧室两相喷雾燃烧的大涡模拟[期刊论文]-航空动力学报2006,21(5)5.胡好生.蔡文祥.赵坚行.吉洪湖.HU Hao-sheng.CAI Wen-xiang.ZHAO Jian-xing.JI Honghu回流燃烧室燃烧过程的三维数值模拟[期刊论文]-航空动力学报2008,23(3)引证文献(7条)1.蔡文祥.赵坚行.胡好生.武晓松数值研究环形回流燃烧室紊流燃烧流场[期刊论文]-航空动力学报2010(5)2.马朝臣.霍学敏.张强一种拓宽微型单管燃烧室流量范围的调节装置[期刊论文]-北京理工大学学报2011(6)3.蔡文祥.赵坚行.胡好生.武晓松环形燃烧室冷态流场数值模拟中的数学方法[期刊论文]-航空动力学报2010(4)4.刘红.解茂昭.王德庆机械搅拌流场中制备闭孔泡沫铝过程的数值模拟[期刊论文]-过程工程学报 2007(1)5.蔡文祥.赵坚行.胡好生.党新宪.武晓松燃烧室贫油熄火极限数值预测[期刊论文]-航空动力学报 2010(7)6.胡好生.蔡文祥.赵坚行.吉洪湖回流燃烧室燃烧过程的三维数值模拟[期刊论文]-航空动力学报 2008(3)7.陈翼.张林进.叶旭初射流混合器内气体湍流扩散过程的CFD数值模拟与实验研究[期刊论文]-过程工程学报 2007(5)本文链接：/Periodical_hkdlxb200505018.aspx。

量子相变中的量子纠缠度量研究-英文版

RG flow
• c-theorem:
cUV cIR
fixed point
(Zamolodchikov, 1986)
UV
fixed point
IR
• Massive theory (off critical) Block entropy saturation
fixed point
RG flow
UV
Irreversibility of RG trajectories
Loss of entanglement
Open Systems & Quantum Information Milano, 10 Marzo 2006
• Local Entropy: when the subsystem A is a single site.
• The concurrence measures the entanglement between two sites after having traced out the remaining sites. • The transition is signaled by the first derivative of the concurrence, which diverges logarithmically (specific heat).

h 1 lnh 1
S1 has the same singularity as
Local (single site) entropy: 1 ρ1 = I + σ z σ z S1 = Tr ρ1lnρ1 2
σz

Local measures of entanglement based on the 2-site density matrix depend on 2-point functions

船舶压载水紫外处理器的CFD数值模拟

船舶压载水紫外处理器的CFD数值模拟付小敏; 刘护平【期刊名称】《《机电设备》》【年(卷),期】2019(036)005【总页数】4页(P76-79)【关键词】船舶压载水; 紫外处理器; CFD【作者】付小敏; 刘护平【作者单位】中国船舶工业集团公司第六三五四研究所江西九江 332000【正文语种】中文【中图分类】U664.50 引言在对船舶压载水[1]处理系统进行设计时，需获得关键部件—紫外处理器流场和光场的准确数据，目前有2种方法：试验和仿真。

因紫外处理器尺寸大、内部结构复杂，进行一次样机试验[1]，须耗费大量的人力、物力，且试验周期在2～3个月，出于成本和项目时效性的考虑，试验法不可取。

随着流体仿真软件和计算机硬件的飞速发展，大规模仿真计算的速度和准确性得到大幅度提高，在许多工程领域得到了成功的应用。

本文采用计算流体动力学（Computational Fluid Dynamics，CFD）仿真软件对紫外处理器进行流场和光场的数值模拟，为反应器的设计提供参考依据。

与试验法相比，仿真计算具有成本低、周期短、可操作性强、易实现等巨大优势。

1 初始条件1.1 设计参数紫外处理器的初始设计参数为：1）流量500 m³/h；2）筒体直径DN500；3）筒体长度1 395 mm；4）进出口管径DN250；5）介质的透光率≥65%（@253.7 nm）；6）灯管参数：a）灯管功率800 W，253.7 nm最大输出功率240 W；b）灯长1 540 mm，弧长1 395 mm；c）外层石英套管直径dq=42 mm，紫外透过率90%。

