The Phonon Boltzmann Equation, Properties and Link to Weakly Anharmonic Lattice Dynamics

For low density gases the validity of the Boltzmann transport equation is well established. The central object is the one-particle distribution function, f, which in the Boltzmann-Grad limit satisfies the Boltzmann equation. Grad and, much refined, Cercignani argue for the existence of this limit on the basis of the BBGKY hierarchy for hard spheres. At least for a short kinetic time span, the argument can be made mathematically precise following the seminal work of Lanford. In this article a corresponding program is undertaken for weakly nonlinear, both discrete and continuum, wave equations. Our working example is the harmonic lattice with a weakly nonquadratic on-site potential. We argue that the role of the Boltzmann f-function is taken over by the Wigner function, which is a very convenient device to filter the slow degrees of freedom. The Wigner function, so to speak, labels locally the covariances of dynamically almost stationary measures. One route to the phonon Boltzmann equation is a Gaussian decoupling, which is based on the fact that the purely harmonic dynamics has very good mixing properties. As a further approach the expansion in terms of Feynman diagrams is outlined. Both methods are extended to the quantized version of the weakly nonlinear wave equation.The resulting phonon Boltzmann equation has been hardly studied on a rigorous level. As one novel contribution we establish that the spatially homogeneous stationary solutions are precisely the thermal Wigner functions. For three phonon processes such a result requires extra conditions on the dispersion law. We also outline the reasoning leading to Fourier’s law for heat conduction.     

The Boltzmann equation with frictional force

The Euler equations with frictional force have been extensively studied. Since the Boltzmann equation is closely related to the equations of gas dynamics, we study, in this paper, the Boltzmann equation with frictional force when the external force is proportional to the macroscopic velocity. It is shown that smooth initial perturbation of a given global Maxwellian leads to a unique global-in-time classical solution which approaches to the global Maxwellian time asymptotically. The analysis is based on the macro-micro decomposition for the Boltzmann equation introduced in Liu et al. [Energy method for the Boltzmann equation, Physica D 188 (3-4) (2004) 178-192] and Liu and Yu [Boltzmann equation: micro-macro-decompositions and positivity of shock profiles, Comm. Math. Phys. 246(1) (2004) 133-179] through energy estimates.     

热力学“时间之箭”

讨论了微观力学可逆而宏观热力学过程不可逆的几种解释.     

LocalH-theorem for the revised Enskog equation

A simple derivation of the local H-theorem for the revised Enskog equation is presented.On leave of absence from the Institute of Theoretical Physics, University of Warsaw, 00-681 Warsaw, Poland.     

Generalized Fokker–Planck equations derived from generalized linear nonequilibrium thermodynamics

Recently, Compte and Jou derived nonlinear diffusion equations by applying the principles of linear nonequilibrium thermodynamics to the generalized nonextensive entropy proposed by Tsallis. In line with this study, stochastic processes in isolated and closed systems characterized by arbitrary generalized entropies are considered and evolution equations for the process probability densities are derived. It is shown that linear nonequilibrium thermodynamics based on generalized entropies naturally leads to generalized Fokker–Planck equations.     

On the H-theorem for polyatomic gases

The H-theorem for a classical gas of polyatomic molecules of arbitrarily complex structure is examined. A simple use of time reversal invariance of the equations of dynamics is used to circumvent the objections which were raised by Lorentz against Boltzmann's proof (nonexistence of inverse collisions).     

H-theorem for a mean field model describing coupled oscillator systems under external forces

This article studies the asymptotic behavior of solutions of Fokker–Planck equations describing mean field approximations of weakly coupled oscillator systems subjected to external forces. Using an H-theorem we show that transient probability densities converge to stationary ones. Furthermore, stability criteria are derived for the stationary solutions of these Fokker–Planck equations. The obtained results are applied to a model that combines the Haken–Kelso–Bunz model and the models of weakly coupled oscillators proposed by Winfree and Kuramoto. The stability criteria based on the H-theorem agree with those derived in our earlier analyses.     

Stationary state of a heated granular gas: Fate of the usual H-functional

We consider the characterization of the nonequilibrium stationary state of a randomly driven granular gas in terms of an entropy-production-based variational formulation. Enforcing spatial homogeneity, we first consider the temporal stability of the stationary state reached after a transient. In connection, two heuristic albeit physically motivated candidates for the nonequilibrium entropy production are put forward. It turns out that none of them displays an extremum for the stationary velocity distribution selected by the dynamics. Finally, the relevance of the relative Kullbach entropy is discussed.     

H-Theorems from Macroscopic Autonomous Equations

The H-theorem is an extension of the Second Law to a time-sequence of states that need not be equilibrium ones. In this paper we review and we rigourously establish the connection with macroscopic autonomy.If for a Hamiltonian dynamics for many particles, the macrostate evolves autonomously, then its entropy is non-decreasing as a consequence of Liouville's theorem. That observation, made since long, is here rigorously analyzed with special care to reconcile the application of Liouville's theorem (for a finite number of particles) with the condition of autonomous macroscopic evolution (sharp only in the limit of infinite scale separation); and to evaluate the presumed necessity of a semigroup property for the macroscopic evolution.     

Global weak solutions to the relativistic BGK equation

Abstract

In this paper, the global existence of weak solutions to the relativistic BGK model for the relativistic Boltzmann equation is analyzed. The proof relies on the strong compactness of the density, velocity, and temperature under minimal assumptions on the control of some moments of the initial condition together with the initial entropy.