Irreversibility in the hard-spheres gas: H theorem and the Gibbs relation

An H-functional is suggested in terms of the single-particle distribution function governed by the Enskog kinetic equation. It is shown to be restricted to situations near the local equilibrium state. A generalized H-theorem valid for all initial situations is established from the point of view of a master equation for the hard-spheres gas. The thermodynamics of irreversible processes is then discussed in relation to the basic foundations of the kinetic theory of dense gases.     

H-theorem for the hard-sphere gas

The approach to equilibrium of the hard-sphere gas is discussed from the master-equation point of view. AnH-theorem is established, which is valid for arbitrary initial conditions.     

H-theorem for a linear kinetic theory

A strongH-theorem is proved for the approximate linear kinetic theory of Bawzdziewicz and Cichocki, obtained by truncating a transformed hierarchy of evolution equations. For an ith truncation we define an entropy functional that is strictly increasing in time, unless the ith reduced distribution function depends on position coordinates only. It also follows that the only stationary solution of the linear kinetic theory is the equilibrium solution. In addition, we show that the usual symmetry properties of equilibrium time correlation functions are preserved by the approximate kinetic theory under consideration.On leave of absence from Institute of Physics, Szczecin University, Wielkopolska 15, Szczecin, Poland.     

Dissipative Linear Boltzmann Equation for Hard Spheres

We prove the existence and uniqueness of an equilibrium state with unit mass to the dissipative linear Boltzmann equation with hard-spheres collision kernel describing inelastic interactions of a gas particles with a fixed background. The equilibrium state is a universal Maxwellian distribution function with the same velocity as field particles and with a non-zero temperature lower than the background one. Moreover thanks to the H-Theorem we prove strong convergence of the solution to the Boltzmann equation towards the equilibrium.     

Potential energy in a three-dimensional vibrated granular medium measured by NMR imaging

We use molecular dynamics simulations to study the swelling of randomly end-cross-linked polymer networks in good solvent conditions. We find that the equilibrium degree of swelling saturates at Q(eq) approximately N(3/5)(e) for mean strand lengths &Nmacr;(s) exceeding the melt entanglement length N(e). The internal structure of the network strands in the swollen state is characterized by a new exponent nu = 0.72+/-0.02. Our findings can be rationalized by a Flory argument for a self-similar structure of mutually interpenetrating network strands, agree partially with the classical Flory-Rehner theory, and are in contradiction to de Gennes' c(*)-theorem.     

On solutions to the linear Boltzmann equation with general boundary conditions and infinite-range forces

This paper considers the linear space-inhomogeneous Boltzmann equation in a convex, bounded or unbounded bodyD with general boundary conditions. First, mildL ¹-solutions are constructed in the cutoff case using monotone sequences of iterates in an exponential form. Assuming detailed balance relations, mass conservation and uniqueness are proved, together with anH-theorem with formulas for the interior and boundary terms. Local boundedness of higher moments is proved for soft and hard collision potentials, together with global boundedness for hard potentials in the case of a nonheating boundary, including specular reflections. Next, the transport equation with forces of infinite range is considered in an integral form. Existence of weakL ¹-solutions are proved by compactness, using theH-theorem from the cutoff case. Finally, anH-theorem is given also for the infinite-range case.     

A Purely Probabilistic Representation for the Dynamics of a Gas of Particles

The aim of the present paper is to give a purely probabilistic account for the approach to equilibrium of classical and quantum gas. The probability function used is classical. The probabilistic dynamics describes the evolution of the state of the gas due to unary and binary collisions. A state change amounts to a destruction in a state and the creation in another state. Transitions probabilities are splittled into destructions terms, denoting the random choice of the colliding particle(s), and creation terms, describing the allocation of the same particle(s) on the final state(s). While the destruction term is the same for all types of particles, the creation one depends upon a parameter bound to the interpraticle correlation. The transition probabilities give rise to a homogeneous Markov chain. The equilibrium distributions satisfy the principle of detailed balance. Relaxation times depend upon the interparticle correlation. Relationships with the Ehrenfest urn model, Brillouin unified method, ensemble interpretation, and quantum H-theorem are considered too.     

Zur Boltzmanngleichung

The existence theory for the nonlinear Boltzmann equation is discussed for an infinite region in the spatially homogeneous case. We show that the solution is given by a nonlinear contraction semigroup. It is found that theH-theorem holds and that the system approaches equilibrium.     

General H-Theorem for Hard Spheres

The maximum entropy formalism is used to investigate the growth of entropy (H-theorem) for an isolated system of hard spheres in an external potential under general boundary geometry. Assuming that only correlations of a finite number of particles are controlled and the rest maximizes entropy, we obtain an H-theorem for such a system The limiting cases such as the modified Enskog equation and linear kinetic theory are discussed.     

Entropy Function Approach to the Lattice Boltzmann Method

In this paper, we present the construction of the Lattice Boltzmann method equipped with the H-theorem. Based on entropy functions whose local equilibria are suitable to recover the Navier–Stokes equations in the framework of the Lattice Boltzmann method, we derive a collision integral which enables simple identification of transport coefficients, and which circumvents construction of the equilibrium. We discuss performance of this approach as compared to the standard realizations.     

LocalH-theorem for a kinetic variational theory

A localH-theorem is derived for a recently proposed extension of Enskog kinetic theory to a dense model fluid composed of particles with interactions extending beyond a hard core.On leave from: Katedra Fizyki, Uniwersytetu Szczecinskiego, 70-451 Szczecin, Poland.     

