统计物理的微观动力学基础

统计的基本出发点是研究系统具有的随机性,不同系统在不同情形下的宏观热力学性质起源于系统内部随机性的差异,通过对宏观热力学系统的微观非线性动力学进行研究探索，我们可以进一步更为深入地理解物态方程、相变等诸多的宏观热力学现象。本文通过哈密顿系统的非线性动力学研究，以及遍历性理论的动力学随机性研究对此问题进行了分析，研究表明，动力学系统的全局性混沌是系统统计成立的根本要素，系统的无限大自由度（热力学极限）已不是决定性的因素，人们可以在此基础上建立少自由度系统的统计力学及热力学。     

Positivity of entropy production in nonequilibrium statistical mechanics

We analyze different mechanisms of entropy production in statistical mechanics, and propose formulas for the entropy production ratee() in a state . When is steady state describing the long term behavior of a system we show thate()0, and sometimes we can provee()>0.     

Mean entropy of states in classical statistical mechanics

The equilibrium states for an infinite system of classical mechanics may be represented by states over AbelianC* algebras. We consider here continuous and lattice systems and define a mean entropy for their states. The properties of this mean entropy are investigated: linearity, upper semi-continuity, integral representations. In the lattice case, it is found that our mean entropy coincides with theKolmogorov-Sinai invariant of ergodic theory.     

Uncertainty relations for information entropy in wave mechanics

New uncertainty relations in quantum mechanics are derived. They express restrictions imposed by quantum theory on probability distributions of canonically conjugate variables in terms of corresponding information entropies. The Heisenberg uncertainty relation follows from those inequalities and so does the Gross-Nelson inequality.     

Nonextensive statistical mechanics of q-bosons based on the q-deformed entropy

Based on the q ↔ q⁻¹ symmetric deformed entropy, we develop a general framework for nonextensive statistical mechanics of ensembles of q-deformed systems. Applying this doubly deformed formalism to q-bosons, a correction to the Planck law is evaluated in the weak deformation regime and its properties are discussed. It is found that at high temperature the dominant part of the correction comes from the deformation of the oscillator dynamics, whereas at low temperature the deformation of the entropy gives a leading contribution. This suggests the nonextensive approach to q-deformed ensembles might be important at low temperature.     

Dynamical entropy of quasi-local algebras in quantum statistical mechanics

We study the dynamical entropy in the sense of Connes, Narnhofer, and Thirring of automorphisms on quasi-local algebras in quantum statistical mechanics. We extend their Kolmogorov-Sinai type theorem for AF-algebras to quasi-local algebras which are not necessarily AF-algebras.Work supported in part by Korean Science Foundation and the Basic Science Research Program, Ministry of Education, 1991     

New microscopic approach to the statistical mechanics of chemical association

We consider a new approach to the statistical mechanics of chemical association, A+BAB. In principle it is exact, and its thermodynamic basis goes back to Gibbs, but its statistical mechanical implementation in terms of molecular models does not appear to have been given before. For practical computations the success of the method hinges on our ability to calculate the free energy for a mixture of A, B, and AB species, where the species concentrations are regarded as independent. We illustrate the method by analysis of some simple limiting cases.     

非广延统计力学与完全开放系统的统计分布

简介了非广延统计力学的Tsallis统计,用其计算了理想气体;推导出了以含有非广延熵常数的Shannon熵为基础和以Tsallis熵为基础的非广延统计力学的完全开放系统的统计分布及计算热力学量的公式;讨论表明:Tsallis熵对应的统计分布及计算热力学量的公式在非广延参量q→1时,完全过渡到了Shannon熵对应的形式.     