1.2 技术指标在上述设计参数下，要求达到以下技术指标：1）紫外剂量不低于30m J/cm2；2）筒体内的平均光照强度≥20 mW/cm2。

在上述初始条件下，通过CFD仿真软件，对紫外处理器的流场、光场进行数值模拟。

2 流场的数值模拟2.1 控制方程与湍流模型自从上个世纪八十年代以来，随着高性能计算机的飞速发展，目前的CFD计算能力与实用方面都发生了深刻的变化，开始在实践中发挥着重要作用，应用领域不断扩大，可以解决的问题与日俱增。

汽车空调出风口啸叫的辨识及机理探究

汽车空调出风口啸叫的辨识及机理探究沈沉;王洋;刘斌;陶泽平【摘要】通过近场测量法获得实车空调出风口啸叫的声压数据,并分析其频谱,又通过耦合气动-声学数值计算研究流场结构并探究啸叫机理.试验结果表明,啸叫的发生与空调出风口的风门位置和内外压差联系密切.当幅值-频域信号中2 kHz以上部分存在明显的频谱峰值时,啸叫明显.啸叫产生的机理在于拟序涡结构脱落引发的有规律的压力脉动,辐射后形成较为规则的声压脉动.为预测啸叫、改进设计、预防啸叫提供理论依据.【期刊名称】《汽车工程学报》【年(卷),期】2015(005)003【总页数】6页(P229-234)【关键词】NVH;噪声辨识;数值模拟;气动噪声;啸叫【作者】沈沉;王洋;刘斌;陶泽平【作者单位】泛亚汽车技术中心,上海201201;泛亚汽车技术中心,上海201201;泛亚汽车技术中心,上海201201;泛亚汽车技术中心,上海201201【正文语种】中文【中图分类】U463.85+1乘用车噪声、振动与声振粗糙度（Noise、Vibration、Harshness，NVH）性能直接影响到乘坐舒适性，空调出风口啸叫是一种常见的NVH问题，会引起乘客不适与疲劳[1]，越来越多的主机厂和供应商开始关心并设法解决该问题。

汽车空调出风口啸叫是一种主观感受，定义并辨识啸叫在出风口研发和认证过程中十分重要。

相关研究已经表明：车内舒适性与响度、尖锐度、粗糙度、波动度等因素相关[2]，而出风口啸叫对车内噪声的响度、尖锐度、粗糙度都有明显的影响，所以研究并改善出风口啸叫具有重要的工程意义。

