Nonequilibrium statistical mechanics of preasymptotic dispersion

Turbulent transport in bulk-phase fluids and transport in porous media with fractal character involve fluctuations on all space and time scales. Consequently one anticipates constitutive theories should be nonlocal in character and involve constitutive parameters with arbitrary wavevector and frequency dependence. We provide here a nonequilibrium statistical mechanical theory of transport which involves both diffusive and convective mixing (dispersion) at all scales. The theory is based on a generalization of classical approaches used in molecular hydrodynamics and on time-correlation functions defined in terms of nonequilibrium expectations. The resulting constitutive laws are nonlocal and constitutive parameters are wavevector and frequency dependent. All results reduce to their convolution-Fickian quasi-Fickian, or Fickian counterparts in the appropriate limits.     

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.     

An alternative approach to calculate the density of states in nonextensive statistical mechanics

A relation between the generalized partition function (Tsallis) and density of states is established by using the method of integral transform which enables reducing some integral equations into the algebraic equations. Inverse Mellin transformation of this equation gives the density of states. Similar relation is also hold the for standard partition function (Boltzmann-Gibbs) and the density of states. Using these relations, we recover the density of states for the classical ideal gas within both statistics.     

Surface-driven instability and enhanced relaxation in the dynamics of a nonequilibrium interface

We determine the stability of a nonequilibrium interface between two coexisting solid phases in the presence of a weak external field. Starting at the coarsegrained (Cahn-Hilliard) level, we use the method of matched asymptotics to derive the macroscopic interfacial dynamics. We then show that the external field leads to an instability due to flux along the interface, in contrast with the more common Mullins-Sekerka type instability, which involves fluxes normal to the interface. We also find that the external field produces an important modification of the Gibbs-Thomson relation. With these results, we perform the linear stability analysis for an approximately flat interface. If the field is tangent to the interface, the modification of the Gibbs-Thomson relation is important and the interface is stabilized. If the field is normal to the interface, the surface flux is important, and the effect can be stabilizing or destabilizing, but the orientational dependence is opposite what would be obtained if the Mullins-Sekerka instability dominates. Numerical simulations are performed to study the effect of the surface current and are in agreement with our analytical results.     

Anomalous Diffusion Limit Induced on a Kinetic Equation

Using a compactness argument based on the velocity averaging lemma of Golse et al., it is shown that the limiting behavior of a kinetic (linearized BGK) gas model confined between two plates with Maxwell boundary conditions, when the distance between the plates goes to zero, under a suitable anomalous scaling, is diffusive. We do not require the use of central limit theorems as in the method of Börgers et al.     

Spatial correlations in bounded nonequilibrium fluid systems

We study the influence of boundaries on the equal-time thermal correlations in a three-dimensional fluid maintained under a constant temperature gradient. Within the confines of the model for an idealized fluid bounded by two infinite, parallel walls, we show that it is crucial to retain the unbounded spatial components in the problem so that the solutions approach meaningful results as we move the walls infinitely far apart. In addition, we consider a composite system by including the dynamics of the walls, and we investigate the conditions for the relevant physical parameters under which the details of wall dynamics may be neglected by employing the simple boundary condition T=0.     

Time-dependent density functional theory for quantum transport

The rapid miniaturization of electronic devices motivates research interests in quantum transport. Recently time-dependent quantum transport has become an important research topic. Here we review recent progresses in the development of time-dependent density-functional theory for quantum transport including the theoretical foundation and numerical algorithms. In particular, the reducedsingle electron density matrix based hierarchical equation of motion, which can be derived from Liouville–von Neumann equation, is reviewed in details. The numerical implementation is discussed and simulation results of realistic devices will be given.     

