On fundamental equation of statistical physics (II)

Some derivations based on the anomalous Langevin equation in Liouville space (i.e. Γ space) or its equivalent Liouville diffusion equation of time-reversal asymmetry are presented. The time rate of change, the balance equation, the entropy flow, the entropy production and the law of entropy increase of Gibbs nonequilibriurn entropy and Boltzmann nonequilibrium entropy are rigorously derived and presented here. Furthermore, a nonlinear evolution equation of Gibbs nonequilibrium entropy density and Boltzmann nonequilibrium entropy density is first derived. The evolution equation shows that the change of nonequilibrium entropy density originates from not only drift, but also typical diffusion and inherent source production. Contrary to conventional knowledge, the entropy production density σ?0 everywhere for all the inbomogeneous systems far from equilibrium cannot be proved. Conversely, σ may be negative in some local space of such systems.     

Irreversibility and the role of an instrument in the functional formulation of classical mechanics

We analyze the role of an instrument in the recently proposed functional formulation of classical mechanics, whose basic equation is the Liouville equation. Its solution has the delocalization (spreading) property, which is interpreted as irreversibility on the microlevel. We show that the reversible and recurrent dynamics for a particle can be observed by tracking the particle dynamics using instruments, but repeated measurements inevitably lead to a heat release and an increase in the entropy of the instrument. The irreversible behavior is thus transported from the system under study to the instrument, which is also a physical system.     

Stochastic dynamics and Boltzmann hierarchy. I

Stochastic dynamics corresponding to the Boltzmann hierarchy is constructed. The Liouville-Itô equations are obtained, from which we derive the Boltzmann hierarchy regarded as an abstract evolution equation. We construct the semigroup of evolution operators and prove the existence of solutions of the Boltzmann hierarchy in the space of sequences of integrable and bounded functions. On the basis of these results, we prove the existence of global solutions of the Boltzmann equation and the existence of the Boltzmann-Grad limit for an arbitrary time interval.     

Stochastic dynamics and Boltzmann hierarchy. III

Stochastic dynamics corresponding to the Boltzmann hierarchy is constructed. The Liouville-Itô equations are obtained, from which we derive the Boltzmann hierarchy regarded as an abstract evolution equation. We construct the semigroup of evolution operators and prove the existence of solutions of the Boltzmann hierarchy in the space of sequences of integrable and bounded functions. On the basis of these results, we prove the existence of global solutions of the Boltzmann equation and the existence of the Boltzmann-Grad limit for an arbitrary time interval.     

Stochastic dynamics and Boltzmann hierarchy. II

Stochastic dynamics corresponding to the Boltzmann hierarchy is constructed. The Liouville-Itô equations are obtained, from which we derive the Boltzmann hierarchy regarded as an abstract evolution equation. We construct the semigroup of evolution operators and prove the existence of solutions of the Boltzmann hierarchy in the space of sequences of integrable and bounded functions. On the basis of these results, we prove the existence of global solutions of the Boltzmann equation and the existence of the Boltzmann-Grad limit for an arbitrary time interval.     

Factorization of a hierarchy of the lattice soliton equations from a binary Bargmann symmetry constraint

Staring from a discrete spectral problem, a hierarchy of the lattice soliton equations is derived. It is shown that each lattice equation in resulting hierarchy is Liouville integrable discrete Hamiltonian system. The binary nonlinearization of the Lax pairs and the adjoint Lax pairs of the resulting hierarchy is discussed. Each lattice soliton equation in the resulting hierarchy can be factored by an integrable symplectic map and a finite-dimensional integrable system in Liouville sense. Especially, factorization of a discrete Kdv equation is given.     

New integrable lattice hierarchies and associated properties

New positive hierarchy and negative hierarchy of Liouville integrable lattice equation and their Hamiltonian structure are generated by use of Tu model. Then, some properties of the obtained equation hierarchies are discussed.     

Methods for derivation of the stochastic Boltzmann hierarchy

We consider different methods for the derivation of the stochastic Boltzmann hierarchy corresponding to the stochastic dynamics that is the Boltzmann-Grad limit of the Hamiltonian dynamics of hard spheres. Solutions of the stochastic Boltzmann hierarchy are the Boltzmann-Grad limit of solutions of the BBGKY hierarchy of hard spheres in the entire phase space. A new concept of reduced distribution functions corresponding to the stochastic dynamics are introduced. They take into account the contribution of the hyperplanes of lower dimension where stochastic point particles interact with one another. The solutions of the Boltzmann equation coincide with one-particle distribution functions of the stochastic Boltzmann hierarchy and are represented by integrals over the hyperplanes where the stochastic point particles interact with one another.     

Functional classical mechanics and rational numbers

The notion of microscopic state of the system at a given moment of time as a point in the phase space as well as a notion of trajectory is widely used in classical mechanics. However, it does not have an immediate physical meaning, since arbitrary real numbers are unobservable. This notion leads to the known paradoxes, such as the irreversibility problem. A “functional” formulation of classical mechanics is suggested. The physical meaning is attached in this formulation not to an individual trajectory but only to a “beam” of trajectories, or the distribution function on phase space. The fundamental equation of the microscopic dynamics in the functional approach is not the Newton equation but the Liouville equation for the distribution function of the single particle. The Newton equation in this approach appears as an approximate equation describing the dynamics of the average values and there are corrections to the Newton trajectories. We give a construction of probability density function starting from the directly observable quantities, i.e., the results of measurements, which are rational numbers.     

