TRANSPORT EQUATIONS FROM LIOUVILLE EQUATIONS FOR FRACTIONAL SYSTEMS

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一类具有阻尼项的分数阶偏微分方程解的振动性

一类具有阻尼项的分数阶偏微分方程解的振动性李伟年【摘要】通过建立分数阶微分不等式,研究了一类具有阻尼项的分数阶偏微分方程解的振动性,并举例说明了主要结果的应用.【期刊名称】《滨州学院学报》【年(卷),期】2016(032)004【总页数】7页(P32-38)【关键词】分数阶偏微分方程;阻尼项;振动性【作者】李伟年【作者单位】滨州学院数学系,山东滨州256603【正文语种】中文【中图分类】O175众所周知，分数阶微分方程在工程、金融、应用数学以及非线性控制等领域都有着非常重要的应用。

近年来，分数阶微分方程理论研究取得了很大的进展，详细情况可参见专著[1-5]。

最近，关于分数阶偏微分方程的研究非常活跃，有关这方面的结果，可参见文献[6-10]及其相应的参考文献。

但是，涉及分数阶偏微分方程解的振动性的研究结果却不多，目前仅有文献[11-17]。

本文考虑以下具有阻尼项的分数阶偏微分方程解的振动性。

其中，Ω是Rn中具有逐片光滑区域∂Ω的有界区域，R+=(0,+∞),α∈(0,1)是一常数，(x,t)是u(x,t)关于t的α阶Liouville分数阶右导数，Δ是Rn上的Laplace变换。

边值条件为或者其中N是∂Ω的单位外法向量，g(x,t)是∂Ω×R+上的非负连续函数。

对于方程(1)，约定下列条件总是成立：(A1) a∈C([0,∞);[0,∞)),p∈C([0,∞);[0,∞));(A2) q(x,t)。

问题(1)～(2)(或者(1)～(3))的解是指函数u(x,t)在上满足方程(1)和边界条件(2)(或者(3))。

问题(1)～(2)(或者(1)～(3))的解u(x,t)称为在上是振动的，如果它既不最终为正也不最终为负。

否则，就称为是非振动的。

定义1[2] 称为函数y(t):R+→R的α阶Liouville分数阶右积分，如果(4)式的右端在R+上是逐点定义的。

这里，α>0为一常数，Γ是通常的Γ函数。

Microcanonical quasi-stationarity of long-range interacting systems in contact with a heat

arXiv:0804.0612v1 [cond-mat.stat-mech] 3 Apr 2008
Microcanonical quasi-stationarity of long-range interacting systems in contact with a heat bath
Fulvio Baldovin,1 Pierre-Henri Chavanis,2 and Enzo Orlandini1 1 Dipartimento di Fisica, CNISM, and Sezione INFN,
HHMF =
N
li2 2
+
1 2N
N
[1 − cos(θi − θj)] ,
ห้องสมุดไป่ตู้
(1)
i=1
i,j=1
where θi ∈ [0, 2π) are the spin angles and li = θ˙i ∈ R
their angular momenta (velocities). Deﬁning the ki-
Universit`a di Padova, Via Marzolo 8, I-35131 Padova, Italy 2 Laboratoire de Physique Th´eorique, Universit´e Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France
Here we demonstrate that a Hamiltonian and a Langevin thermostat provide in fact an equivalent description of the nonequilibrium behavior of a paradigmatic long-range system. This result is established, both analytically and numerically, on the basis of the scaling properties of the QSSs lifetimes. Generalizing the classical Zwanzig [10] derivation, we are able to recast

Superconducting qubits II Decoherence

present address: Physics Department and Insitute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada; fwilhelm@iqc.ca ‡ mgeller@
The transition from quantum to classical physics, now known as decoherence, has intrigued physicists since the formulation of quantum mechanics (Giulini et al., 1996; Leggett, 2002; Peres, 1993; Feynman and Vernon, 1963; Zurek, 1993). It has been put into the poignant Schr¨ odinger cat paradox (Schr¨ odinger, 1935) and was considered an open fundamental question for a long time.
and compare it to the corresponding classical mixture leading to the same expectation value of σz 1 1 0 ρmix = (2) 2 0 1 we can see that the von-Neumann entropy ρ = −kB Tr [ρ log ρ] rises from Spure = 0 to Smix = kB ln 2. Hence, decoherence taking ρpure to ρmix creates entropy and is irreversible. Quantum mechanics, on the other hand, is always reversible. It can be shown, that any isolated quantum system is described by the Liouville von-Neumann equation i¯ hρ ˙ = [H, ρ] (3)

CHIRALLY INVARIANT TRANSPORT EQUATIONS FOR QUARK MATTER

hep-ph/9505407 30 May 95
Abstract: Transport equations for quark matter are derived in the mean eld approxi-
1
2Hale Waihona Puke 31. Introduction
The ultimate goal of the ultra-relativistic heavy-ion collisions being currently carried out at CERN is the observation of the phase transition from hadronic matter to the so-called quark-gluon plasma 1, 2]. One expects that during such a decon nement phase transition, in which the structures of individual hadrons are destroyed and replaced by a highly-energetic system of quarks and gluons, chiral symmetry is additionally restored. Lattice simulations of QCD 3] indicate that the chiral symmetry restoration phase transition and decon nement phase transitions may indeed occur at the same temperature. Looking for the signatures of these two phase transitions is one of the most challenging problems of high-energy nuclear physics, both from the experimental and the theoretical point of view 4]. Since the matter produced in high energy nuclear collisions lives only for a short while, it is natural to expect that its space-time evolution proceeds far away from equilibrium. Consequently, there exists a growing interest in applying and developing transport theories, which are suitable for the description of non-equilibrium processes. A kinetic theory based directly on QCD has been formulated in papers by Heinz 5], and by Elze, Gyulassy and Vasak 6]. However this approach, with a few exceptions 7], is still far removed from practical applications. In the context of the two phase transitions that have been mentioned, it is desirable and important to be able to include some of their aspects into the kinetic approach, which can otherwise be based on some phenomenological assumptions. The decon nement phase transition seems at present to be too di cult to deal with, since we still do not yet understand this phenomenon fully. On the other hand, the chiral phase transition has been studied in terms of e ective chiral models of QCD, for which phenomenological Lagrangians can be given. Such an approach has been quite successful in clarifying this transition. One such e ective approach is based on the Nambu { Jona-Lasinio (NJL) model; within its framework we can study the temperature (density) dependence of various physical quantitites like, e.g., the quark condensate, the pion decay constant, or the quark and meson masses 8, 9]. It is also possible to study the change of the thermodynamic properties of the quark-meson plasma as the system passes the chiral phase transition 10]. With few exceptions however 11, 12, 13], the considerations based on the NJL model are usually restricted to equilibrium situations. We refer the reader to reviews on the NJL model given in 14]. In the present paper, we wish to investigate how the concept of chiral symmetry can be explicitly included into the quantum and classical transport theory based on the NJL Lagrangian. In our investigations, we shall restrict ourselves to the mean eld or Hartree 2

刘维尔方程范文

刘维尔方程范文刘维尔方程（Liouville’s equation）是经典力学中的一个重要方程，描述了质点在势能场中的运动，并且可以用来解析地表面高度分布、气体分子运动等现象。

本文将详细介绍刘维尔方程的背景、推导过程、应用和现代发展。

一、背景刘维尔方程是由法国物理学家刘维尔于1838年提出的，旨在解决经典力学中的一个困难问题：对于有多个自由度的系统，如何确定系统在相空间中的运动。

在经典力学中，系统的运动状态由广义坐标和广义动量决定，描述了系统所有粒子的位置和动量。

然而，在现实问题中，系统的自由度往往非常多，使得直接求解运动方程十分困难。

二、推导过程我们以一个简单的二维例子来推导刘维尔方程。

假设有一个质点在二维势能场中运动，其势能函数为V(x,y)。

根据哈密顿原理，系统的运动满足下列哈密顿方程：dx/dt=dH/dpydy/dt=-dH/dpx其中，H是系统的哈密顿量，p是广义动量。

为了简化问题，我们将势能函数V(x,y)展开成泰勒级数：V(x,y)=V(0,0)+x*(∂V/∂x)，0,0+y*(∂V/∂y)，0,0+...将势能函数的泰勒级数代入哈密顿方程中，我们得到：dx/dt=∂H/∂py+O(x,y)dy/dt=-∂H/∂px+O(x,y)上式的O(x,y)表示高阶无穷小。

接下来，我们对上述方程组进行平均处理。

首先，对于x的时间导数，我们有：d(x^2)/dt=2x*(dx/dt)=2x*(∂H/∂py+O(x,y))利用x^2的平均值与平方平均值之间的关系，我们得到：d<x^2>/dt=2<x*∂H/∂py>+O(x^2) (1)同理，对于y的时间导数，我们有：d<y^2>/dt=2<y*(-∂H/∂px)>+O(y^2) (2)由于系统的演化是以经典概率平均的方式进行的，因此我们还需要对动量进行平均处理。

我们定义f(x,y,p,t)为处于状态（x,y,p）的粒子的概率密度函数，则概率密度函数的时间演化满足：dp/dt=-∂H/∂x对上式两边同时乘以p并对p积分，我们得到：d<p>/dt=-<∂H/∂x>=-<∂V/∂x>根据定义，p的平方平均值等于动能的平均值：<p^2>/m=<p·p>/m=2K/m其中，K是系统的动能。