Biased Diffusion with Correlated Noise

The diffusion of hard-core particles subject to a global bias is described by a nonlinear, anisotropic generalization of the diffusion equation with conserved, local noise. Using renormalization group techniques, we analyze the effect of an additional noise term, with spatially long-ranged correlations, on the long-time, long-wavelength behavior of this model. Above an upper critical dimension d _LR, the long-ranged noise is always relevant. In contrast, for d<d _LR, we find a weak noise regime dominated by short-range noise. As the range of the noise correlations increases, an intricate sequence of stability exchanges between different fixed points of the renormalization group occurs. Both smooth and discontinuous crossovers between the associated universality classes are observed, reflected in the scaling exponents. We discuss the necessary techniques in some detail since they are applicable to a much wider range of problems.     

Smooth Dynamics and New Theoretical Ideas in Nonequilibrium Statistical Mechanics

This paper reviews various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics. We adopt a new point of view which has emerged progressively in recent years, and which takes seriously into account the chaotic character of the microscopic time evolution. The emphasis is on nonequilibrium steady states rather than the traditional approach to equilibrium point of view of Boltzmann. The nonequilibrium steady states, in presence of a Gaussian thermostat, are described by SRB measures. In terms of these one can prove the Gallavotti–Cohen fluctuation theorem. One can also prove a general linear response formula and study its consequences, which are not restricted to near-equilibrium situations. At equilibrium one recovers in particular the Onsager reciprocity relations. Under suitable conditions the nonequilibrium steady states satisfy the pairing theorem of Dettmann and Morriss. The results just mentioned hold so far only for classical systems; they do not involve large size, i.e., they hold without a thermodynamic limit.     

The initial distribution in classical statistical mechanics

This paper presents arguments proving that several kinds of experimental preparation procedures for classical systems lead in certain limits to initial distributions that are functions only of macroscopic variables.This research was supported by the U.S. Air Force Office of Scientific Research under Contract F44620-72-C-0072.     

On the statistical mechanics approach in the random matrix theory: Integrated density of states

We consider the ensemble of random symmetricn×n matrices specified by an orthogonal invariant probability distribution. We treat this distribution as a Gibbs measure of a mean-field-type model. This allows us to show that the normalized eigenvalue counting function of this ensemble converges in probability to a nonrandom limit asn and that this limiting distribution is the solution of a certain self-consistent equation.     

Spontaneous synchronisation and nonequilibrium statistical mechanics of coupled phase oscillators

Spontaneous synchronisation is a remarkable collective effect observed in nature, whereby a population of oscillating units, which have diverse natural frequencies and are in weak interaction with one another, evolves to spontaneously exhibit collective oscillations at a common frequency. The Kuramoto model provides the basic analytical framework to study spontaneous synchronisation. The model comprises limit-cycle oscillators with distributed natural frequencies interacting through a mean-field coupling. Although more than forty years have passed since its introduction, the model continues to occupy the centre stage of research in the field of non-linear dynamics and is also widely applied to model diverse physical situations. In this brief review, starting with a derivation of the Kuramoto model and the synchronisation phenomenon it exhibits, we summarise recent results on the study of a generalised Kuramoto model that includes inertial effects and stochastic noise. We describe the dynamics of the generalised model from a different yet a rather useful perspective, namely, that of long-range interacting systems driven out of equilibrium by quenched disordered external torques. A system is said to be long-range interacting if the inter-particle potential decays slowly as a function of distance. Using tools of statistical physics, we highlight the equilibrium and nonequilibrium aspects of the dynamics of the generalised Kuramoto model, and uncover a rather rich and complex phase diagram that it exhibits, which underlines the basic theme of intriguing emergent phenomena that are exhibited by many-body complex systems.     

Action-angle variables in quantum statistical mechanics

Utilizing the facts (i) that the number of particles in the many-boson system is conserved and (ii) that the Hamiltonian is Hermitian, a new set of variables comprising action and angle variables has been introduced. These variables are conjugate in the mean and provide a rigorous approach to introducing phase variables for total-number-conserving many-boson systems.Lecture given at the Saha Institute of Nuclear Physics, Calcutta, June 1971.     