目前对于出风口啸叫的评估主要采取台架试验和整车试验两种方式，而对啸叫产生的机理缺少有效的分析方法。

啸叫的成因较为复杂，其影响因素包括风机、风门、风道、叶片等。

运用计算流体动力学（Computational Fluid Dynamics，CFD）分析啸叫形成机理对识别影响因素、改进设计、预防啸叫具有指导意义。

金兹堡朗道理论

Ginzburg–Landau theoryFrom Wikipedia, the free encyclopediaIn physics, Ginzburg–Landau theory, named after Vitaly Lazarevich Ginzburg and Lev Landau, is a mathematical physical theory used to describe superconductivity. In its initial form, it was postulated as a phenomenological model which could describe type-I superconductors without examining their microscopic properties. Later, a version of Ginzburg–Landau theory was derived from the Bardeen-Cooper-Schrieffer microscopic theory by Lev Gor'kov, thus showing that it also appears in some limit of microscopic theory and giving microscopic interpretation of all its parameters.Contents•1Introduction•2Simple interpretation•3Coherence length and penetration depth•4Fluctuations in the Ginzburg–Landau model•5Classification of superconductors based on Ginzburg–Landau theory•6Landau–Ginzburg theories in string theory•7See also•8References•8.1PapersIntroduction[edit]Based on Landau's previously-established theory of second-order phase transitions, Ginzburg and Landau argued that the free energy, F, of a superconductor near the superconducting transition can be expressed in terms ofa complex order parameter field, ψ, which is nonzero below a phase transition into a superconducting state and isrelated to the density of the superconducting component, although no direct interpretation of this parameter was given in the original paper. Assuming smallness of |ψ| and smallness of its gradients, the free energy has the form ofa field theory.where F n is the free energy in the normal phase, α and β in the initial argument were treated as phenomenologicalparameters, m is an effective mass, e is the charge of an electron, A is the magnetic vector potential, and is the magnetic field. By minimizing the free energy with respect to variations in the order parameter and the vector potential, one arrives at the Ginzburg–Landau equationswhere j denotes the dissipation-less electric current density and Re the real part. The first equation — which bears some similarities to the time-independent Schrödinger equation, but is principally different due to a nonlinear term —determines the order parameter, ψ. The second equation then provides the superconducting current.Simple interpretation[edit]Consider a homogeneous superconductor where there is no superconducting current and the equation for ψ simplifies to:This equation has a trivial solution: ψ = 0. This corresponds to the normal state of the superconductor, that is for temperatures above the superconducting transition temperature, T>T c.Below the superconducting transition temperature, the above equation is expected to have a non-trivial solution (that is ψ ≠ 0). Under this assumption the equation above can be rearranged into:When the right hand side of this equation is positive, there is a nonzero solution for ψ (remember that the magnitude of a complex number can be positive or zero). This can be achieved by assuming the following temperature dependence of α: α(T) = α0 (T - T c) with α0/ β > 0:•Above the superconducting transition temperature, T > T c, the expression α(T) / β is positive and the right hand side of the equation above is negative. The magnitude of a complex number must be a non-negative number, so only ψ = 0 solves the Ginzburg–Landau equation.•Below the superconducting transition temperature, T < T c, the right hand side of the equation above is positive and there is a non-trivial solution for ψ. Furthermorethat is ψ approaches zero as T gets closer to T c from below. Such a behaviour is typical for a second order phase transition.In Ginzburg–Landau theory the electrons that contribute to superconductivity were proposed to forma superfluid.[1] In this interpretation, |ψ|2 indicates the fraction of electrons that have condensed into a superfluid.[1] Coherence length and penetration depth[edit]The Ginzburg–Landau equations predicted two new characteristic lengths in a superconductor which wastermed coherence length, ξ. For T > T c (normal phase), it is given bywhile for T < T c (superconducting phase), where it is more relevant, it is given byIt sets the exponential law according to which small perturbations of density of superconducting electrons recover their equilibrium value ψ0. Thus this theory characterized all superconductors by two length scales. The second one is the penetration depth, λ. It was previously introduced by the London brothers in their London theory. Expressed in terms of the parameters of Ginzburg-Landau model it iswhere ψ0 is the equilibrium value of the order parameter in the absence of an electromagnetic field. The penetration depth sets the exponential law according to which an external magnetic field decays inside the superconductor. The original idea on the parameter "k" belongs to Landau. The ratio κ = λ/ξ is presently known asthe Ginzburg–Landau parameter. It has been proposed by Landau that Type I superconductors are those with 0 < κ< 1/√2, and Type II superconductors those with κ> 1/√2.The exponential decay of the magnetic field is equivalent with the Higgs mechanism in high-energy physics. Fluctuations in the Ginzburg–Landau model[edit]Taking into account fluctuations. For Type II superconductors, the phase transition from the normal state is of second order, as demonstrated by Dasgupta and Halperin. While for Type I superconductors it is of first order as demonstrated by Halperin, Lubensky and Ma.Classification of superconductors based on Ginzburg–Landau theory[edit]In the original paper Ginzburg and Landau observed the existence of two types of superconductors depending on the energy of the interface between the normal and superconducting states.The Meissner state breaks down when the applied magnetic field is too large. Superconductors can be divided into two classes according to how this breakdown occurs. In Type I superconductors, superconductivity is abruptly destroyed when the strength of the applied field rises above a critical value H c. Depending on the geometry of the sample, one may obtain an intermediate state[2] consisting of a baroque pattern[3] of regions of normal material carrying a magnetic field mixed with regions of superconducting material containing no field. In Type II superconductors, raising the applied field past a critical value H c1 leads to a mixed state (also known as the vortex state) in which an increasing amount of magnetic flux penetrates the material, but there remains no resistance to the flow of electric current as long as the current is not too large. At a second critical field strength H c2, superconductivity is destroyed. The mixed state is actually caused by vortices in the electronic superfluid, sometimes called fluxons because the flux carried by these vortices is quantized. Most pure elemental superconductors, except niobium and carbon nanotubes, are Type I, while almost all impure and compound superconductors are Type II.The most important finding from Ginzburg–Landau theory was made by Alexei Abrikosov in 1957. He used Ginzburg–Landau theory to explain experiments on superconducting alloys and thin films. He found that in a type-II superconductor in a high magnetic field, the field penetrates in a triangular lattice of quantized tubes offlux vortices.[citation needed]Landau–Ginzburg theories in string theory[edit]In particle physics, any quantum field theory with a unique classical vacuum state and a potential energy witha degenerate critical point is called a Landau–Ginzburg theory. The generalization to N=(2,2) supersymmetric theories in 2 spacetime dimensions was proposed by Cumrun Vafa and Nicholas Warner in the November 1988 article Catastrophes and the Classification of Conformal Theories, in this generalization one imposes thatthe superpotential possess a degenerate critical point. The same month, together with Brian Greene they argued that these theories are related by a renormalization group flow to sigma models on Calabi–Yau manifolds in thepaper Calabi–Yau Manifolds and Renormalization Group Flows. In his 1993 paper Phases of N=2 theories intwo-dimensions, Edward Witten argued that Landau–Ginzburg theories and sigma models on Calabi–Yau manifolds are different phases of the same theory. A construction of such a duality was given by relating the Gromov-Witten theory of Calabi-Yau orbifolds to FJRW theory an analogous Landau-Ginzburg "FJRW" theory in The Witten Equation, Mirror Symmetry and Quantum Singularity Theory. Witten's sigma models were later used to describe the low energy dynamics of 4-dimensional gauge theories with monopoles as well as brane constructions. Gaiotto, Gukov & Seiberg (2013)See also[edit]•Domain wall (magnetism)•Flux pinning•Gross–Pitaevskii equation•Husimi Q representation•Landau theory•Magnetic domain•Magnetic flux quantum•Reaction–diffusion systems•Quantum vortex•Topological defectReferences[edit]1.^ Jump up to:a b Ginzburg VL (July 2004). "On superconductivity and superfluidity (what I have and havenot managed to do), as well as on the 'physical minimum' at the beginning of the 21 st century". Chemphyschem.5 (7): 930–945. doi:10.1002/cphc.200400182. PMID15298379.2.Jump up^ Lev D. Landau; Evgeny M. Lifschitz (1984). Electrodynamics of Continuous Media. Course ofTheoretical Physics8. Oxford: Butterworth-Heinemann. ISBN0-7506-2634-8.3.Jump up^ David J. E. Callaway (1990). "On the remarkable structure of the superconductingintermediate state". Nuclear Physics B344 (3): 627–645. Bibcode:1990NuPhB.344..627C.doi:10.1016/0550-3213(90)90672-Z.Papers[edit]•V.L. Ginzburg and L.D. Landau, Zh. Eksp. Teor. Fiz.20, 1064 (1950). English translation in: L. D. Landau, Collected papers (Oxford: Pergamon Press, 1965) p. 546• A.A. Abrikosov, Zh. Eksp. Teor. Fiz.32, 1442 (1957) (English translation: Sov. Phys. JETP5 1174 (1957)].) Abrikosov's original paper on vortex structure of Type-II superconductors derived as a solution of G–L equations for κ > 1/√2•L.P. Gor'kov, Sov. Phys. JETP36, 1364 (1959)• A.A. Abrikosov's 2003 Nobel lecture: pdf file or video•V.L. Ginzburg's 2003 Nobel Lecture: pdf file or video•Gaiotto, David; Gukov, Sergei; Seiberg, Nathan (2013), "Surface Defects and Resolvents" (PDF), Journal of High Energy Physics。