ReynoldsTransportTheorem：雷诺兹运输定理

Reynolds Transport TheoremOctober 1,2003Consider a system and a control volume (C.V.)as follows:•the system occupies C.V.(I+II)at time t•the same system occupies volume (II+III)at t +∆t•ﬂuid particles are free to enter and leave C.V.•C.V.may move as time passes but does notdeformReynolds Transport Theorem:The total rate of change of any extensive property N s (=ηdm =ηρd ∀)of a system occupying a control volume C.V.at time t is equal to the sum ofa)the temporal rate of change of N within the C.V.b)the net ﬂux of N through the control surface C.S.that surrounds theC.V.dN s dt =∂∂t ZZZ C.V.ηρd ∀+ZZ C.S.ηρ−→V rel ·d −→A (1)This theorem applies to any transportable property,including mass,momen-tum and energy.1Proof:The change of property N s of system during∆t is∆N s=N s|t+∆t−N s|t=(N II+N III)|t+∆t−(N I+N II)|t[add and subtract N I|t+∆t]=(N II+N III)|t+∆t−(N I+N II)|t+N I|t+∆t−N I|t+∆t=(N I+N II)|t+∆t−(N I+N II)|t+N III|t+∆t−N I|t+∆t=N C.V.|t+∆t−N C.V.|t+N III|t+∆t−N I|t+∆t The rate of change of property N s of the system isdN s dt =lim∆t→0N C.V.|t+∆t−N C.V.|t∆t+lim∆t→0N III|t+∆t∆t−lim∆t→0N I|t+∆t∆t=a+b+c(2)We will evaluate the three terms a,b,c separately.First term:This is the rate of change of property N contained within the C.V.a=lim∆t→0N C.V.|t+∆t−N C.V.|t∆t≡∂N C.V.∂t=∂∂t ZZZ C.V.ηρd∀(3)Second term:This refers to the amount of property N contained in the volume III,which is the part of the volume occupied by the system at time t+∆t,which is outside the original C.V.Consider aﬂuid element in the C.V., in contact with an elementary area dA on the part of the control surface(C.S.I) which is between the C.V.and volume III.During a small∆t,this element will move through volume III and reach its surface on the other side,moving by a distance∆l=V rel∆t,thus deﬁning a volumed∀=∆l(dA cosα)=(V rel cosαdA)∆t=³−→V rel·−→n dA´∆t=³−→V rel·−→dA´∆t2This volume containsﬂuid that leaves the C.V.during∆t(outﬂow).Thus,−π2<α<π2,so that0<cosαand the volume d∀is positive,as it should be. The amount of property N within d∀at time t+∆t isdN III|t+∆t=ηρd∀|t+∆twhile the total amount of N within volume III at time t+∆t isN III|t+∆t=ZZZ III dN III=ZZZ IIIηρd∀= ZZC.S.IIIηρ−→V rel·−→dA ∆tThen,b=lim∆t→0N III|t+∆t∆t=ZZ C.S.IIIηρ−→V rel·−→dAThis term represents the outﬂow of property N from the C.V.Third term:This refers to the amount of property N contained in volume I at t+∆t;volume I is the part of the C.V.that is occupied by newﬂuid(not part of the original system)at time t+∆t.Thisﬂuid enters the volume from outside(inﬂow),so thatπ2<α<π,or−π<α<−π2,which means that cosα<0.Consider aﬂuid element in the C.V.,in contact with an elementary area dA on the part of the control surface(C.S.I)which will remain outside volume II during time∆t.During a small∆t,this element will move through volume I and reach the interface between volumes I and II,moving by a distance∆l=V rel∆t, thus deﬁning a volume(volume must always be positive)d∀=∆l(−dA cosα)=(−V rel cosαdA)∆t=−³−→V rel·−→n dA´∆t=−³−→V rel·−→dA´∆tThe amount of property N within d∀at time t+∆t isdN I|t+∆t=ηρd∀|t+∆t3The total amount of N within volume I at t+∆t,which represents the total inﬂow ofﬂuid into the C.V.during∆t,isN I|t+∆t=ZZZ I dN I=ZZZ Iηρd∀=−ZZ C.S.Iηρ−→V rel·−→dA∆tThen,c=−lim∆t→0N I|t+∆t∆t=+ZZ C.S.Iηρ−→V rel·−→dAThe sum b+c represents the totalﬂux ofﬂuid through the entire surface C.S. of the C.V.,i.e.b+c=ZZ C.S.IIIηρ−→V rel·−→dA+ZZ C.S.Iηρ−→V rel·−→dA=ZZ C.S.ηρ−→V rel·−→dA(4)Substituting equations(3)and(4)into equation(2),we get equation(1), which proves the Reynolds transport theorem.4。

常微分方程中的英文单词和短语

常微分方程中的英文单词和短语
常微分方程
Ordinary Differential Equtions
第一章基本概念
Chapter 1 Basic Concepts
常微分方程
阶线性非线性偏微分方程
通解特解初值条件初值问题柯西问题几何解释积分曲线线素线素场方向场等斜线对称形式奇异点通积分
boundary finite closed interval open set finite semi-open interval
Lagrange formula
l ocal finite covering theorem
bar area monotone decreasing sectorial area comparability solution minimum solution maximum solution sl ope
topological dynamical system
differential dynamical system
stability of solution Liapunov stability
asymptotic stability asymptotic stability field attraction domain globe asymptotic stability
第六章线性微分方程组
Chapter 6 Linear Differential Equations
存在和唯一性定理齐次线性微分方程组基本解组朗斯基行列式刘维尔公式解矩阵基解矩阵常数矩阵非齐次线性微分方程组常数列向量常数变异公式计算公式结构公式常系数线性微分方程组矩阵指数函数标准基解矩阵若尔当标准型矩阵函数实值解复值解代数余子式特征方程算子式算子多项式拉普拉斯逆变换

NRL_FORMULARY_11

3
Euler-Mascheroni constant1 γ = 0.57722 Gamma Function Γ(x + 1) = xΓ(x): Γ(1/6) Γ(1/5) Γ(1/4) Γ(1/3) Γ(2/5) Γ(1/2) = = = = = = 5.5663 4.5908 3.6256 2.6789 2.2182 √ 1.7725 = π Γ(3/5) Γ(2/3) Γ(3/4) Γ(4/5) Γ(5/6) Γ(1) = = = = = = 1.4892 1.3541 1.2254 1.1642 1.1288 1.0
C
dl · A dl × A f dg = − gdf
C
(35)
S
(d S × ∇) × A = d S · (∇f × ∇g ) =
C
(36)
S
C
DIFFERENTIAL OPERATORS IN CURVILINEAR COORDINATES5 Cylindrical Coordinates Divergence ∇·A= Gradient (∇f )r = Curl (∇ × A)r = ∂Aφ 1 ∂Az − r ∂φ ∂z ∂Ar ∂Az − ∂z ∂r 1 ∂ 1 ∂Ar (rAφ ) − r ∂r r ∂φ ∂f ; ∂r (∇f )φ = 1 ∂f ; r ∂φ (∇f )z = ∂f ∂z 1 ∂Aφ ∂Az 1 ∂ (rAr ) + + r ∂r r ∂φ ∂z
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 . . . . . . . . . . . . . . . . . . . . 30 . . . . . . . . . . . . . . . . . . . . . 31 . . . . . . . . . . 40 . . . . . . . . . . . . . . . . . . . . . . 42 . . . . . . . . . . . . . . . . . . . . . 43 . . . . . . . . . . . . . . . . . . . . . . 44 . . . . . . . . . . . . . . . . . . . . 46

国际贸易中的引力方程【外文翻译】

外文翻译原文The Gravity Equation in International TradeMaterial Source: KELLY SCHOOL OF BUSINESS INDIANA UNIVERSITY Department of Business Economics and Public Policy Author: Michele FratianniAbstractThis chapter offers a selective survey of the gravity equation (GE) in international trade. This equation started in the Sixties as a purely empirical proposition to explain bilateral trade flows, without little or no theoretical underpinnings. At the end of the Seventies, the GE was “legitimized” by a series of theoretical articles that demonstrated that the basic GE form was consistent with various models of trade flows. Empirical applications of GE expanded to cover a variety of issues, such as the impact of regional trade agreements, national borders and currency unions on trade, as well as the use of the equation to sort out the relative merit of alternative trade theories. A new wave of studies is now concentrating on the general equilibrium properties of the GE and finer econometrics points. The renewed interest of the academic profession in the development of the GE is undoubtedly driven by the equation’s empirical success.Keywords: gravity equation, trade theories, borders, regional trade agreement, currency unions.JEL Classification: E58, F15, F33, G15Draft date: August, 2007I. INTRODUCTIONInternational economics and international business have common interests but somewhat different research agendas. The former emphasizes cross-border trade and capital flows, whereas the latter looks predominantly at foreign direct investment. Part of this difference results from the emphasis that scholars in international business place on the study of the multinational firm and part is due to intellectualspecialization. It is worth recalling that the yearly flows of international trade are a large multiple of the yearly flows of foreign direct investment, while the stock of foreign direct investment has only recently approached annual trade flows (see Figure 1). Furthermore, total real exports have grown faster, on average, than the world real GDP since the mid-1980s (see Figure 2). Finally, it is widely believed that exports are an engine of economic growth; see Krueger (2006). For all these reasons, international trade economists spend a great deal of time and resources understanding and explaining trade flows.With this brief background, I can state the objectives and outline of the chapter. The objective is to explain trade flows in terms of the gravity equation (GE). The reason for focusing on GE is two fold. The first is that GE, unlike other frameworks, has had great empirical success in explaining bilateral trade flows. For a long time, however, GE was a child without a father in the sense that it was thought to have no theoretical support. Since the late 1970s, this state of affairs has changed radically. Now, the gravity equation has strong theoretical support and can be derived from a variety of models of international trade. The second is that GE can be used to sort out alternative hypotheses of international trade.In its simplest form, the gravity equation (GE) explains flows of a good between pairs of countries in terms of the countries’ incomes, distance and a host of idiosyncratic factors--such as common border, common language, and common money-- that enhance or reduce bilateral trade flows:（1） 123ijk 0k k k k ik jk ij ijk M A Y Y d U ααα=，wheredenotes that the k good is exported by country i and imported by country j,andare expenditures on the k good by thetwo countries , and d is distance; A and αs are coefficients, and U is a well behaved error term. The vector of idiosyncratic factors has been omitted in (1) because these factors are more control variables than theoretically derived variables. Aggregating over all k goods, the GE of a given product can be transformed into a GE of total exports of country i:（2） 312ijk 0i j ij ij M A Y Y d U ααα=，where the k subscript has been suppressed and Y is the country’s income (forexample, nominal gross domestic product or GDP). The implications of GE –which we develop and discuss below-- are such that α1 and α2 are positive and in some instances equal t o one and that α3 is negative. Typically, equation (2) is specified in log linear form and estimated either with cross-section or panel data. In the latter case, a time subscript τ is added, except for the time -invariant physical distance:（3） 034ln()ln()ln()ijt jt ij ij ijt M Y d F u αααα=++++，Where in stands for the natural logarithm, 00ln()A α= and ln()ijt ijt u U =. The vector of idiosyncratic factors,, has also been added to equation (3). These factors are typically measured as dummy variables that acquire the value of one for the existence of the phenomenon and zero for its absence. The coefficients α1 through α3 are interpreted as elasticity’s or as percentage changes in bilateral trade for one percentage change in income and distance. The coeffic ient α4 is positive if the factor is trade enhancing (e.g., common language) and negative if trade reducing (e.g., terrorism).In the following section I will explore different models of international trade from which GE can be derived, ranging from models of complete specialization and identical consumers’ preferences (Anderson 1979; Bergstrand 1985; Deardorff 1998) to models of product differentiation in a regime of monopolistic competition (Helpman 1987) to hybrid models of different factor proportions and product differentiation (Bergstrand 1989; Evenett and Keller 2002) to models of incomplete specialization and trading costs (Haveman and Hummels 2004).II. TRADE THEORY AND THE GEComplete specializationSpecialization is at the heart of trade theory; it is complete or deepest when each country specializes in the production of its own output and consumers purchase the output of each country according to identical and homothetic preferences. Furthermore, trade occurs without friction, meaning that it is not impeded either by transport costs, tariffs or tariff-equivalent border obstacles. This idealized set-up serves the purpose of creating a benchmark of maximum trade flows. Each country imports and consumes a share of the goods produced by all other countries, as well as a share of its own output. These shares are the same for all countries. Consider, for example, two countries, country 1 and country 2, producing differentiatedproducts by country of origin. Country 1 will export its own good to country 2 in the amount of 1212M b Y =, where 1b =marginal propensity to import good 1 in country 2. Country 1 will also sell 12b Y amount of the good it produces to domestic consumers. Note that the propensity to consume good 1 is the same across all consumers regardless of location. Income of country 1 is the sum of purchases by consumers located in country 1 and consumers located in country 2, i.e., 111121w Y bY bY bY =+=, where w Y = world income = 12Y Y +. Thus, identical and homotheticpreferences imply that the propensity to import and consume good 1 is equal to country 1’s share of world income. Replacing 1b with 1/w Y Y , 1212/w M YY Y =. This is the simple GE derived by Anderson (1979, p. 108):译文国际贸易中的引力方程资料来源：印地安那大学凯莱商学院校刊作者：米歇尔·弗拉蒂安尼摘要本文章精心选择并调查了在国际贸易中的重力方程（简称GE ）。