RNA分子折叠的统计力学

RNA分子折叠对决定分子(基因)水平上的生命活动是至关重要的.文章对RNA分子折叠(特别是二级结构的折叠)的热力学和动力学性质作一简单回顾与介绍.因为静电相互作用在RNA折叠过程中的特殊重要作用, 文章作者对RNA(与DNA)折叠的电相互作用单独作较为详细的讨论.     

RNA分子折叠的统计力学

RNA分子折叠对决定分子（基因）水平上的生命活动是至关重要的．文章对RNA分子折叠（特别是二级结构的折叠）的热力学和动力学性质作一简单回顾与介绍．因为静电相互作用在RNA折叠过程中的特殊重要作用，文章作者对RNA（与DNA）折叠的电相互作用单独作较为详细的讨论．     

Phase transitions in a driven lattice gas in two planes

We report on a Monte Carlo study of ordering in a nonequilibrium system. The system is a lattice gas that comprises two equal, parallel square lattices with stochastic particle-conserving irreversible dynamics. The particles are driven along a principal direction under the competition of the heat bath and a large, constant external electric field. There is attraction only between particles on nearest-neighbor sites within the same lattice. Particles may jump from one plane to the other; therefore, density fluctuations have an extra mechanism to decay and build up. It helps to obtain the steady-state accurately. Spatial correlations decay with distance according to a power law at high enough temperature, as for the ordinary two-dimensional case. We find two kinds of nonequilibrium phase transitions. The first one has a critical point for half occupation of the lattice, and seems to be related to the anisotropic phase transition reported before for the plane. This transition becomes discontinuous for low enough density. The difference of density between the planes changes discontinuously for any density at a lower temperature. This seems to correspond to a phase transition that does not have a counterpart in equilibrium nor in the two-dimensional nonequilibrium case.     

Entropy Production in Nonlinear,Thermally Driven Hamiltonian Systems

We consider a finite chain of nonlinear oscillators coupled at its ends to two infinite heat baths which are at different temperatures. Using our earlier results about the existence of a stationary state, we show rigorously that for arbitrary temperature differences and arbitrary couplings, such a system has a unique stationary state. (This extends our earlier results for small temperature differences.) In all these cases, any initial state will converge (at an unknown rate) to the stationary state. We show that this stationary state continually produces entropy. The rate of entropy production is strictly negative when the temperatures are unequal and is proportional to the mean energy flux through the system     

Nonequilibrium statistical mechanics of finite classical systems-II

The work of the previous paper is applied to the study of weakly interacting systems. Either by quasilinear techniques or by analyzing the perturbation series for the smoothed probability density, it is possible to derive a master equation equivalent to that of Brout and Prigogine without requiring the size of the system to become infinite. The properties of this equation are discussed. The equation is self-consistent provided the interactions are weak enough; however, examination of higher terms in the perturbation series shows that their effect might make the master equation invalid for times longer than that taken by a typical particle to cross the containing vessel. In many physical cases, the relaxation time will be shorter than this; also, further studies may show the higher terms to be less important than they seem.Formerly at Department of Applied Mathematics and Theoretical Physics, University of Cambridge, England.     

Nonequilibrium statistical mechanics of finite classical systems-I

It is pointed out that the fine-grained probability density of statistical mechanics is of interest only through coarse-grained densities—integrals over nonzero volumes of phase space. This suggests the definition of a smoothed probability density: the unsmoothed density convoluted with a kernel having a small spread aroundzero velocity. If this kernel is of Gaussian form, the smoothed density satisfies a closed and exact equation for its evolution differing from the Liouville equation by the addition of one term. This equation is applied to the simple example of a noninteracting system. We need make no assumption about the size of the system in our discussion, though if the system is large enough, the assumption that it is infinite gives the same results. Reduced distribution functions are then discussed, and a treatment of the Landau damping of electron plasma oscillations is given that is free from the usual difficulties occasioned by the breakdown of the linearization.Formerly at Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, England.