Stochastic Dynamics and Hierarchy for the Boltzmann Equation with Arbitrary Differential Scattering Cross Section

The stochastic dynamics for point particles that corresponds to the Boltzmann equation with arbitrary differential scattering cross section is constructed. We derive the stochastic Boltzmann hierarchy the solutions of which outside the hyperplanes of lower dimension where the point particles interact are equal to the product of one-particle correlation functions, provided that the initial correlation functions are products of one-particle correlation functions. A one-particle correlation function satisfies the Boltzmann equation. The Kac dynamics in the momentum space is obtained.__________Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 12, pp. 1629–1653, December, 2004.     

A Generalized Coordinate-Momentum Representation in Quantum Mechanics

We obtain a one-parameter family of (q, p)-representations of quantum mechanics; the Wigner distribution function and the distribution function we previously derived are particular cases in this family. We find the solutions o the evolution equations or the microscopic classical and quantum distribution functions in the form of integrals over paths in a phase space. We show that when varying canonical variables in the Green’s function of the quantum Liouville equation, we must use the total increment o the action functional in its path-integral representation, whereas in the Green’s function of the classical Liouville equation, the linear part o the increment is sufficient. A correspondence between the classical and quantum schemes holds only under a certain choice of the value of the distribution family parameter. This value corresponds to the distribution unction previously found.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 3, pp. 401–416, June, 2005.     

Spatially-Homogeneous Boltzmann Hierarchy as Averaged Spatially-Inhomogeneous Stochastic Boltzmann Hierarchy

We introduce the stochastic dynamics in the phase space that corresponds to the Boltzmann equation and hierarchy and is the Boltzmann–Grad limit of the Hamiltonian dynamics of systems of hard spheres. By the method of averaging over the space of positions, we derive from it the stochastic dynamics in the momentum space that corresponds to the space-homogeneous Boltzmann equation and hierarchy. Analogous dynamics in the mean-field approximation was postulated by Kac for the explanation of the phenomenon of propagation of chaos and derivation of the Boltzmann equation.     

Classical and quantum transport in random media

We study in this article the transport of particles in time-dependent random media, in the so-called weak coupling limit. We show the convergence of a Liouville equation to a Fokker–Planck equation. We also obtain the semi-classical limit of Schrödinger equations. This limit is described by a linear Boltzmann equation. In both cases, the ratio between a typical time scale and the scale of the media determines whether the limit diffusion and the collision process are elastic or not.     

新的Liouville完全可积的广义Burgers方程族及其Hamilton结构

基于屠格式,从一个新的等谱问题,本文获得了一族广义Ｂｕｒｇｅｒｓ 方程及其Ｈａｍ ｉｌｔｏｎ 结构．最后证明了该族方程是Ｌｉｏｕｖｉｌｌｅ 完全可积的,并且有无穷多个彼此对合的公共守恒密度     

HAMILTONIAN STRUCTURE AND INFINITE NUMBER OF CONSERVATION LAWS FOR THE COUPLED DISCRETE KdV EQUATIONS

A new discrete isospectral problem is introduced, from which the coupled discrete KdV hierarchy is deduced and is written in its Hamiltonian form by means of the trace identity.It is shown that each equation in the resulting hierarchy is Liouville integrable. Furthermore,an infinite number of conservation laws are shown explicitly by direct computation.     

非线性玻尔兹曼方程的傅里叶谱方法

玻尔兹曼方程作为空气动理学中最基本的方程之一,是连接微观牛顿力学和宏观连续介质力学的重要桥梁.该方程描述了一个由大量粒子组成的复杂系统的非平衡态时间演化:除了基本的输运项,其最重要的特性是粒子间的相互碰撞由一个高维,非局部且非线性的积分算子来描述,从而给玻尔兹曼方程的数值求解带来非常大的挑战.在过去的二十年间,基于傅里叶级数的谱方法成为了数值求解玻尔兹曼方程的一种很受欢迎且有效的确定性算法.这主要归功于谱方法的高精度及它可以被快速傅里叶变换加速的特质.本文将回顾玻尔兹曼方程的傅里叶谱方法,具体包括方法的导出,稳定性和收敛性分析,快速算法,以及在一大类基于碰撞的空气动理学方程中的推广.     

Burgers and Kadomtsev-Petviashvili hierarchies: A functional representation approach

Functional representations of (matrix) Burgers and potential Kadomtsev-Petviashvili (pKP) hierarchies (and others), as well as some corresponding Bäcklund transformations, can be obtained surprisingly simply from a “discrete” functional zero-curvature equation. We use these representations to show that any solution of a Burgers hierarchy is also a solution of the pKP hierarchy. Moreover, the pKP hierarchy can be expressed in the form of an inhomogeneous Burgers hierarchy. In particular, this leads to an extension of the Cole-Hopf transformation to the pKP hierarchy. Furthermore, these hierarchies are solved by the solutions of certain functional Riccati equations.     

一个Lie代数的子代数及其相关的两类Loop代数

本文构造了Lie代数A2的一个子代数A2,通过选取恰当的基元阶数得到相应的一个loop代数A2,由此设计一个等谱问题,利用屠格式得到了一个新的Liouville可积的Hamilton方程族.作为其约化情形,得到了一个非线性有理分式型演化方程.再由一个矩阵变换,得到了换位运算与A2等价的Lie代数A1的一个子代数A1,将A1再扩展成一个新的高维loop代数G,利用G获得了所得方程族的一类扩展可积系统.