精馏塔气相流动分布的CFD模拟及分析

第三章进气结构气流流场的数值模拟１．将结构的所有面以三角形方式画网格２．若边界层与任何体积面相连，则在与边界层相连的区域分别产生含有四边形、二角形元素的六面体后棱柱形网格。

３．若模型面上（或与其相连的边界层项部）存在任何四边形面元素，则产生一个棱锥元素来创建一个从想关联六面体／四边形面元素的转换，以补充余下模型内部的体积。

４，将余下的模型体积以四面体形式划分网格。

应用ＴＧｒｉｄＭｅｓｈｉｎｇＳｃｈｅｍｅ对巾８００塔的进气管结构以间隔４５单位进行网格计算其结果如图３－２。

（ａ）（ｂ）（ｃ）图３－２管式进气结构计算网格Ｆｉｇｕｒｅ３－２ＧｒｉｄｏｆＰｉｐｅｇａｓｄｉｓｔｒｉｂｕｔｏｒ２７（ａ）（ｂ）（ｃ）图３－３管式进气结构速度向量分布Ｆｉｇｕｒｅ３－３ＶｅｌｏｃｎｙｖｅｃｔｏｒｏｆＰｉｐｅｇａｓｄｉｓｔｒｉｂｕｔｏｒ距离太远，从设备角度考虑，增加了设备投资，同时也降低了应有的气体动能，同样影响填料的效率发挥。