Towards transport theory of hadron gases

An important role of hadron resonances for determining the characteristics of hadron gases is argued. A kinetic theory model of hadron gas is developed. A classical, nonquantum, distribution function of a resonance is defined with the help of the profile function being an analogue of the mass shell delta function of stable particles. The Boltzmann equation is generalized to include the resonance decay and resonance formation processes. To determine the unknown profile function, the transition rates are assumed to satisfy the bilateral normalization or the detailed balance condition. The profile function is expressed through the resonance formation cross section and the decay width. The H-theorem is proved, and it is shown that the form of the equilibrium distribution function of a resonance coincides with that of a stable particle. Macroscopic equilibrium characteristics are studied. Significance of the resonance mass smearing effect is demonstrated.     

Statistical mechanics of a nonlinear stochastic model

A multivariable Fokker-Planck equation (FPE) is used to investigate the equilibrium and dynamical properties of a nonlinear stochastic model. The model displays a phase transition. The equilibrium distributions are found to be non-Gaussian; the deviation from Gaussian is especially significant near the transition point. To study the nonequilibrium behavior of the model, a self-consistent dynamic mean field (SCDMF) theory is derived and used to transform the FPE to a systematic hierarchy of equations for the cumulant moments of the time-dependent distribution function. These equations are numerically solved for a variety of initial conditions. During the time evolution of the system from an initial unstable equilibrium state to the final equilibrium state, three distinct time stages are found.Supported by a grant from the National Research Council of Canada (to RCD) and by the Sherman Fairchild Foundation (to RZ).Also Sherman Fairchild Distinguished Scholar, 1974–75, at the California Institute of Technology, where the early part of this research was done.     

Boltzmann equation for a dissociating gas

The incorporation of three-body collisions for dissociation/recombination into the Boltzmann equation is discussed. Conditions are assumed such that collisions are completed in the sense of scattering theory, so the collision operator is determined by scattering and reaction cross sections. The resulting equation has anH-theorem, and the equilibrium solution requires the law of mass action in addition to the Maxwellian dependence on momentum. A brief discussion is given of the normal solution and the transport coefficients.This paper is dedicated to Prof. E. G. D. Cohen on the occasion of his 65th birthday.     

液晶胆甾相中的分子短程关联

向列相液晶的二粒子集团理论被推广应用于研究胆甾相二维模型.手征性分子固定在三维简单立方晶格的格点上,而分子取向限制在二维.理论结果表明,平衡态螺旋波矢依赖于温度的变化,且存在胆甾相到向列相相变.通过考虑分子间短程关联,二粒子集团理论的数值结果较平均场理论更接近Monte Carlo模拟结果.     

Ideal magnetofluid turbulence in two dimensions

A continuum model of coherent structures in two-dimensional magnetohydro-dynamic turbulence is developed. These structures are macroscopic states which persist among the turbulent microscopic fluctuations, typically as magnetic islands with flow. They are modeled as statistical equilibrium states for the non-dissipative dynamics, which conserves energy and families of cross-helicity and flux integrals. The model predicts that from a given initial state an ideal magnetofluid will evolve into a final state having steady mean magnetic and velocity fields, and Gaussian local fluctuations in these fields. Excellent qualitative and quantitative agreement is found with the known results of direct numerical simulations. A rigorous justification of the theory is also provided, in the sense that the continuum model is derived from a lattice model in a fixed-volume, small-spacing limit. This construction uses the discrete Fourier transform to link the discretization ofx-space with the truncation ofk-space. Under the ergodic hypothesis and a separation-of-scales hypothesis, the lattice model is defined by a mean-field approximation to the Gibbs measure on the discretized phase space. A concentration property shows that this measure is equivalent to the microcanonical measure in the continuum limit.     

Entropy considerations in numerical simulations of non-equilibrium rarefied flows

Non-equilibrium rarefied flows are encountered frequently in supersonic flight at high altitudes, vacuum technology and in microscale devices. Prediction of the onset of non-equilibrium is important for accurate numerical simulation of such flows. We formulate and apply the discrete version of Boltzmann’s H-theorem for analysis of non-equilibrium onset and accuracy of numerical modeling of rarefied gas flows. The numerical modeling approach is based on the deterministic solution of kinetic model equations. The numerical solution approach comprises the discrete velocity method in the velocity space and the finite volume method in the physical space with different numerical flux schemes: the first-order, the second-order minmod flux limiter and a third-order WENO schemes. The use of entropy considerations in rarefied flow simulations is illustrated for the normal shock, the Riemann and the two-dimensional shock tube problems. The entropy generation rate based on kinetic theory is shown to be a powerful indicator of the onset of non-equilibrium, accuracy of numerical solution as well as the compatibility of boundary conditions for both steady and unsteady problems.     

The a-theorem

The c-theorem is a profound result applying to statistical mechanical theories or quantum field theories in two dimensions. Such theories may be described by a set of parameters which vary as we increase or decrease the scale on which we observe the system, until we reach a fixed point or critical point where the couplings have fixed values. The c-theorem defines a quantity (the c-function) which always increases (or is constant) with increasing scale and thereby gives a valuable insight into the ‘flows’ of the couplings between fixed points. The a-theorem is a proposed generalisation of the c-theorem to higher dimensions, especially four. In this article, we describe the c-theorem, starting in the simpler statistical mechanical context and then showing how in quantum field theory the theorem is most easily formulated in a curved spacetime. We then sketch how these concepts are applied in the more technically complex scenario of four dimensions.