The time-local view of nonequilibrium statistical mechanics. I. Linear theory of transport and relaxation

The various approaches to nonequilibrium statistical mechanics may be subdivided into convolution and convolutionless (time-local) ones. While the former, put forward by Zwanzig, Mori, and others, are used most commonly, the latter are less well developed, but have proven very useful in recent applications. The aim of the present series of papers is to develop the time-local picture (TLP) of nonequilibrium statistical mechanics on a new footing and to consider its physical implications for topics such as the formulation of irreversible thermodynamics. The most natural approach to TLP is seen to derive from the Fourier-Laplace transform ) of pertinent time correlation functions, which on the physical sheet typically displays an essential singularity at z= and a number of macroscopic and microscopic poles in the lower half-plane corresponding to long- and short-lived modes, respectively, the former giving rise to the autonomous macrodynamics, whereas the latter are interpreted as doorway modes mediating the transfer of information from relevant to irrelevant channels. Possible implications of this doorway mode concept for socalled extended irreversible thermodynamics are briefly discussed. The pole structure is used for deriving new kinds of generalized Green-Kubo relations expressing macroscopic quantities, transport coefficients, e.g., by contour integrals over current-current correlation functions obeying Hamiltonian dynamics, the contour integration replacing projection. The conventional Green-Kubo relations valid for conserved quantities only are rederived for illustration. Moreover, may be expressed by a Laurent series expansion in positive and negative powers ofz, from which a rigorous, general, and straightforward method is developed for extracting all macroscopic quantities from so-called secularly divergent expansions of as obtained from the application of conventional many-body techniques to the calculation of . The expressions are formulated as time scale expansions, which should rapidly converge if macroscopic and microscopic time scales are sufficiently well separated, i.e., if lifetime (memory) effects are not too large.     

A unified approach for deriving kinetic equations in nonequilibrium statistical mechanics. I. Exact results

A unified method for deriving exact kinetic equations for dynamical quantities of a many-body system is presented. The well-known results of Mori and Zwanzig are recovered as special cases. Furthermore, it is shown that they differ only by the way in which the system is prepared at the initial time. Connections between this method and others recently developed are also discussed.     

The Langevin dynamics of vibrated powders

We present a microscopic theory of the relaxational behaviour of a granular pile submitted to vibration, elucidating the different roles of collective and independent-particle relaxation. We write down and solve Langevin equations for these processes, which have an explicit coupling. The analysis of the solution in terms of independent-particle and collective relaxations provides a consistent framework for the interpretation of experimental results.     

A unified approach for deriving kinetic equations in nonequilibrium statistical mechanics. II. Approximate results

The exact form for the kinetic equation derived by Mori, Fujisaka, and Shigematsu (MFS) is used to obtain several approximations better suited to be compared with macroscopic transport equations. Three approximations are discussed, namely, those known as the diagonal, the slow process, and the Markovian. The corresponding results are emphasized and their relationship is established. In particular, the Kramers-Moyal expansion for the Markovian kinetic equation is obtained from a microscopic basis.     

On the nonequilibrium statistical mechanics of a binary mixture. II. The transport coefficients

This paper is devoted to the study of the hydrodynamic stage of a two-component dense fluid. Starting from the BBGKY hierarchy obtained earlier, we first derive the expressions for the generalized fluxes. We proceed to set up the generalized kinetic equations, using Bogoliubov's functional assumption. Then we solve these equations by means of a Chapman-Enskog method. The generalized expressions for the transport coefficients are thus obtained. All our results are independent of the existence of density expansions of the relevant quantities.     

Intersecting disks (and spheres) and statistical mechanics. II. the hard-disk system

A system of hard disks (diameter ) is studied by considering the corresponding exclusion disks (radius ). Thus, the results of previous papers on overlapping disks can be used for a geometrical analysis of the system. The concept of fluctuating free volume is compared with free volume theory. Finally, a series of computer experiments on hard disks is analyzed geometrically, especially with respect to the fluid-solid transition.     

The quasiclassical Langevin equation and its application to the decay of a metastable state and to quantum fluctuations

We report on investigations on the consequences of the quasiclassical Langevin equation. This Langevin equation is an equation of motion of the classical type where, however, the stochastic Langevin force is correlated according to the quantum form of the dissipation-fluctuation theorem such that ultimately its power spectrum increases linearly with frequency. Most extensively, we have studied the decay of a metastable state driven by a stochastic force. For a particular type of potential well (piecewise parabolic), we have derived explicit expressions for the decay rate for an arbitrary power spectrum of the stochastic force. We have found that the quasiclassical Langevin equation leads to decay rates which are physically meaningful only within a very restricted range. We have also studied the influence of quantum fluctuations on a predominantly deterministic motion and we have found that there the predictions of the quasiclassical Langevin equations are correct.     