非线性分数阶微分方程耦合系统三点边值问题解的存在性

非线性分数阶微分方程耦合系统三点边值问题解的存在性姜小霞;欧阳自根;彭湘凌【摘要】讨论了非线性分数阶微分方程耦合系统的三点边值问题，利用Green函数的性质，将其转化为等价的积分方程耦合系统，应用Schauder不动点定理得到解的存在的充分条件。

%In this paper,we study the three-point boundary value problem to a coupled system of nonlinear fractional differential equations. By the means of the Green’s function,the system can be reduced to the equivalent integral equation. Then we obtain some sufficient conditions for the existence of the solutions for the system by using the Schauder fixed-point theorem.【期刊名称】《南华大学学报（自然科学版）》【年(卷),期】2015(000)001【总页数】6页(P94-99)【关键词】耦合系统;边值问题;Riemann-Liouville分数阶导数;Schauder不动点定理【作者】姜小霞;欧阳自根;彭湘凌【作者单位】南华大学数理学院，湖南衡阳421001;南华大学数理学院，湖南衡阳421001;南华大学数理学院，湖南衡阳421001【正文语种】中文【中图分类】O175key words：coupled system;boundary value problem;Riemann-Liouville fractional derivative;Schauder fixed-point theorem近几十年来，分数阶微分方程在物理、机械、化学、工程等学科的应用越来越广泛，许多学者对分数阶微分方程进行了研究[1-6]，还有些学者对分数阶微分方程耦合系统进行了研究[7-11].例如Su[9]研究了以下分数阶微分方程耦合系统两点边值问题其中是Caputo型分数阶导数，f,g:[0,1]×R→R是连续函数，作者应用Schauder不动点定理证明了其解的存在性.在本文中，将对下面一类非线性分数阶微分方程耦合系统三点边值问题解的存在性进行研究其中1<αi<2,i=1,2,p,q,γ>0,0<η<1,α1-q≥1,α2-p≥1,γηαi-1<1,D是标准的Riemann-Liouville分数阶导数，且f,g:[0,1]×R×R→R是连续函数.利用Green函数的性质和Schauder不动点定理，得到分数阶微分方程耦合系统(1)～(2)解存在的充分条件.令I=[0,1],C(I)表示为定义在I上的所有连续函数所构成的集合，并令，则X为Banach空间，其中范数定义为为Banach空间，其范数为，这里0<p,q<1，那么(X×Y,‖·‖X×Y)也是Banach空间，且范数为‖(x,y)‖X×Y=max{‖x‖X,‖y‖Y}.由此容易得到对于任意αi>1,若x(t)∈X，则tα1-1x(t)∈X，若y(t)∈Y，则tα2-1y(t)∈Y.定义1[1,10] 函数f:(0,)→R的α>0阶Riemann-Liouville分数阶积分为其中Γ(α)为Gamma函数，右端在R+上逐点有定义.定义2[1,10] 连续函数f:(0,)→R的α阶Riemann-Liouville分数阶导数为其中α>0，n=[α]+1，Γ(α)为Gamma函数，右端在R+上逐点有定义.由定义2，有引理3[1,5] 令α>0，如果u∈C(0,1)∩L(0,1)，那么分数阶微分方程有一个解为u(t)=c1tα-1+c2tα-2+…+cNtα-N,ci∈R,i=1,2,…,N,N为大于或等于α的最小整数引理4[1,5] 假设u∈C(0,1)∩L(0,1)，且分数阶导数α>0，那么IαDαu(t)=u(t)+c1tα-1+c2tα-2+…+cNtα-N,ci∈R,i=1,2,…,N,N为大于或等于α的最小整数定理5(Schauder不动点定理)[12] 设P是E中有界凸闭集，T:P→P全连续，则T在P中必具有不动点.在本文中，假设f(t,x,y)，g(t,x,y)满足下列条件：(H1)f(t,x,y)，g(t,x,y)∈C(I×R×R,R)，f(t,x,y)，g(t,x,y)都是关于x和y的连续函数，且f(t,x,y)，g(t,x,y)对t∈I是可测的.(H2)f(t,x,y)，g(t,x,y)都是关于x和y的单调不减函数，存在非负函数a(t)，b(t)，c(t)，d(t)∈L(0,1)，使得其中p1,p2,q1,q2≥0，p1+p2<1，q1+q2<1.为方便起见，引入以下记号：，，，，，s.此外给出一些本文需要用到的引理.引理6[10] 问题(1)～(2)等价于以下列积分方程其中令那么问题(1)～(2)等价于下列积分方程记那么因此，只需研究积分方程(12)解的存在性.引理7 对任意是连续函数，且.证明下面只证明(t,s)>0，其他证明是相类似的，这里就不重复证明.很容易得到当max{t,η}<s≤1时，；当t<s≤η时，；当；当0≤s≤min{t,η}<1时，有于是.由式(8)～式(9)，容易得到引理对变量t∈(0,1)是单调不增的函数，当s≤t≤1时，；当0≤t≤s时，.即对任意的t∈(0,1)，有.由引理8，有下面令其中那么‖u(t)‖X≤r,‖v(t)‖Y≤r,也就意味着定义K上的算子T：Tw(t)=(T1v(t),T2u(t)),其中下面给出本文的主要结论和证明.定理9 若条件(H1)，(H2)成立，则问题(1)～(2)有一个解.证明证明算子T:K→K是一个完全连续算子. 第一步，证明算子T:K→K.ds于是同理，容易得到‖T2u(t)‖Y≤ν+Λ2rq1+q2≤r，因此‖Tw(t)‖X×Y≤r.第二步，证明算子T是连续算子.令wn(t)=(un(t),vn(t))是K中的序列，且满足，对t∈[0,1]，有tα1-1un(t)，tα1-1u(t)∈X，tα2-1vn(t)，tα2-1v(t)∈Y，因为f(t,x,y)，g(t,x,y)对于x和y都是连续函数，Gi(t,s)对于s∈[0,1]是一致连续的且满足因此，对任意的ε>0，t∈I，存在正整数N1，当n>N1时，有由式(16)～式(17)，有又因为T1vn(t)，T1v(t)∈X，于是对于上述ε，存在N2>0，当n>N2时，有由式(20)～式(22)有取N=max{N1,N2}，结合式(19)和式(23)，对于上述ε，当n>N，有即用同样的方式可得到结合上述两式，有所以T是K上的连续算子.最后，证明T是等度连续的.只需证明对于任意w(t)∈K，和任意的0<t1<t2<1，当t1→t2时，有Tw(t1)→Tw(t2)即可.接下来分以下三种情况来考虑：1)0<t1<t2<η；2)0<t1<η<t2；3)0<η<t1<t2.下面只对1)进行证明，2)和3)证明与1)完全类似，不再重复. 令，则相类似的可以得到于是算子T是完全连续算子.由Schaudar不动点定理可知，方程(1)～(2)存在一个解.[1] Kilbas A A,Srivastava H M,Trujillo J J.Theory and applications of fraction al differential equations[M].Amsterdam:Elsevier Science B V,2006.[2] Lakshmikantham V.Theory of fractional functional differential equations [J].Nonlinear Anal,2008,69(10):3337-3343.[3] Agarwal R P,Lakshmikantham V,Nieto Juan J.On the concept of solution for fractional differential equations with uncertainty[J].Nonlinear Anal,201 0,72(6):2859-2862.[4] Podlubny I.Fractional differential equations[M].San Diego:Academic Pre ss,1999.[5] Miller K S,Ross B.An introduction the fractional Calculus and fractional e quations[M].New York:Wiley,1993.[6] Agrawal R P,Zhou Y,He Y.Existence of fractional neutral differential equa tions[J].Compul.And Math.with appl,2010,59(3):1095-1110.[7] Bai C Z,Fang J X.The existence of a positive solution for a singular coupl ed system of nonlinear fractional differential equations[J]p ut,2004,150(3):611-621.[8] Chen Y,An H L.Numerical solutions of coupled Burgers equations with ti me-and space-fractional derivatives[J]put,2008,200(1):215-225.[9] Su X W.Boundary value problem for a coupled system of nonlinear fract ional differential eququations[J].Appl.Math.Lett,2009,22(1):64-69.[10] Ahmad B,Nieto J J.Existence results for a coupled system of nonlinear f ractional differential equations with three-point boundary conditions[J].Comput.Math.Appl,2009,58(9):1838-1843. [11] Zhou Y.Existence and uniqueness of solutions for a system of fractiona l differential equations[J].J.Frac.Calc.Appl.Anal,2009,12(2):195-204. [12] 郭大钧.非线性泛函分析[M].2版.济南:山东科学技术出版社,2001.。

