Nonequilibrium statistical mechanics II. Basic equations

The derivation and property of the fundamental equation of statistical mechanics are analyzed. The conditions for which a macroscopic system can be described with the help of average values are investigated. It is shown how the law of increasing entropy is obtained from the general formulation of the theory.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 14–17, December, 1981.     

Nonlinear transport equations in statistical mechanics

A simple projection operator method is developed for computing nonequilibrium ensemble averages for systems that are close to a state of local equilibrium. The formalism used here is a straight-forward generalization of the Mori-Zwanzig techniques used in linear response theory and it avoids many of the technical difficulties associated with time-dependent projection operators. The method is used here to derive gradient expansions for nonequibrium average values about their values in local equilibrium. This is used to derive the nonlinear hydrodynamic equations for a pure fluid, to Burnett order.     

Generalized canonical statistical operator and entropy of partially coherent radiation

Using information-theoretical concepts and Glauber's correlation functions, the statistical operator of an Nth-order partially coherent radiation field is constructed. Special consideration is given to the cases N = 1 and N = 2. The entropy for N = 1 yields the early formula of von Laue. Our results for N > 1 shed new light on the relationship between thermodynamical order of a radiation field, as measured by its entropy, and optical coherence, especially in the quantum-mechanical case.     

Solitary wave solutions of selective nonlinear diffusion-reaction equations using homogeneous balance method

An auto-Bäcklund transformation derived in the homogeneous balance method is employed to obtain several new exact solutions of certain kinds of nonlinear diffusion-reaction (D-R) equations. These equations arise in a variety of problems in physical, chemical, biological, social and ecological sciences.     

Exact solutions of some coupled nonlinear diffusion-reaction equations using auxiliary equation method

Travelling and solitary wave solutions of certain coupled nonlinear diffusion-reaction equations have been constructed using the auxiliary equation method. These equations arise in a variety of contexts not only in biological, chemical and physical sciences but also in ecological and social sciences.     

Generalized gradient expansions in quantum transport equations

Gradient expansions in quantum transport equations of a Kadanoff-Baym form have been reexamined. In a consistent approach the expansion should be performed also inside the self-energy in the scattering integrals of these equations. In the first perturbation order this internal expansion gives new correction terms to the generalized Boltzmann equation. These correction terms are found here for several typical systems. Possible corrections to the theory of a linear response to weak electric fields are also discussed.     

Nonequilibrium charge susceptibility and dynamical conductance: identification of scattering processes in quantum transport

We calculate the nonequilibrium charge transport properties of nanoscale junctions in the steady state and extend the concept of charge susceptibility to the nonequilibrium conditions. We show that the nonequilibrium charge susceptibility is related to the nonlinear dynamical conductance. In spectroscopic terms, both contain the same features versus applied bias when charge fluctuation occurs in the corresponding electronic resonances. However, we show that, while the conductance exhibits features at biases corresponding to inelastic scattering with no charge fluctuations, the nonequilibrium charge susceptibility does not. We suggest that measuring both the nonequilibrium conductance and charge susceptibility in the same experiment will permit us to differentiate between different scattering processes in quantum transport.     

Nonequilibrium mesoscopic transport: a genealogy

Models of nonequilibrium quantum transport underpin all modern electronic devices, from the largest scales to the smallest. Past simplifications such as coarse graining and bulk self-averaging served well to understand electronic materials. Such particular notions become inapplicable at mesoscopic dimensions, edging towards the truly quantum regime. Nevertheless a unifying thread continues to run through transport physics, animating the design of small-scale electronic technology: microscopic conservation and nonequilibrium dissipation. These fundamentals are inherent in quantum transport and gain even greater and more explicit experimental meaning in the passage to atomic-sized devices. We review their genesis, their theoretical context, and their governing role in the electronic response of meso- and nanoscopic systems.     

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.     

Chaotic and fractal properties of deterministic diffusion-reaction processes

We study the consequences of deterministic chaos for diffusion-controlled reaction. As an example, we analyze a diffusive-reactive deterministic multibaker and a parameter-dependent variation of it. We construct the diffusive and the reactive modes of the models as eigenstates of the Frobenius-Perron operator. The associated eigenvalues provide the dispersion relations of diffusion and reaction and, hence, they determine the reaction rate. For the simplest model we show explicitly that the reaction rate behaves as phenomenologically expected for one-dimensional diffusion-controlled reaction. Under parametric variation, we find that both the diffusion coefficient and the reaction rate have fractal-like dependences on the system parameter. (c) 1998 American Institute of Physics.     

Some new solitary and travelling wave solutions of certain nonlinear diffusion-reaction equations using auxiliary equation method

Attempts are made to look for the soliton content in the exact solutions of certain types of nonlinear diffusion-reaction (DR) equations with the quadratic and cubic nonlinearities. Such equations may arise in a variety of contexts in physical problems. In this Letter using the auxiliary equation method, some new solitary and travelling wave solutions of such nonlinear DR equations are obtained in a very general form. Several interesting special cases of these general solutions are also discussed.     

Application of the projection operator method in equilibrium statistical physics

In the framework of the Liouville equation formalism the projecton operator method is used in equilibrium statistical physics.     

Book review:Statistical physics II: Nonequilibrium statistical mechanics

Journal of Statistical Physics -     

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.     

Nonequilibrium processes in condensed media: Part 1. Experimental studies in light of nonlocal transport theory

Based on experimental research in shock loading of solid-state materials it is shown that among the important dynamic characteristics of the process, like spatial-temporal mass velocity profiles of shock waves, are the mass velocity variation, velocity defect, and structural instability threshold recorded in real time. Analysis of these characteristics depending on the strain rate, target thickness, and structural state of material demonstrates that conventional approaches of continuum mechanics fail to provide their adequate interpretation and simulation of shock wave processes. A new concept of shock wave processes in condensed media is proposed. The concept, being based on nonlocal nonequilibrium transport theory, allows describing the transition from elastic to hydrodynamic response of a medium depending on the loading rate and time. A nonstationary elastoplastic wave model is proposed for describing the relaxation of an elastic precursor and formation of a retarded plastic front during the wave propagation in a medium with regard to structural evolution. Analysis of the experimental data shows that the division of stresses and strains into elastic and plastic components is incorrect for shock loading.