因此，选择适宜的分布器与填料层间距离是卜分必要的。

３．６．１气速不均匀度的定义本文采用近几年常用的不均匀度定义：【４肘协刊。

５式中：ｎ一轴向截面上计算气速点数ｕ，～轴向截面上ｉ点气速，ｍ／ｓ五一轴向截面上平均气速３．６．２不均匀度的计算及计算结果分析（３—１５）根据实际设计的需要，在对各结构的流场进行模拟的过程中，选定沿塔轴向从进气处至与进气处相距２米范围内每隔０，２米设置一平面如图３—７，并对每一个平面上的速度进行不均匀度计算，经模拟计算，在每一个平面上得到的速度值为２００～２４０个。

本次计算的初始进气速度选择了工程上常用的３０、４５、６０ｍ／ｓ三种情况进行模拟计算。

图３．７管各结构轴向截面Ｆｉｇｕｒｅ３－７ＡｘｉｓａｌｓｅｃｔｉｏｎｏｆＰｉｐｅｇａｓｄｉｓｔｒｉｂｕｔｏｒ共生成网格４２，５２１４个，总体积为２．５０９２８７ｘｌＯｍ３，最小网格体积４．５５５８６０ｅ．０９ｍ３，最大网格体积为１．３０９９８１ｅ一０４ｒｌｌ３，见图３－１６。

维生素D对急性脑梗死老年患者血脂水平及脑血流动力学的影响

第42卷第1期J ournal of MuDanJiang Medical U niversity Vol.42No.12021-105-维生素D对急性脑梗死老年患者血脂水平及脑血流动力学的影响刘娟，齐锞，刘丽杰（许昌市中医院中风一科，河南许昌461000）摘要：目的探讨维生素D对急性脑梗死老年患者血脂水平及脑血流动力学的影响。

方法选取收治的老年急性脑梗死患者124例，按随机数字表法分为对照组和观察组，每组62例，对照组给予标准化治疗，观察组在对照组的基础上加用维生素D滴剂胶囊口服,6个月后对比两组患者神经功能、脑中动脉血流变化情况（最大峰值流速、舒张未流速、平均血流速度、血流阻力指数）、血脂水平［总胆固醇（TC）、三酰甘油（TG）、低密度脂蛋白胆固醇（LDL-C）］。

结果两组治疗后NIHSS评分、Fugl-Meyer评分、Barthel指数较治疗前明显好转,观察组治疗效果优于对照组（P<0.05）。

两组治疗后最大峰值流速、舒张未流速、平均血流速度、血流阻力指数水平较治疗前升高（P<0.05）；观察组治疗效果优于对照组（P<0.05）o两组治疗后TC、TG、LDL-C均比治疗前降低，观察组与对照组相比效果显著，差异有统计学意义（P<0.05）o结论急性脑梗死患者加服维生素D治疗后，患者的NIHSS评分明显降低、Fugl-Meyer评分和Barthel指数明显升高，不仅可以提高患者的最大峰值流速、舒张未流速、平均血流速度和血流阻力指数水平，而且可以降低患者的血脂水平,对患者早日康复意义重大,值得在临床上广泛运用。