The future of transportation Urban air mobility

The future of transportation Urban airmobilityUrban air mobility (UAM) is a concept that refers to the transportation of people and goods in urban areas using air vehicles. With the rapid growth of urban populations and the increasing congestion on the roads, there is a growing needfor alternative transportation solutions. UAM has the potential to address these challenges by providing a faster, more efficient, and environmentally friendly mode of transportation. However, there are also several significant challengesthat need to be addressed in order for UAM to become a viable and safe transportation option in the future. One of the main challenges facing the future of UAM is the development of the necessary infrastructure. Unlike traditional airports, UAM vehicles will require a network of vertiports and charging stations throughout urban areas. This will require significant investment and coordination between various stakeholders, including government agencies, private companies, and local communities. Without the proper infrastructure in place, the potential of UAM to alleviate urban congestion and provide efficient transportation will not be realized. Another challenge is the integration of UAM into existing airtraffic management systems. With the potential for a large number of UAM vehicles operating in urban airspace, it will be essential to ensure the safe and efficient integration of these vehicles with traditional aircraft. This will require the development of new technologies and regulations to manage the increased complexity of urban airspace and prevent potential safety hazards. Furthermore, the public acceptance of UAM will also be crucial for its future success. While UAM has the potential to revolutionize urban transportation, there may be concerns regarding noise pollution, safety, and privacy. It will be important for UAM developers and regulators to address these concerns through effective communication and transparency, as well as the implementation of appropriate regulations and standards. In addition, the development of UAM vehicles with the necessary safety features and reliability will be essential for gaining public trust and regulatory approval. This will require significant advancements in technology, including the development of autonomous flight systems, collision avoidance technology, andadvanced battery and propulsion systems. Ensuring the safety and reliability of UAM vehicles will be crucial for their widespread adoption and integration into urban transportation systems. Moreover, the environmental impact of UAM will also need to be carefully considered. While UAM vehicles have the potential to be more environmentally friendly than traditional modes of transportation, their widespread adoption could also lead to increased energy consumption and emissions. It will be essential to develop UAM vehicles and infrastructure that minimizetheir environmental impact and contribute to overall sustainability goals. In conclusion, while UAM has the potential to revolutionize urban transportation, there are several significant challenges that need to be addressed for its successful implementation. The development of necessary infrastructure,integration into existing air traffic management systems, public acceptance, safety and reliability, and environmental impact are all crucial considerationsfor the future of UAM. Addressing these challenges will require collaboration and innovation from various stakeholders, including government agencies, industry leaders, and the public, in order to realize the full potential of UAM as a future mode of transportation.。

运筹学课件 Transportation Model

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Operations Research
Ch5 Transportation Model
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Example 5.1-2 Suppose that the Detroit plant capacity is 1300 (instead of 1500). Then due to demand exceed the supply, a dummy source with a capacity of 200 should be added. The case where the supply exceeds the demand can be demonstrated by assuming that the demand at Denver is 1900. In this case, we need to add a dummy distribu#39;
A solution to the transportation looks like
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Balancing the Transportation Model. The transportation algorithm is based on the assumption that the total demand equals the total supply. If the model is unbalanced, we can always add a dummy source or a dummy destination.
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Operations Research
Ch5 Transportation Model

Chapter 1 Derivation of Reaction-Diffusion Equations

where J is the ﬂux of P , d(x) is called diﬀusion coeﬃcient at x, and ∇x is the gradient operator ∇x f (x) = (∂f /∂x1 , ∂f /∂x2 , · · · , ∂f /∂xn ). On the other hand, the number of particles at any point may change because of other reasons like birth, death, hunting, or chemical reactions. We assume that the rate of change of the density function due to these reasons is f (t, x, P ), which we usually call the reaction rate. Now we derive a diﬀerential equation using the balanced law. We choose any region O, then the total population in O is O P (t, x)dx, and the rate of change of the total population is d dt P (t, x)dx.
Chapter 1
Derivation of Reaction-Diﬀusion Equations
1.1 Fick’s Law
Diﬀusion mechanism models the movement of many individuals in an environment or media. The individuals can be very small such as basic particles in physics, bacteria, molecules, or cells, or very large objects such as animals, plants, or certain kind of events like epidemics, or rumors. The particles reside in a region, which we call Ω, and we assume that Ω is open subset of Rn (the n-th dimensional space with Cartesian coordinate system) with n ≥ 1. In particular, we are interested in the cases of n = 1, 2 and 3, but most material here are true regardless of the dimensions of the space (sometimes n = 1 and n ≥ 2 may be diﬀerent as we will see.) The main mathematical variable we consider here is the density function of the particles: P (t, x), where t is the time, and x ∈ Ω is the location. The dimension of the population density usually is number of particles or organisms per unit area (if n = 2) or unit volume (if n = 3). For example, the human population density is often expressed in number of people per square kilometer. A list of world population and population density can be found at /wiki/List_of_countries_by_population_density. However in such data table, we can only ﬁnd the population density for countries, and a country is not a “point” on our map. In reality, population density is always associated with a scale, like country, city, county, town, and street. But as in many other mathematical models, we will assume that the function P (t, x) has nicer properties, like continuity and diﬀerentiability, which is in fact reasonable, when a population with a large number of organisms is considered. Technically, we deﬁne the population density function P (t, x) as follows: let x be a point in the habitat Ω, and let {On }∞ n=1 be a sequence of spatial regions (which have the same dimension as Ω) surrounding x; here On is chosen in a way that the spatial measurement |On | of On (length, area, volume, or 1

Death to the Log-Linearized Consumption Euler Equation! (And Very Poor Health to the Second

