On the Boltzmann equation for the Lorentz gas

We consider the Boltzmann-Grad limit for the Lorentz, or wind-tree, model. We prove that if is a fixed configuration of scatterer centers belonging to a set of full measure with respect to the Poisson distribution with parameter >0, then the evolution of an initial a.c. particle density tends in the Boltzmann-Grad limit to the solution of the Boltzmann equation for the model. As an intermediate step we prove that the process of the free path lengths and impact parameters induced by the Lebesgue measure on a small region tends to a limiting independent process.     

Exact and approximate generalized diffusion equation for the Lorentz gas

The Lorentz model of a rarefied gas is used for testing two different methods of solving the Boltzman kinetic equation. It is shown that the Zwanzig-Mori method gives the generalized diffusion equation which agrees with the exact Hauge solution. The Zubarev-Khonkin approach gives a series expansion of the exact generalized diffusion coefficient. Their method is compared with the Chapman-Enskog method.     

Kinetic equation for a dense soliton gas

We propose a general method to derive kinetic equations for dense soliton gases in physical systems described by integrable nonlinear wave equations. The kinetic equation describes evolution of the spectral distribution function of solitons due to soliton-soliton collisions. Owing to complete integrability of the soliton equations, only pairwise soliton interactions contribute to the solution, and the evolution reduces to a transport of the eigenvalues of the associated spectral problem with the corresponding soliton velocities modified by the collisions. The proposed general procedure of the derivation of the kinetic equation is illustrated by the examples of the Korteweg-de Vries and nonlinear Schr?dinger (NLS) equations. As a simple physical example, we construct an explicit solution for the case of interaction of two cold NLS soliton gases.     

Kinetic equation for a weakly nonideal Bose gas

A kinetic equation for a weakly nonideal Bose gas is derived in the range where the thermal wavelength (λ_T) is less than or on the order of the mean interatomic distance (r ₀). The case when λ_T > r ₀ is discussed.     

Kinetic equation for a weakly interacting classical electron gas

A kinetic equation is derived for the two-time phase space correlation function in a dilute classical electron gas in equilibrium. The derivation is based on a density expansion of the correlation function and the resummation of the most divergent terms in each order in the density. It is formally analogous to the ring summation used in the kinetic theory of neutral fluids. The kinetic equation obtained is consistent to first order in the plasma parameter and is the generalization of the linearized Balescu-Guersey-Lenard operator to describe spatially inhomogeneous equilibrium fluctuations. The importance of consistently treating static correlations when deriving a kinetic equation for an electron gas is stressed. A systematic derivation as described here is needed for a further generalization to a kinetic equation that includes mode-coupling effects. This will be presented in a future paper.     

Kinetic equation in the kinetic region of the dilute and nonuniform electron plasma

Mori's scaling method is used to derive the kinetic equation for a dilute, nonuniform electron plasma in the kinetic region where the space-time cutoff (b, t _c) satisfies _Dbl _f , _D t _{c f} , with _D the Debye length, _D ^–1= _p the plasma frequency, andl _f and _f the mean free path and time, respectively. The kinetic equation takes account of the nonuniformity of the order ofl _f and _D for the single-and the two-particle distribution function, respectively. Thus the Vlasov term associated with the two-particle distribution function is retained. This kinetic equation is deduced from the kinetic equation in the coherent region obtained by Morita, Mori, and Tokuyama, where the space-time cutoff of the coherent region satisfies _Dbr ₀, _Dt _{c 0}, withr ₀ the Landau length and ₀ the corresponding time scale.     

Lorentz gas shear viscosity via nonequilibrium molecular dynamics and Boltzmann's equation

When nonequilibrium molecular dynamics is used to impose isothermal shear on a two-body periodic system of hard disks or spheres, the equations of motion reduce to those describing a Lorentz gas under shear. In this shearing Lorentz gas a single particle moves, isothermally, through a spatially periodic shearing crystal of infinitely massive scatterers. The curvilinear trajectories are calculated analytically and used to measure the dilute Lorentz gas viscosity at several strain rates. Simulations and solutions of Boltzmann's equation exhibit shear thinning resembling that found inN-body nonequilibrium simulations. For the three-dimensional Lorentz gas we obtained an exact expression for the viscosity which is valid at all strain rates. In two dimensions this is not possible due to the anisotropy of the scattering.     