On the nonequilibrium statistical mechanics of a binary mixture. I. The distribution functions

In this paper, we obtain the generalization of the BBGKY hierarchy for a binary mixture of chemically neutral particles. Using modified boundary conditions different from the ones proposed by Bogoliubov, we solve the hierarchy, and obtain explicitly the set of two-particle distribution functions for the several species of the mixture, up to first order in the density.     

Statistical mechanics of nonlinear wave equations. 3. Metric transitivity for hyperbolic sine-gordon

McKean and Vaninsky proved that the canonical measuree ^–H d Q d P based upon the Hamiltonian of the wave equation ² Q/t ² - ² Q/x ² +f(Q) = 0 with restoring forcef(Q)=F'(Q) is preserved by the associated flow ofQ andP =Q , and they conjectured that metric transitivity prevails,always on the whole line, and likewise on the circleunless f(Q)=Q orf(Q)=shQ. Here, the metric transitivity is proved for the whole line in the second case. The proof employs the beautiful d'Alembert formula of Krichever.     

Analogue simulation of quantum mechanical systems

An extension of the technique of analogue simulation to the treatment of quantum mechanical systems, based on an analogue variant of the method of stochastic quantization, is reported. The analogue stochastic quantization (ASQ) technique is introduced by application to the quantum harmonic oscillator, a particularly simple system for which all the answers are already known. ASQ measurements of the lowest eigenvalues and eigenfunctions of the latter system are presented and compared with theoretical predictions. The future potential of the ASQ technique in relation to some more complicated quantum systems of topical interest is discussed.     

Studies in nonlinear stochastic processes. I. Approximate solutions of nonlinear stochastic differential equations by the method of statistical linearization

A method based on a variational procedure is presented which provides simple and useful approximate solutions to a wide variety of nonlinear stochastic differential equations. This method of statistical linearization is most successful when the stochasticity of the differential equation is due to excitations which are normally distributed or harmonic with random phase. Effects due to deviations from normality can be corrected for in a systematic fashion. Comments regarding existence and uniqueness are given and some error bounds arising from the use of statistical linearization are computed.Supported by the National Science Foundation under Grants NSF GP 32031X and MPS72-04353 A03.     

Master and Langevin equations for electromagnetic dissipation and decoherence of density matrices

We set up a forward - backward path integral for a point particle in a bath of photons to derive a master equation for the density matrix which describes electromagnetic dissipation and decoherence. We also derive the associated Langevin equation. As an application, we recalculate the Wigner-Weisskopf formula for the natural line width of an atomic state at zero temperature and find, in addition, the temperature broadening caused by the decoherence term. Our master equation also yields the correct Lamb shift of atomic levels. The two equations may have applications to dilute interstellar gases or to few-particle systems in cavities. Received 29 November 2000 and Received in final form 11 February 2001     

Corner transfer matrices of the chiral Potts model. II. The triangular lattice

We consider a two-dimensional edge-interaction model satisfying the star-triangle relations. For the triangular lattice, the corner transfer matrices are functions of three rapidities: we show that they possess various factorization properties and satisfy certain equations. We indicate how these equations can be solved for the Ising model. We then consider the three-state chiral Potts model and obtain low-temperature solutions to the equations. The conjectured formula for the order parameter (the spontaneous magnetization) is verified to one more order in a series expansion.     

Bringing statistical mechanics into chemistry: The early scientific work of Karl F. Herzfeld

The physicist Karl F. Herzfeld was instrumental in introducing the techniques of statistical mechanics to physical chemists in the years between 1920 and 1935. Herzfeld's work included an early derivation of the rate constant for a first order reaction. This was similar to and fifteen years earlier than Eyring's calculation of the rate constant. Herzfeld also carried out the first calculation of a relaxation equation for vibrational energy transfer. Herzfeld's success lay in his classical approach to physics, which, it is suggested, made his work accessible to chemists.     

Corner transfer matrices of the eight-vertex model. II. The Ising model case

In a previous paper certain corner transfer matrices were defined. It was conjectured that for the zero-field, eight-vertex model these matrices have a very simple eigenvalue spectrum. In this paper these conjectures are verified for the case when the eight-vertex model reduces to two independent and identical square-lattice Ising models. The Onsager-Yang expression for the magnetization follows immediately.