Death to the Log-Linearized Consumption Euler Equation!(And Very Poor Health to the Second-Order Approximation)Christopher D.Carroll†ccarroll@Published as:Carroll,Christopher D.(2001)”Death to the Log-Linearized Consumption Euler Equation!(And Very Poor Health to the Second-Order Approximation)”,Advances in Macroeconomics:Vol.1:No.1,Article6./bejm/advances/vol1/iss1/art6Keywords:Euler equation,uncertainty,consumption,excess sensitivityJEL Classiﬁcation Codes:C6,D91,E21AbstractThis paper shows that standard methods for estimating log-linearized consumption Euler equations using micro data cannot successfully uncover structural parameters like the co-eﬃcient of relative risk aversion from a dataset of simulated consumers behaving exactly according to the standard model.Furthermore,consumption growth for the simulated con-sumers is very highly statistically related to predictable income growth–and thus standard ‘excess sensitivity’tests would reject the hypothesis that consumers are behaving according to the model.Results are not much better for the second-order approximation to the Eu-ler equation.The paper concludes that empirical estimation of consumption Euler equations should be abandoned,and discusses some alternative empirical strategies that are not subject to the problems of Euler equation estimation.Thanks to participants in the1997NBER Summer Institute Consumption workshop,and to Orazio Attanasio,Martin Browning,John Campbell,Sydney Ludvigson,and Chris Paxson for comments.All of the programs used to generate the results in this paper are available at the author’s website,/People/ccarroll/carroll.html.†Department of Economics,Johns Hopkins University,Baltimore,MD21218-2685,410-516-7602(oﬃce),410-516-7600(fax).1IntroductionEstimation of Euler equations has occupied a central place in consumption research over the more than twenty years since Hall(1978)ﬁrst derived and tested the consumption Euler equation.Unfortunately,despite scores of careful empirical studies using household data, Euler equation estimation has not fulﬁlled its early promise to reliably uncover preference parameters like the intertemporal elasticity of substitution.Even more frustrating,the model does not even seem to fail in a consistent way:Some studiesﬁnd strong evidence of‘excess sensitivity’of consumption to predictable income growth,while othersﬁnd little or no excess sensitivity.This paper oﬀers an explanation for the conﬂicting empirical results,by showing that when the Euler equation estimation methods that have been widely used on household data are applied to a set of data generated by simulated consumers behaving exactly according to the standard consumption model,those methods are incapable of producing an econometri-cally consistent estimate of the intertemporal elasticity of substitution.Furthermore,‘excess sensitivity’tests canﬁnd either high or low degrees of sensitivity,depending on the exact nature of the test.In principle,the theoretical problems with Euler equation estimation stem from approx-imation error.The standard procedure has been to estimate a log-linearized,orﬁrst-order approximated,version of the Euler equation.This paper shows,however,that the higher-order terms are endogenous with respect to theﬁrst-order terms(and also with respect to omitted variables),rendering consistent estimation of the log-linearized Euler equation impos-sible.Unfortunately,the second-order approximation fares only slightly better.The paper concludes that empirical estimation of approximated consumption Euler equations should be abandoned,and discusses some alternative empirical methods for studying consumption behavior that are not subject to the problems of Euler equation estimation.The paper begins by presenting the speciﬁc version of the dynamic optimization problem that is solved and simulated.The next section describes the standard empirical methodology for estimating Euler equations and summarizes the results that have been reported in the literature.Section4describes the details of the simulations which generate the data to be analyzed.Section5is the heart of the paper:It shows that the standard empirical methods cannot produce consistent estimates of true model parameter values.The penultimate section describes several empirical strategies that are candidates to replace Euler equation estimation, and theﬁnal section concludes.2The ModelConsider a consumer solving the following maximization problem(essentially the same as the model in Carroll(1992,1997)and Zeldes(1989)):1max {C t }u (C t )+E tT s =t +1βs −t u (˜C s ) (1)s.t.X s +1=R (X s −C s )+Y s +1Y s +1=P s +1V s +1P s +1=GP s N s +1u (C )=C 1−ρP t c t −ρ =1.(3)For a given value of x T −1this equation can be solved numerically to ﬁnd the optimal value of c T −1.This is done for a grid of possible values for x T −1and a numerical optimal con-sumption rule c ∗T −1(x T −1)is constructed by linear interpolation between these points.Givenc ∗T −1(x T −1)the same methods can be used to construct c ∗T −2(x T −2)and so on to any arbitrarynumber of periods from the end of life.2Carroll(1996)shows that if Deaton’s‘impatience’condition RβE t[(G˜N t+1)−ρ]<1holds,these successive optimal consumption rules will con-verge as the horizon recedes,and consumers behaving according to the converged rule can be described as engaging in‘buﬀer-stock’saving.I will denote the optimal consumption rule forany period t as c∗t(x t)and the converged consumption rule as c∗(x)=limn→∞c∗T−n(x),whichcorresponds to the inﬁnite horizon solution.All numerical and simulation results in the paper will be generated from the converged consumption rule.Carroll(1997)argues that empirical evidence for US households suggests that even consumers withﬁnite horizons behave like impatient(but inﬁnite-horizon)‘buﬀer-stock’for much of their working lifetimes;that paper argues that the transition from buﬀer-stock behavior to something more closely resembling classical life cycle behavior(where the problems emphasized in this paper would be lessened)happens for the median household somewhere between ages45and50.Cagetti’s(1999)recent paper implies a similar age for the transition;Gourinchas and Parker(1999)argue that the transition occurs somewhere around age40.3Since most empirical microeconometric work has restricted the sample to households between the ages of25and60(to avoid including students and others who have not formed a permanent attachment to the labor force on the young end,and early retirees on the older end),under any of these estimates buﬀer-stock saving behavior should be expected to obtain for a large proportion of the households in the data that has been used in empirical studies.To verify accuracy of the numerical solution,Figure1plots RβE t[(C t+1/C t)−ρ]as a function of x t.Errors in the numerical solution will lead the function to diﬀer from one at points away from the gridpoints chosen for x t(where equality is imposed by the solution method).Theﬁgure shows that the errors involved in numerical solution are very small; the function is so close to one over the entire plotted range(which encompasses the range of values of wealth that actually arise when the model is simulated)that it appears to be a solid line exactly at one.Thisﬁgure serves to illustrate the point that the problems with Euler equation estimation documented in the rest of the paper are in a sense attributable to the use of approximations to the Euler equation,since(as theﬁgure shows)the true nonlinear Euler equation always holds by construction.3The Standard Procedure3.1Derivation of the Log-Linearized Consumption Euler EquationThe“Log-Linearized”consumption Euler equation of this paper’s title is obtained by taking aﬁrst-order Taylor expansion of the nonlinear Euler equation(2),and making some approxi-mations.For every possible C t and C t+1there will be someηt+1for which C t+1=(1+ηt+1)C t (assuming that consumption is always positive).Since we rarely expect to see consumption0.5 1.0 1.5 2.02.53.0x 0.951.001.051.10True Euler EquationFigure 1:Numerical Value of RβE t˜P t +1c ∗(˜x t +1)whereǫt+1is iid and the law of iterated expectations implies that it is uncorrelated with any variable known at time t(Hall(1978)).Those authors made uncomfortable by theﬁrst-order approximations involved in deriving equation(7)have sometimes been reassured by a well-known result that suggests that the second-order approximation leads to the same estimating equation.The second-order Taylor approximation of(1+ηt+1)−ρaroundηt+1=0is(1+ηt+1)−ρ≈1−ρηt+1+ρ(ρ+1)2 E t[˜η2t+1]+ǫt+1,(8) and ifη2t+1is uncorrelated with r andδ,then the E t[˜η2t+1]term will be absorbed in the constant term of a regression estimate of(7).43.2Previous Empirical ResultsTo keep the notation simple,the derivations thus far have implicitly assumed thatρ,δ,and r are constants.Of course,if these parameters were constant across all times,places,and people then it would be impossible to estimate a coeﬃcientρin an equation like(7).In practice, Euler equations like(7)have mainly been estimated in two ways.In microeconomic data, the most common procedure has been to estimate the equation across diﬀerent consumers at a point in time,by identifying groups of consumers for whom diﬀerent interest rates apply.In macroeconomic data,the equation has been estimated by exploiting time-variation in the aggregate interest rate.5The principal purpose of this paper is to show that the usual cross-section procedures for microeconomic estimation of this equation do not work; the penultimate section brieﬂy discusses whether time series estimation methods are similarly problematic.The instrumental variables approach to estimating the model using microeconomic data can be usefully thought of as equivalent to taking means within groups of consumers with similar characteristics,and identifying parameter values by diﬀerences in these group-means. For example,typical instruments used in the empirical literature are education group or occupation group.Henceforth I will denote distinct groups by the subscript j and the group-mean value of a variable X whose value diﬀers across members of the group will be designated (X)j.For example,if we were to designate the growth rate of consumption for an individ-ual household as∆log C i,t+1then the group-mean value of consumption growth across all consumers in group j would be designated(∆log C t+1)j which would be calculated(assum-ing there are m consumers in group j who happen to have index numbers i=1...m)as (∆log C t+1)j=(1/m) m i=1∆log C i,t+1.Parameters which are assumed to take a commonvalue for all members of the group are unobtrusively indicated by a subscript j,e.g.ρj,r j, andδj.In this notation,equation(7)becomes:(∆log C t+1)j≈ρ−1j(r j−δj)+(ǫt+1)j(9) Thus,the standard log-linearized empirical Euler equation has been estimated using re-gression equations of the form(∆log C t+1)j=α0+α1r j+(ǫt+1)j(10) where the understanding has been thatα1,the coeﬃcient on r,should be a consistent estimate of the intertemporal elasticity of substitution,ρ−1.According to equation(9),this will be true if three conditions hold:ﬁrst,the approximations involved in deriving equation(7)are not problematic;second,any diﬀerences inδj across groups are uncorrelated with whatever diﬀerences there may be in r j;and,ﬁnally,there are no diﬀerences across groups inρj.Empirical results for estimating equations like(10)have been ually theα1term is estimated to be insigniﬁcantly diﬀerent from zero;only a few studies have found signiﬁcantly positive values ofρ.6However,the poor results in estimatingρhave often been interpreted as reﬂecting poor identifying information about exogenous diﬀerences in r across groups,rather than as important rejections of the Euler equation itself.7The potential empirical problems with identifying exogenous variation in interest rates across households have led many authors to focus on another feature of the model:Hall’s ‘random walk’proposition.Hall(1978)showed that in a model with quadratic utility,con-sumption should follow a random walk and no information known at time t should help to forecast the change in consumption between t and t+1.The alterative hypothesis has usu-ally been that consumption is‘excessively sensitive’to forecastable income growth.Formally, denoting the expected growth rate of income as E t[∆log˜Y t+1],the equation most commonly estimated has been:(∆log C t+1)j=α0+α1r j+α2(E t[∆log˜Y t+1])j+ǫj,(11) and the‘random walk’proposition implies thatα2=0when the expected growth rate of income is instrumented using information known by the group j consumers at time t.Empirical results estimating equation(11)using micro data have been hardly better than those estimating the baseline equation(10).8In a comprehensive survey article,Browning and Lusardi(1996)cite roughly twenty studies that have estimated the coeﬃcient on predictable income growth.Estimates of the marginal propensity to consume out of predictable income growth ranged from zero(consistent with the CEQ LC/PIH model)up to2.An apologist for the model might note that most estimates are in the range between0and0.6.3.3The Explanation?Carroll(1992,1996,1997)has challenged the foregoing empirical methodology on the grounds that theory implies that the higher-order terms in the approximation cannot be ignored because they are endogenous and in particular are correlated withρj,δj,and,fatally,r j and (E t[∆log˜Y t+1])j.Those papers show show that‘impatient’consumers behaving according to the standard CRRA intertemporal optimization model will engage in‘buﬀer-stock’or target saving behavior,9and that,among a collection of buﬀer-stock consumers with the same parameter values,if the distribution of x across consumers has converged to its ergodic distribution,then average consumption growth across the members of the group will be equal to average permanent income growth.Thus,if we have j groups of consumers such that within each group j all consumers have the same parameter values,and x has converged to its ergodic distribution within each group, then(∆log C t+1)j=(∆log P t+1)j=g j.