Kinetic perturbation theory for dilute gases

We prove, for two classes of smooth, repulsive interparticle potentials ø(r) = ø₀(r) + δø₁(r), that the collision integ rals of the linearized Boltzmann equation are analytic functions of λ in the neighborhood of λ = 0. It then follows, for example, that the first Enskog approximation for the transport coefficients can be represented by a power series in λ.     

Periodic orbit expansions for the Lorentz gas

We apply the periodic orbit expansion to the calculation of transport, thermodynamic, and chaotic properties of the finite-horizon triangular Lorentz gas. We show numerically that the inverse of the normalized Lyapunov number is a good estimate of the probability of an individual periodic orbit. We investigate the convergence of the periodic orbit expansion and compare it with the convergence of the cycle expansions obtained from the Ruelle dynamical -function. For this system with severe pruning we find that applying standard convergence acceleration schemes to the periodic orbit expansion is superior to the dynamical -function approach. The averages obtained from the periodic orbit expansion are within 8% of the values obtained from direct numerical time and ensemble averaging. None of the periodic orbit expansions used here is computationally competitive with the standard simulation approaches for calculating averages. However, we believe that these expansion methods are of fundamental importance, because they give a direct route to the phase space distribution function.     

Measures with infinite Lyapunov exponents for the periodic Lorentz gas

We study invariant measures for the periodic Lorentz gas which are supported on the set of points with infinite Lyapunov exponents. We construct examples of such measures which are measures of maximal entropy and ones which are not.     

Kinetic equation for an extremely rarefied gas of fast particles interacting with a crystal

The statistical equations of Bogolyubov are used to examine a two-component system consisting of a gas and a crystal. A kinetic equation describing the elastic interaction of an extremely rarefied gas of high-energy particles with a crystalline structure is systematically derived from these equations.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 79–83, April, 1977.     

Invariance principle for the stochastic Lorentz lattice gas

We prove scaling to nondegenerate Brownian motion for the path of a test particle in the stochastic Lorentz lattice gas on ^d under a weak ergodicity assumption on the scatterer distribution. We prove that recurrence holds almost surely ind2. Transience ind3 remains open.     

A dynamical partition function for the Lorentz gas

In this paper we introduce a dynamically defined partition function for the Lorentz gas and investigate its connection with the classical ensembles and the phase-space probability measure derived from periodic orbit expansions. Numerical evidence is presented to support the equivalence of these measures and to link them to the thermodynamic quantities for the Lorentz gas. This also suggests a new dynamical basis for the assumption of equala priori probabilities in the microcanonical ensemble.     

The velocity correlation function for the Lorentz gas

The results of variational solutions of the repeated ring and self-consistent repeated ring equations for the two-and three-dimensional overlapping Lorentz gas (LG), as formulated in a previous report, are presented. Calculations of the full velocity correlation function (VCF) for the 2D LG, including long-time tails, are compared with those from molecular dynamics. The trial functions chosen lead to predictions for the long-time tails that improve as the density of the scatterers is increased. At a value of 0.24 for* (= ², where is the density and the radius of scatterers), the self-consistent amplitudes of the long-time tail are within 40% of the molecular dynamics. A limited number of 3D results for the short-time behavior of the repeated ring VCF are presented. The 3D solutions agree with the molecular dynamics to within 10%.     

Quasi-relativistic Boltzmann equation for systems with short-distance forces

Systems are referred to as quasirelativistic if terms up to the order of v²/c² suffice to describe them; v is the particle velocity, c is the light velocity. Systems of neutral particles are considered with nonvanishing interaction forces at such short distances that the interaction delay can be ignored. Equations are derived for the correlation functions using the Lagrange function which is known in the quasirelativistic approximation; hence using the N. N. Bogolyubov method the quasirelativistic analog is obtained of the Boltzmann equation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, Vol. 17, No. 2, pp. 78–82, February, 1974.In conclusion the authors consider it their pleasant duty to express their thanks to N. N. Bogolyubov, B. L. Bonch-Bruevich, and N. A. Chernikov for discussing with them the preprint of this article [9].     

Asymptotic fields for the quantum nonlinear Schrödinger equation with attractive coupling

The quantum nonlinear Schrödinger equation with attractive coupling is considered through the quantum inverse scattering method. The asymptotic fields arising in this model are characterized in terms of scattering data operators.