关键词：维生素D；急性脑梗死；血脂；脑血流动力学中图分类号:R743.33文献标识码:A文章编号：1001-7550（2021）01-0105-04Effects of vitamin D on blood lipid levels and cerebral hemodynamics in elderly patients with acute cerebral infarctionLIU Juan et al(Department of Stroke,Xuchang City Hospital of Traditional Chinese Medicine,Xuchang461000,China)Abstract：Objective To investigate the effect of vitamin D on blood lipid levels and cerebral hemodynamics in elderly patients with acute cerebral infarction.Methods124elderly patients with acute cerebral infarction admitted to our hospital from June2017to June2019were selected and divided into a control group and an observation group according to a random number table,with62cases in each group.The control group was given standardized treatment,and the observation group was treated on the basis of the control group, vitamin D drops capsules were taken orally.After6months,the changes in nerve function and middle cerebral artery blood flow(maximum peak flow velocity,diastolic flow velocity,average blood flow velocity,blood flow resistance index),blood lipid levels[(total cholesterol(TC),triacylglycerol(TG),low-density lipoprotein cholesterol(LDL-C)]of the two groups were compared.Results After treatment,the NIHSS score,Fugl-Meyer score,and Barthel index of the two groups improved significantly compared with before treat-ment.The treatment effect of the observation group was better than that of the control group(P<0.05).After treatment,the maximum peak flow velocity,undiastolic flow velocity,average blood flow velocity,and blood flow resistance index levels in the two groups were higher than those before treatment(P<0.05).The treatment effect of the observation group was better than that of the control group(P <0.05).After treatment,TC,TG,LDL-C of the two groups were lower than those before pared with the control group,the observation group had a significant effect and the difference was statistically significant(P<0.05).Conclusion After the addition of vitamin D treatment in patients with acute cerebral infarction,the patient's NIHSS score is significantly reduced,Fugl-Meyer score and Barthel index are significantly increased,which can not only increase the patient's maximum peak flow velocity,diastolic flow velocity, and average blood flow velocity And blood flow resistance index level,and can reduce the patient's blood lipid level,which is of great significance to the patient's early recovery and is worthy of extensive clinical use.Key words：vitamin D；acute cerebral infarction；blood lipids；cerebral hemodynamics急性脑梗死是临床上最常见的多发病之一，可发生于任何年龄段，约占急性脑血管病的70%,多见于45~70岁中老年人,男女比例约为1：1o急性脑梗死主要是因脑部组织缺血、缺氧所致的缺血性坏死、软化，其临床症状与脑损害部位、脑缺血性血管大小、缺血的严重程度、发病前有无基础病等有作者简介：刘娟（1989-），女，本科，主治医师，研究方向：神经内科。

Flow equation renormalization of a spin-boson model with a structured bath

a r X i v :c o n d -m a t /0302357v 1 [c o n d -m a t .m e s -h a l l ] 18 F eb 2003Flow equation renormalization of a spin-boson model with astructured bathSilvia Kleﬀa ,1,Stefan Kehrein b ,Jan von Delft aa Lehrstuhl f¨u r Theoretische Festk¨o rperphysik,Ludwig-Maximilians Universit¨a t,Theresienstr.37,80333M¨u nchen,Germany bTheoretische Physik III –Elektronische Korrelationen und Magnetismus,Universit¨a t Augsburg,86135Augsburg,Germany1.Introduction -ModelRecently a new strategy for performing measure-ments on solid state (Josephson)qubits was proposed which uses the entanglement of the qubit with states of a damped oscillator [1],with this oscillator repre-senting the plasma resonance of the Josephson junc-tion.This system of a spin coupled to a damped har-monic oscillator (see Fig.1)can be mapped to a stan-dard model for dissipative quantum systems,namely the spin-boson model [2].Here the spectral function governing the dynamics of the spin has a resonance peak.Such structured baths were also discussed in con-nection with electron transfer processes [2].We use the ﬂow equation method introduced by Wegner [3]to an-alyze the system shown in Fig.1,consisting of a two-2σx +ΩB †B +g (B †+B )σz +k˜ωk ˜b †k ˜b k+(B †+B )kκk (˜b †k +˜b k )+(B †+B )2 kκ2k2σx +1(Ω2−ω2)2+(2πΓωΩ)2with α=8Γg 2inﬁnitesimal unitary transformations.The continuous sequence of unitary transformations U (l )is labelled by a ﬂow parameter l .Applying such a transformation to a given Hamiltonian,this Hamiltonian becomes a function of l :H (l )=U (l )H U †(l ).Here H (l =0)=H is the initial Hamiltonian and H (l =∞)is the ﬁnal diagonal ually it is more convenient to work with a diﬀerential formulation d H (l )dlU −1(l ).(3)Using the ﬂow equation approach one can decouple system and bath by diagonalizing H (l =0)[4]:H (l =∞)=−∆∞2∆−ωk2q ,kλk λq I (ωk ,ωq ,l )(b k +b †k )(b q −b †q ),(5)with I (ωk ,ωq ,l )=ωqωk +∆+ωq −∆∂l=−2(ω−∆)2J (ω,l )(6)+2∆J (ω,l )d ω′J (ω′,l )I (ω,ω′,l ),d ∆ω+∆.(7)The unitary ﬂow diagonalizing the Hamiltonian gener-ates a ﬂow for σz (l )which takes the structure σz (l )=h (l )σz +σxkχk (l )(b k +b †k ),(8)where h (l )and χk (l )obey the diﬀerential equations dhωk +∆,(9)dχkωk +∆+qχq λk λq ∆I (ωk ,ωq ,l ).(10)One can show that the function h (l )decays to zero asl →∞.Therefore the observable σz decays completely into bath operators [4].J (ω)/Ωω/∆0C (ω)∆0Fig.2.(a)Diﬀerent eﬀective spectral functions J (ω,l =0)and(b)the corresponding C (ω)for ΩΓ=0.06and α=0.15.The inset shows the term scheme of a two-level system coupled to a harmonic oscillator for the two limits ∆0≪Ωand ∆0≫Ω.We integrated the ﬂow equations numerically in or-der to calculate the Fourier transform,C (ω),of the spin-spin correlation function C (t )≡1[3]F.Wegner,Ann.Phys.3,77(1994).[4]S.Kehrein and A.Mielke,Ann.Phys.6,90(1997).3。