(12) The intuition for this result is fairly simple:If consumers are behaving according to a buﬀer-stock model with a target wealth¯w,then it is impossible for consumption growth to be permanently diﬀerent from underlying income growth.If consumption growth were forever greater than permanent labor income growth,consumption would eventually exceed labor income by an arbitrarily large amount,driving wealth to negative inﬁnity.If consumption growth were permanently less than labor income growth,labor income would eventually exceed consumption by an arbitrarily large amount,driving wealth to inﬁnity.Thus,in a model where there is an ergodic distribution of wealth across consumers,it is impossible for average consumption growth to diﬀer permanently from average income growth.10 As an aid to understanding the nature of the endogeneity problem,suppose that the second-order approximation equation(8)captures all of the important endogeneity so that the terms of third order and higher can safely be ignored(we will examine this assumption carefully below).Assume thatρdoes not diﬀer across the groups,and rewrite the second order approximation equation(8)in the new notation:(∆log C t+1)j≈ρ−1(r j−δj)+ 1+ρ9The term‘impatient’here and henceforth refers to the condition RβE t[(G˜N t+1)−ρ[<1.Note that,so long as income is growing over time G>1,consumers can be impatient in the required sense even ifβ=1so that they do not discount future utility at all.10For much more careful discussion and arguments,see Carroll(1996,1997).7must take:(η2t+1)j≈ 211This is an implication of the concavity of the consumption function proven by Carroll and Kimball(1996). 12This statement assumes that the second-order approximation holds exactly.The more general statement would be that all of the higher-order terms together should take on values that make(∆log C t+1)j=g j.8Low High0.000.04δ0.040.020.06ρ30.050.15σv0.1013Thus,there are13groups altogether:the baseline group plus one positive and one negative deviation from the baseline parameter value for each of six parameters.90.75 1.00 1.25 1.50 1.75 2.00 2.25 2.500.20.40.60.81.0Steady State CDFCDF H x 10LFigure 2:Distribution of x After 10Years (Solid)vs.Ergodic Distribution (Dashing)The goal is to characterize the kinds of regression results that an econometrician would obtain using a sample of data drawn from these simulations.The appropriate strategy is therefore a Monte Carlo procedure which reports both the mean parameter estimates that would be obtained by a large number of studies on such data,and the variation in parameter estimates that would be found across the diﬀerent studies.My Monte Carlo procedure is as follows.For each ‘group’to be included in a regression,I draw a random sample of 1000observations from the 10,000available for that group.I then perform the regressions and record the coeﬃcient estimates and standard errors.I then draw another sample of 1000observations for each group,perform another regression,and record the results.I repeat this procedure 10,000times to obtain a distribution of parameter estimates and standard error estimates.Note that there are several respects in which the econometrician examining the simulated data is better oﬀthan his counterpart using actual data.First,there is no measurement error in the simulated data for either income or consumption;estimates of the fraction of measurement error in the PSID data on food consumption range up to 92percent.Second,the econometrician working with simulated data can directly observe the interest rate that applies for each household.In empirical work there is rarely a really convincing way to identify exogenous diﬀerences in interest rates across the diﬀerent households in the sample.Third,the diﬀerent ‘groups’in the simulations diﬀer from the baseline parameter values in only a single dimension (parameter)at a time.In reality,occupation or education may be correlated with several parameters;for example,education is highly correlated with the growth rate of income,but may also be correlated with the time preference rate.Finally,the typical10empirical dataset probably has fewer than a hundred consumers in any given instrumented age/occupation or age/education cell,while I have a thousand consumers for each possible combination of parameter values.The purpose of these simulations is to show that even in such ideal circumstances,Euler equation estimation by standard microeconometric IV methods does not work.There is even less reason to expect it to work under the less than ideal circumstances faced in actual data.5Estimating Consumption Euler Equations on the Simulated Data5.1The Log-Linearized Euler EquationTable2presents the results when the log-linearized Euler equation(10)is estimated on the simulated data.Theﬁrst of the six panels presents baseline results when equal numbers of consumers from each possible parametric combination(except for deviations ofρfrom baseline)are included.14(This sample selection is indicated by the text‘All butρ’under the‘Consumers in Sample’column).The second column indicates the set of instruments used for predicting all instrumented variables in the regression.Since r is the only explanatory variable included in the regression reported in panel,the dummy variable indicating interest rate group(RDUM) is the only instrument that makes sense in these two regressions.I exclude from the regressions all consumers for whom income was zero in either period of observation,V t=0or V t+1=0,for two reasons.First,such data are typically excluded from the empirical regressions whose methods I am trying to duplicate.Second,extreme income shocks tend to interact strongly with the nonlinearities of the model,so even a relatively small number of such extreme events could heavily inﬂuence the results.It is therefore a more compelling indictment of the estimation method if it performs badly even when such extreme events are excluded.As noted above,I estimate the regressions10,000times with10,000diﬀerent randomly-chosen collections of1000simulated consumers.For each variable,the table presents the mean(across the10,000regressions)of the coeﬃcient estimates and the mean of the estimated standard errors.Next to the means are theﬁfth and ninety-ﬁfth percentiles in the distribution of coeﬃcient estimates and standard error estimates.The last column indicates the average number of observations in each regression.Because the probability that either V t=0or V t+1=0is0.01,this number should on average be equal to0.99*1000*(number of groups included in regression).For example,one would expect a sample size of0.99*1000*11= 10890for theﬁrst row,since there are11distinct possible combinations of parameter values excluding combinations whereρdiﬀers from baseline.The actual average value of NOBS is almost exactly right,at10889.Coeﬀ.on(∆log Y t+1)jPanel Instruments‡[.05-.95]Range[.05-.95]Range1RDUM[-0.23,0.24]ρ(0.14)BASE+0.002970(0.15,0.15)3RDUM[-0.20,0.20][0.79,1.20]ρ(0.14)(0.14)BASE+0.000.974950 +GDUM(0.10,0.18)(0.07,0.25)5RDUM[-0.21,0.21][0.07,0.10]ρ(0.12)(0.01)BASE+0.000.112970+V t(0.12,0.14)(0.02,0.02)which permanent-income-growth group the consumer belongs to(RDUM remains in the instrument set to instrument for the interest rate).Again the equation is estimated for two samples,one which includes members with all appropriate parametric combinations,and one containing only consumers who are members of the R and G groups.In panel3,the mean coeﬃcient on the predictable growth rate of income is0.97,highly signiﬁcantly diﬀerent from zero,but not signiﬁcantly diﬀerent from one.Results are similar in panel4,which restricts the sample to the set of consumers for whom one might expect the best results for the Euler equation method.Furthermore,in the typical regression the coeﬃcient on the interest rate term is again estimated to be zero.This result,consumption growth equal to predictable permanent income growth but independent of the interest rate,is precisely what the analysis in Section3and in Carroll(1996,1997)showed holds if consumers are distributed according to the ergodic distribution.Apparently,at least under the parameter values considered here, 9years of presample simulation for1000consumers suﬃce to create a sample that generates behavior very similar to that under the ergodic distribution.As noted in the literature survey above,empirical point estimates of the excess sensitivity of consumption growth to predictable income growth have mostly fallen in the range from0.0 to about0.6.Although many of the studies could not reject a coeﬃcient of1on the income growth term,the bulk of the estimates were closer to zero than to one.It might seem, then,that these results rescue the Euler equation from the Scylla of a(rejected)prediction thatα1=0only to smash against the Charybdis of a(rejectable)prediction thatα1=1. Fortunately,there is an escape hatch.The theoretical arguments and simulation evidence presented thus far do not necessarily imply a coeﬃcient of1on E t[∆log˜Y t+1]–they imply a coeﬃcient of one on E t[∆log˜P t+1].That is,consumption should on average grow at the rate of permanent income growth.None of the theoretical or simulation work up to this point in the paper has indicated what the coeﬃcient should be on predictable transitory movements in income.The last two panels of the table present the model’s predictions about the coeﬃcient on the predictable transitory movements in income.(Transitory movements in income are predictable because the level of the transitory shock is white noise.Thus,if income’s level is temporarily low today,income growth between today and tomorrow is likely to be high, and vice versa.Hence the instrument used for E t[∆log˜Y t+1]is V t.)Panels5and6reveal that the coeﬃcient on predictable transitory movements in income is statistically signiﬁcantly diﬀerent from zero,but,at around0.10,is much closer to zero than to one.As before,the coeﬃcient on the interest rate term is insigniﬁcantly diﬀerent from zero.These very diﬀerent results for predictable transitory and for predictable permanent in-come growth imply that there is little we can say about the model’s prediction for the coef-ﬁcient on predictable income growth,if we have not decomposed that growth into the part representing transitory growth and the part representing permanent growth.15Essentially all we can say is that(under this range of parameter values),the coeﬃcient on predictable income growth should be somewhere between0.10and1.0.Of the roughly twenty studiescited by Browning and Lusardi(1996),none(to my knowledge)attempts to decompose pre-dictable income growth into predictable transitory and predictable permanent components.16 Since the conﬁdence intervals forα1in virtually all of these papers overlap the range between 0.10and1.0,if‘excess sensitivity’is deﬁned as a degree of sensitivity inconsistent with un-constrained intertemporal optimization,none of the‘excess sensitivity’tests summarized by Browning and Lusardi(1996)provides any evidence on whether consumption actually exhibits excess sensitivity to predictable changes in income.These results also bear on theﬁnding of Campbell and Mankiw(1989)that regressions of aggregate consumption growth on predictable aggregate income growthﬁnd a coeﬃcient of roughly0.5.Although Campbell and Mankiw interpreted theirﬁndings as suggesting that about half of consumers behave according to a‘rule-of-thumb’and set their consumption equal to their income,they did not decompose their predictable income growth term into a predictable permanent growth term and a predictable transitory term,so it is quite possible that their results are consistent with an optimizing model like the one considered here without the need for introducing‘rule-of-thumb’consumers.Aﬁnal category of tests should be mentioned brieﬂy:Empirical estimates of the rate of time wrance(1991),for example,estimates an equation like(11)using data from the PSID,but including dummy variables for education in the estimating equation. Sheﬁnds that consumers with more education have higher rates of consumption growth,and concludes that consumers with more education must be more patient.This conclusion would be warranted if the log-linearized consumption Euler equation were valid,because−ρ−1δj is omitted from the baseline empirical speciﬁcation sinceδj is unobserved.However,given that a positive correlation between permanent income growth and education is a bedrock empirical result in labor economics,an obvious alternative explanation of Lawrance’s results is that the higher consumption growth for more educated consumers reﬂects their faster predictable permanent income growth,not a greater degree of patience.To summarize,when the log-linearized consumption Euler equation is estimated on house-hold data generated by consumers behaving exactly according to the standard model,using the methods that have been used by most of the existing cross-section empirical studies,the results provide no information on either the coeﬃcient of relative risk aversion or on whether consumption exhibits‘excess sensitivity’to predictable income growth.5.2The Second Order ApproximationA few empirical studies,of which Dynan(1993)is one of the earliest and best,have avoided the log-linearized Euler equation and instead used the second-order approximation to the。