On the statistical mechanics of constrained biophysical systems

A well-known class of biophysical models, first introduced by Kerner, is shown to admit a convenient Hamiltonian formulation in which motion through the phase space of system variables involves explicit constraints. To treat the macroscopic properties of such models, we develop an ensemble theory of systems subjected to phase space constraints. For such systems we obtain a generalized Hamiltonian statistical mechanics which preserves much of the structure and efficacy of the corresponding physical theory. In a first application of the method, we recover Kerner's original biological ensemble as a special case involving information optimality and conservative biosystems.Funding for this project was provided through the generosity of the National Research Council of Canada. The work reported here was carried out while CJL was a graduate scholar of the National Research Council.     

Formulation of statistical mechanics for chaotic systems

We formulate the statistical mechanics of chaotic system with few degrees of freedom and investigated the quartic oscillator system using microcanonical and canonical ensembles. Results of statistical mechanics are numerically verified by considering the dynamical evolution of quartic oscillator system with two degrees of freedom.      

Random matrix theory and the statistical mechanics of disordered systems

A new method, that systematically combines results of random matrix theory and the usual statistical mechanics, is described to study thermodynamic properties of disordered systems. Two exactly solvable models are examined in this formulation to illustrate the usefulness of this method for systems described by random as well as non-random Hamiltonian.     

小系统的非平衡统计力学与随机热力学

热力学是一个古老的课题,古典热力学以宏观的具有大粒子数的系统为研究对象,自17世纪以来,科学家们构建了热力学的完备公理化体系。将热力学推广至小系统是近三十年来的研究前沿。文章介绍小系统的非平衡统计力学以及小系统的随机热力学。作为研究案例,利用时间依赖的谐振子势场控制单个粒子来构造随机热机的类卡诺循环,并发现该热机最大功率对应的效率等于1-,其中T_c和T_h分别对应于低温热库和高温热库的温度。     

Nonequilibrium statistical mechanics of systems with long-range interactions

Systems with long-range (LR) forces, for which the interaction potential decays with the interparticle distance with an exponent smaller than the dimensionality of the embedding space, remain an outstanding challenge to statistical physics. The internal energy of such systems lacks extensivity and additivity. Although the extensivity can be restored by scaling the interaction potential with the number of particles, the non-additivity still remains. Lack of additivity leads to inequivalence of statistical ensembles. Before relaxing to thermodynamic equilibrium, isolated systems with LR forces become trapped in out-of-equilibrium quasi-stationary states (qSSs), the lifetime of which diverges with the number of particles. Therefore, in the thermodynamic limit LR systems will not relax to equilibrium. The qSSs are attained through the process of collisionless relaxation. Density oscillations lead to particle–wave interactions and excitation of parametric resonances. The resonant particles escape from the main cluster to form a tenuous halo. Simultaneously, this cools down the core of the distribution and dampens out the oscillations. When all the oscillations die out the ergodicity is broken and a qSS is born. In this report, we will review a theory which allows us to quantitatively predict the particle distribution in the qSS. The theory is applied to various LR interacting systems, ranging from plasmas to self-gravitating clusters and kinetic spin models.     

To fluctuation theory in statistical mechanics of nonextensive systems

New inequalities and uncertainty relations are derived for physical quantities being measured simultaneously in equilibrium and nonequilibrium quantum nonextensive systems and depending on the value of a quantum analog of the Havrda-Charvat-Daroczy entropy. Superoperators are introduced and equations for the density and distribution operators are derived. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 14–19, October, 2005.     

Classical mechanics and entropy

This note addresses a problem of nineteenth century applied mathematics—is it possible in the context of Hamiltonian mechanics to define a functionS of the generalized coordinates and momenta which is monotonically increasing along orbits? The question is of interest, because, for a sytem not in thermodynamic equilibrium, entropy should increase strictly monotonically along an orbit, and a negative answer implies that mechanical principles different from those of Hamiltonian mechanics must be introduced to explain thermodynamics. This note answers the question rigorously for Hamiltonian systems confined to an invariant region of finite volume in phase space; it is not possible to define a continuous function which increases monotonically along orbits. An appendix gives a translation of an 1889 paper of Poincaré addressing the same issue.