Fluent讲义-计算模型及网格划分介绍(哈工大培训)

Gambit中有三类几何体：
Real:
Virtual:
根据一个或多个实体(real，称为宿主)来确定其几何描述
Faceted geometry(有小面的几何体):
象virtual 一样处理
一些实几何操作对虚几何体不能正常使用
CFD-FVM
20
7/5/2004
多相流动
Multiphase flow in fluent
大多数附加的模型是在主体方程组上补充一些附加源项、附加输运方程与关系式.
CFD-FVM
2
7/5/2004
CFD-FVM
3
7/5/2004
求解器
针对各种复杂流动的物理现象， FLUENT软件采用的不同的数值解法，以期在计算速度、稳定性和精度等方面达到优化组合，
形成多种解算器，可根据实际应用选择恰当的求解器
Mixing plane (MPM): 在旋转和静止区域的交界面使用混合平面来考虑相邻区域的影响，如多级的叶轮机械
Sliding mesh (SMM): 用网格移动算法来考虑特定区域的运动
CFD-FVM
29
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User defined functions (UDF)
链接在求解器上的用户自己编制的C语言程序 UDF的主要应用：
纯金属或二元合金的液/固凝固/融化过程连续浇铸的铸造过程凝固材料与壁面的接触热阻
CFD-FVM
25
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ቤተ መጻሕፍቲ ባይዱ
Pollutant formation(污染物的形成过程)
氧化氮（NO, NO2, N2O）
只能采用segregated solver 可以与预混燃烧模型同时使用

Renormalization of the vacuum angle for a particle on a ring

Renormalization of the vacuum angle for a particle on a ring
S. M. Apenko
Theory Department, Lebedev Physics Institute, Moscow, 117924, Russia∗ (Dated: February 6, 2008)
1 2
exp −
q
m (φi+1 − φi + 2a
+ 2πq )2 + i
θ (φi+1 − φi + 2πq ) 2π
2 where the sum is over diﬀerent winding numbers q = 0 ± 1 . . .. Then P [φi ] ∼
i
G(φi , φi+1 ; a) ∼ exp(−Sef f )
β
˙ 2 (τ )dτ − i n
0
θ 2π
β
ǫab na (τ )n ˙ b (τ )dτ, (1)
0
where ǫab is the two dimensional antisymmetric tensor and β is the inverse temperature. Since n(0) = n(β ) the model is actually deﬁned on a circle. The last term in (1) has the form iθQ where Q is the topological charge which distinguishes inequivalent mappings S 1 → S 1 and takes integer values (equal to a number of rotations the particle make in time β ), making the theory periodic in θ. This (0 + 1)-dimensional ﬁeld theory may seem trivial, since in terms of the polar angle the action (1) is quadratic, but this is not so due to topological eﬀects (like in compact electrodynamics [1]). This model may be used also to describe a single electron box (SEB)[8], which is essentially a metallic island coupled to the outside circuit by a tunnel junction of capacitance C and resistance R. If R → ∞ the action for the SEB reduces to Eq. (1) where the ﬁrst term ac-