c ○ World Scientific Publishing Company LIOUVILLE AND BOGOLIUBOV EQUATIONS WITH FRACTIONAL

Modern Physics Letters B,Vol.21,No.5(2007)237–248c World Scientiﬁc Publishing CompanyLIOUVILLE AND BOGOLIUBOV EQUATIONS WITHFRACTIONAL DERIV ATIVESV ASILY E.TARASOVSkobeltsyn Institute of Nuclear Physics,Moscow State University,Moscow119992,Russiatarasov@theory.sinp.msu.ruReceived14March2006The Liouville equation,ﬁrst Bogoliubov hierarchy and Vlasov equations with derivativesof non-integer order are derived.Liouville equation with fractional derivatives is obtainedfrom the conservation of probability in a fractional volume element.This equation is usedto obtain Bogoliubov hierarchy and fractional kinetic equations with fractional deriva-tives.Statistical mechanics of fractional generalization of the Hamiltonian systems isdiscussed.Fractional kinetic equation for the system of charged particles are considered.Keywords:Liouville equation;Bogoliubov equation;fractional derivatives;fractionalkinetics.PACS Number(s):05.20.-y,05.20.Dd,45.10.Hj1.IntroductionFractional equations1are equations that contain derivatives of non-integer order.2,3 The theory of derivatives of non-integer order goes back to Leibniz,Liouville,Rie-mann,and Letnikov.3Derivatives and integrals of fractional order have found many applications in recent studies in mechanics and physics.In a short period of time the list of applications have become long.For example,it includes chaotic dynamics,4,5 mechanics of fractal media,6–8physical kinetics,4,9–12plasma physics,13–15astro-physics,16long-range dissipation,17–19mechanics of non-Hamiltonian systems,20,21 theory of long-range interaction,22–24and many others physical topics.In this paper,we derive Liouville equation with fractional derivatives with re-spect to coordinates and momenta.To derive the fractional Liouville equation,we consider the conservation of probability toﬁnd a system in the fractional diﬀerential volume ing the fractional Liouville equation,we derive the fractional generalization of the Bogoliubov hierarchy equations.These equations can be used to derive fractional kinetic equations.4,9,10,12A linear fractional kinetic equation for the system of charged particles is suggested.In Sec.2,we derive the Liouville equation with fractional derivatives from the conservation of probability toﬁnd a system in the fractional volume element of the phase space.In Sec.3,we obtain theﬁrst Bogoliubov hierarchy equation with237238V.E.Tarasovfractional derivatives in the phase space from the fractional Liouville equation.In Sec.4,the Vlasov equation with fractional derivatives in phase space is considered. In Sec.5,a linear fractional kinetic equation for the system of charged particles is suggested.Finally,a short conclusion is given in Sec.6.2.Liouville Equation with Fractional DerivativesA basic principle of statistical mechanics is the conservation of probability.The Liouville equation is an expression of this basic principle in a convenient form for the analysis.In this section,we derive the Liouville equation with fractional derivatives from the conservation of probability in a fractional volume element.For the phase space R2n with coordinates(x1,...,x2n)=(q1,...,q n, p1,...,p n),we consider a fractional diﬀerential volume elementdαV=dαx1···dαx2n.(1) Here,dαis a fractional diﬀerential.25For the function f(x),dαf(x)=2nk=1Dαxkf(x)(dx k)α,(2)where Dαxkis a fractional derivative3of orderαwith respect to x k.The Caputo derivative6,26is deﬁned byDαx f(x)=1(x−z)α+1−mdz,(3)where m−1<α<m,f(m)(τ)=d m f(τ)/dτm,andΓ(z)is the Euler gamma-function.For Caputo and Riesz3fractional derivatives,we have Dαxk1=0,and D xkx l=0(k=l).Using Eq.(2),we obtaindαx k=Dαxkx k(dx k)α,(α>0).(4) Then(dx k)α=(Dαxkx k)−1dαx k.(5) For Caputo derivatives,Dαxk xβk=Γ(β+1)∂t=d[ρ(t,x)(u,d S)],(8)Liouville and Bogoliubov Equations with Fractional Derivatives239 for the usual volume element(α=1),and−dαV∂ρ(t,x)∂t =dαV2nk=1(Dαxkx k)−1Dαxk[ρu k].(14)As the result,we obtain∂ρ∂t =−Γ(2−α)2nk=1xα−1kDαxk[ρu k].(16)Equation(15)is a Liouville equation that contains the derivatives of fractional orderα.Fractional Liouville equation(15)describes the probability conservation toﬁnd a system in the fractional volume element(1)of the phase space.For the coordinates(x1,...,x2n)=(q1,...,q n,p1,...,p n),Eq.(15)is ∂ρ240V.E.Tarasovwhere V k=u k,and F k=u k+n(k=1,...,n).The functions V k=V k(t,q,p)are the components of velocityﬁeld,and F k=F k(t,q,p)are the components of force ﬁeld.In general,Dαpk [ρF k]=ρDαpkF k+F k Dαpkρ.(18)Suppose that F k does not depend on p k,and the k th component V k of the velocity ﬁeld does not depend on k th component q k of coordinates.In this case,Eq.(17) gives∂ρdt =V k(t,q,p),dq k∂p j−∂V j∂q i+∂F i∂q j−∂F j dt=∂Hdt=−∂H∂t +nk=1(Dαqkq k)−1DαpkHDαqkρ−nk=1(Dαpkp k)−1DαqkHDαpkρ=0.(24)We can deﬁne{A,B}α=nk=1((Dαqkq k)−1DαqkADαpkB−(Dαpkp k)−1DαqkBDαpkA).(25)Forα=1,Eq.(25)deﬁnes Poisson brackets.Note that the brackets(25)satisfy the relations{A,B}α=−{B,A}α,{1,A}α=0.Liouville and Bogoliubov Equations with Fractional Derivatives241 In general,the Jacoby identity cannot be satisﬁed.The property{1,A}α=0is sat-isﬁed only for Caputo and Riesz fractional derivatives(Dαx1=0).For the Riemann–Liouville derivative,Dαx1=ing Eq.(25),we get Eq.(24)in the form∂ρ∂t =−Nk=1(Dαqk(V kρN)+Dαpk(F kρN)),(27)whereDαqk V k=(Dαqkq k)−1DαqkV k=ms=1(Dαqksq ks)−1DαqksV ks,(28)Dαpk F k=(Dαpkp k)−1DαpkF k=ms=1(Dαqksp ks)−1DαpksF ks.(29)The one-particle reduced density of probabilityρ1can be deﬁned byρ1(q,p,t)=ρ(q1,p1,t)=ˆI[2,...,N]ρN(q,p,t),(30) whereˆI[2,...,N]is an integration with respect to variables q2,...,q N,p2,...,p N. Obviously,that one-particle density of probability satisﬁes the normalization con-ditionˆI[1]ρ1(q,p,t)=1.(31) The Bogoliubov hierarchy equations27–30describe the evolution of the reduced density of probability.They can be derived from the Liouville equation.Let us derive theﬁrst Bogoliubov equation with fractional derivatives from the fractional Liouville equation(27).Diﬀerentiation of Eq.(30)with respect to time gives∂ρ1∂t ˆI[2,...,N]ρN=ˆI[2,...,N]∂ρN242V.E.Tarasov Using Eq.(27),we get∂ρ1∂t =−ˆI[2,...,N](Dαq1(V1ρN)+Dαp1(F1ρN)).(35)Since the variable q1is an independent of q2,...,q N and p2,...,p N,theﬁrst term in Eq.(35)can be written asˆI[2,...,N]Dαq k (V1ρN)=Dαq1V1ˆI[2,...,N]ρN=Dαq1(V1ρ1).The force F1acts on theﬁrst particle.It is a sum of the internal forcesF1k=F(q1,p1,q k,p k,t),and the external force F e1=F e(q1,p1,t).In the case of binary interaction,we haveF1=F e1+Nk=2F1k.(36)Using Eq.(36),the second term in Eq.(35)can be rewritten in the formˆI[2,...,N]Dαp1(F1ρN)=ˆI[2,...,N](Dαp1(F e1ρN)+Nk=2Dαp1(F1kρN))=Dαp1(F e1ρ1)+Nk=2Dαp1ˆI[2,...,N](F1kρN).(37)We assume thatρN is invariant under the permutations of identical particles:31ρN(...,q k,p k,...,q l,p l,...,t)=ρN(...,q l,p l,...,q k,p k,...,t).Liouville and Bogoliubov Equations with Fractional Derivatives243 In this case,ρN is a symmetric function,and all(N−1)terms of sum(37)are identical.Therefore the sum can be replaced by one term with the multiplier(N−1):N k=2ˆI[2,...,N]Dαp1s(F1kρN)=(N−1)ˆI[2,...,N]Dαp1(F12ρN).(38)UsingˆI[2,...,N]=ˆI[2]ˆI[3,...,N],we rewrite the right-hand side of(38)in the formˆI[2]Dαp1(F12ˆI[3,...,N]ρN)=Dαp1ˆI[2](F12ρ2),(39)whereρ2is two-particle density of probability that is deﬁned by the fractional integration of the N-particle density of probability over all q k and p k,except k= 1,2:ρ2=ρ(q1,p1,q2,p2,t)=ˆI[3,...,N]ρN(q,p,t).(40) Since p1is independent of q2,p2,we can change the order of the integrations and the diﬀerentiations:ˆI[2]Dαp1(F12ρ2)=Dαp1ˆI[2]F12ρ2.Finally,we obtain∂ρ1∂q1,V1=∂H(q1,p1)244V.E.TarasovIn this case,we can rewrite Eq.(45)in the formI(ρ2)=−Dαp1(ρ1F eﬀ).(46) Substituting of Eq.(46)into Eq.(41),we obtain∂ρ1∂t+(v,Dαq f)+e(E,Dαp f)=0,(48) where f=ρ1is the one-particle density of probability,and(v,Dαq f)=ms=1(v s,Dαqsf).(49)If we take into account the magneticﬁeld(B=0),then we must use the fractional generalization of Leibnitz rules:Dαp(fg)=∞r=0Γ(α+1)mc Dαp([p,B]f)=emc klmεklm B m Dαp k(p l f) =eΓ(r+1)Γ(α−r+1)[Dα−ip kf]δkl p r l =emc klmεklm B m p l[Dαp k f]=eLiouville and Bogoliubov Equations with Fractional Derivatives245 where f0is a homogeneous stationary density of probability that satisﬁes Eq.(48) for E=0.Substituting of Eq.(52)into(48),we get for theﬁrst perturbation∂δf2π +∞−∞dke−ikx e−|k|α(55)is the Levy stable density of probability.40Forα=1,the function(55)gives the Cauchy distributionL1[x]=1x2+1,(56)and Eq.(54)is1q2s(g s t)−2+1.(57) Forα=2,we get the Gauss distribution:L2[x]=1πe−x2/4,(58)and the function(54)is(g s t)−1/21πe−q2s/(4g s t).(59)For1<α≤2,the function Lα[x]can be presented as the expansionLα[x]=−1n!sin(nπ/2).(60)The asymptotic(x→∞,1<α<2)is given byLα[x]∼−1n!sin(nπ/2).(61)As the result,we arrive at the asymptotic of the solution,which exhibits power-like tails for x→∞.The tail is the important property for the solutions of fractional equations.246V.E.Tarasov6.ConclusionIn this paper,we consider equations with derivatives of non-integer order that can be used in statistical mechanics and kinetic theory.We derive the Liouville,Bogoli-ubov and Vlasov equations with fractional derivatives with respect to coordinates and momenta.To derive the fractional Liouville equation,we consider the conser-vation of probability toﬁnd a system in the fractional diﬀerential volume element. Using the fractional Liouville equation,we obtain the fractional generalization of the Bogoliubov hierarchy equations.Fractional Bogoliubov equations can be used to derive fractional kinetic equations.4,9,10The fractional kinetic is related to equa-tions that have derivatives of non-integer orders.Fractional equations appear in the description of chaotic dynamics and fractal media.For the fractional linear oscil-lator,the physical meaning of the derivative of orderα<2is dissipation.Note that fractional derivatives with respect to coordinates can be connected with the long-range power-law interaction of the systems.22–24References1.I.Podlubny,Fractional Diﬀerential Equations(Academic Press,New York,1999).2.K.B.Oldham and J.Spanier,The Fractional Calculus(Academic Press,New York,1974).3.S.G.Samko,A.A.Kilbas and O.I.Marichev,Fractional Integrals and DerivativesTheory and Applications(Gordon and Breach,New York,1993).4.G.M.Zaslavsky,Chaos,fractional kinetics,and anomalous transport,Phys.Rep.371(2002)461–580.5.G.M.Zaslavsky,Hamiltonian Chaos and Fractional Dynamics(Oxford UniversityPress,Oxford,2005).6. A.Carpinteri and F.Mainardi,Fractals and Fractional Calculus in ContinuumMechanics(Springer,New York,1997).7.R.Hilfer,Applications of Fractional Calculus in Physics(World Scientiﬁc,Singapore,2000).8.V.E.Tarasov,Continuous medium model for fractal media,Phys.Lett.A336(2005)167–174;Fractional Fokker–Planck equation for fractal media,Chaos15 (2005)023102;Possible experimental test of continuous medium model for fractal media,Phys.Lett.A341(2005)467–472;Fractional hydrodynamic equations for fractal media,Ann.Phys.318(2005)286–307;Wave equation for fractal solid string, Mod.Phys.Lett.B19(2005)721–728;Dynamics of fractal solid,Int.J.Mod.Phys.B19(2005)4103–4114;Fractional Chapman–Kolmogorov equation,Mod.Phys.Lett.B21(4)(2007).9.G.M.Zaslavsky,Fractional kinetic equation for Hamiltonian chaos,Physica D76(1994)110–122.10. A.I.Saichev and G.M.Zaslavsky,Fractional kinetic equations:Solutions andapplications,Chaos7(1997)753–764.11.G.M.Zaslavsky and M. A.Edelman,Fractional kinetics:From pseudochaoticdynamics to Maxwell’s demon,Physica D193(2004)128–147.12.R.R.Nigmatullin,Fractional kinetic equations and universal decoupling of a memoryfunction in mesoscale region,Physica A363(2006)282–298;A.V.Chechkin,V.Yu.Gonchar and M.Szydlowsky,Fractional kinetics for relaxation and superdiﬀusion in magneticﬁeld,Physics of Plasmas9(2002)78–88;R.K.Saxena,A.M.Mathai andLiouville and Bogoliubov Equations with Fractional Derivatives247H.J.Haubold,On fractional kinetic equations,Astrophysics and Space Science282(2002)281–287.13. B.A.Carreras,V.E.Lynch and G.M.Zaslavsky,Anomalous diﬀusion and exit timedistribution of particle tracers in plasma turbulence model,Physics of Plasmas8 (2001)5096–5103.14.L.M.Zelenyi and ovanov,Fractal topology and strange kinetics:Frompercolation theory to problems in cosmic electrodynamics,Physics Uspekhi47(2004) 749–788.15.V. E.Tarasov,Electromagneticﬁeld of fractal distribution of charged particles,Physics of Plasmas12(2005)082106;Multipole moments of fractal distribution of charges,Mod.Phys.Lett.B19(2005)1107–1118;Magnetohydrodynamics of fractal media,Physics of Plasmas13(2006)052107;Electromagneticﬁelds on fractals,Mod.Phys.Lett.A21(2006)1587–1600.16.V.E.Tarasov,Gravitationalﬁeld of fractal distribution of particles,Celes.Mech.Dynam.Astron.19(2006)1–15.17. F.Mainardi and R.Gorenﬂo,On Mittag-Leﬄer-type functions in fractional evolutionprocesses,put.Appl.Math.118(2000)283–299.18. F.Mainardi,Fractional relaxation-oscillation and fractional diﬀusion-wavephenomena,Chaos,Solitons and Fractals7(1996)1461–1477.19.V.E.Tarasov and G.M.Zaslavsky,Dynamics with low-level fractionality,Physica A368(2006)399–415.20.V.E.Tarasov,Fractional generalization of Liouville Equation,Chaos14(2004)123–127;Fractional systems and fractional Bogoliubov hierarchy equations,Phys.Rev.E71(2005)011102;Fractional Liouville and BBGKI equations,J.Phys.Conf.Ser.7(2005)17–33;Transport equations from Liouville equations for fractional systems,Int.J.Mod.Phys.B20(2006)341–354;Fokker–Planck equation for frac-tional systems,Int.J.Mod.Phys.B21(2007)accepted.21.V. E.Tarasov,Fractional generalization of gradient and Hamiltonian systems,J.Phys.A38(2005)5929–5943;Fractional generalization of gradient systems,Lett.Math.Phys.73(2005)49–58;Fractional variations for dynamical systems:Hamilton and Lagrange approaches,J.Phys.A39(2006)8409–8425.skin and G.M.Zaslavsky,Nonlinear fractional dynamics on a lattice with long-range interactions,Physica A368(2006)38–54.23.V. E.Tarasov and G.M.Zaslavsky,Fractional dynamics of coupled oscillatorswith long-range interaction,Chaos16(2006)023110;Fractional dynamics of sys-tems with long-range interaction,Commun.Nonlin.Sci.Numer.Simul.11(2006) 885–898.24.N.Korabel,G.M.Zaslavsky and V.E.Tarasov,Coupled oscillators with power-lawinteraction and their fractional dynamics analogues,Commun.Nonlin.Sci.Numer.Simul.11(2006)accepted(math-ph/0603074).25.K.Cottrill-Shepherd and M.Naber,Fractional diﬀerential forms,J.Math.Phys.42(2001)2203–2212; F.B.Adda,The diﬀerentiability in the fractional calculus, Nonlinear Analysis47(2001)5423–5428.26.M.Caputo and F.Mainardi,A new dissipation model based on memory mechanism,Pure and Appl.Geophys.91(1971)134–147.27.N.N.Bogoliubov,Kinetic equations,Zh.Exper.Teor.Fiz.16(1946)691;J.Phys.USSR10(1946)265.28.N.N.Bogoliubov,in Studies in Statistical Mechanics,Vol.1,eds.J.de Boer and G.E.Uhlenbeck(North-Holland,Amsterdam,1962),p.1.29.K.P.Gurov,Foundation of Kinetic Theory:Method of N.N.Bogoliubov(Nauka,Moscow,1966)in Russian.248V.E.Tarasov30. D.Ya.Petrina,V.I.Gerasimenko and P.V.Malishev,Mathematical Basis of ClassicalStatistical Mechanics(Naukova dumka,Kiev,1985)in Russian.31.N.N.Bogoliubov and N.N.Bogoliubov,Jr.,Introduction to Quantum StatisticalMechanics(World Scientiﬁc,Singapore,1982),Sec.1.4.32. A.A.Vlasov,Vibrating properties of electronic gas,Zh.Exper.Teor.Fiz.8(1938)291;On the kinetic theory of an assembly of particles with collective interaction, SR9(1945)25.33. A.A.Vlasov,Many-particle Theory and its Application to Plasma(Gordon andBreach,New York,1961).34.G.Ecker,Theory of Fully Ionized Plasmas(Academic Press,New York,1972).35.N.A.Krall and A.W.Trivelpiece,Principles of Plasma Physics(McGraw-Hill,NewYork,1973).36.V. E.Tarasov,Stationary solution of Liouville equation for non-Hamiltoniansystems,Ann.Phys.316(2005)393–413;Classical canonical distribution for dis-sipative systems,Mod.Phys.Lett.B17(2003)1219–1226.37. A.Isihara,Statistical Physics(Academic Press,New York,London,1971),Appendix IV,Sec.7.5.38.P.Resibois and M.De Leener,Classical Kinetic Theory of Fluids(John Wiley andSons,New York,London,1977),Sec.IX.4.39. D.Forster,Hydrodynamics Fluctuations,Broken Symmetry,and CorrelationFunctions(W.E.Benjamin Inc.,London,Amsterdam,1975),Sec.6.4.40.V.Feller,An Introduction to Probability Theory and its Applications,Vol.2(Wiley,New York,1971).。

Bernoulli‘s equation

Bernoulli’s equation
Bernoulli desses this principle quantitatively:
This is Bernoulli’s equation.
Bernoulli’s equation can be written as:
Bernoulli’s Equation
Bernoulli’s principle
Where the velocity of a fluid is high, the pressure is low, and where the velocity is low, the pressure is high.
To derive Bernoulli’s equation, we assume the flow is steady and laminar, the fluid is incompressible, and the viscosity is small enough to be ignored.
Lift on an airplane wing. We are in the reference frame of the wing, seeing the air flow by.
Looking down on a pitched baseball heading toward home plate. We are in the reference frame of the baseball, with the air flowing by.
To be general, we assume the fluid is flowing in a tube of nonuniform cross section that varies in height above some reference level.

A Semiclassical Transport Model for Two-dimensional Thin Quantum Barriers

2
Particle behavior at a quantum barrier
To model quantum dynamics, we consider a top-down multiscale approach by considering the quantum eﬀects as local corrections to the global classical particle dynamics. In order to isolate and simplify the problem, we make the following assumptions/limitations: (1) The eﬀective width of a barrier is O(ε). On the classical scale, this means that we may approximate the barrier as having zero width; on the quantum scale, this means that we may typify it as a single scattering center and we may neglect particle dwell time in the quantum region in the semiclassical limit. (2) The distance between neighboring barriers is O(1) and hence each barrier may be considered independently. (3) The change in the potential ∇V (x) is O(1) except at quantum barriers. (4) The coherence time is suﬃciently short and therefore we may neglect interference away from the barrier. We begin with the classical Hamiltonian system d x = ∇p H (x, p), dt d p = −∇x H (x, p). dt (1)