The stochastic Boltzmann equation and hydrodynamic fluctuations

Based on the assumption of a kinetic equation in space, a stochastic differential equation of the one-particle distribution is derived without the use of the linear approximation. It is just the Boltzmann equation with a Langevin-fluctuating force term. The result is the general form of the linearized Boltzmann equation with fluctuations found by Bixon and Zwanzig and by Fox and Uhlenbeck. It reduces to the general Landau-Lifshitz equations of fluid dynamics in the presence of fluctuations in a similar hydrodynamic approximation to that used by Chapman and Enskog with respect to the Boltzmann equation.This work received financial support from the Alexander von Humboldt Foundation.     

Note on Boltzmann Langevin equation and hydrodynamic fluctuations

A microscopic derivation presented of the generalization of the linearized Boltzmann equation with the Langevin fluctuation force, which has earlier been postulated by Bixon and Zwanzig and by Fox and Uhlenbeck in their kinematical discussions on the hydrodynamic fluctuations.     

The Boltzmann equation

An abstract form of the spatially non-homogeneous Boltzmann equation is derived which includes the usual, more concrete form for any kind of potential, hard or soft, with finite cutoff. It is assumed that the corresponding gas is confined to a bounded domain by some sort of reflection law. The problem then considered is the corresponding initial-boundary value problem, locally in time.Two proofs of existence are given. Both are constructive, and the first, at least, provides two sequences, one converging to the solution from above, the other from below, thus producing, at the same time as existence, approximations to the solution and error bounds for the approximation.The solution is found within a space of functions bounded by a multiple of a Maxwellian, and, in this space, uniqueness is also proved.Research supported, in part, by the National Research Council of Canada (NRC A8560)     

The Boltzmann equation with a soft potential

The initial value problem for the linearized spatially-homogeneous Boltzmann equation has the form ?f/?t+Lf=0 withf(ξ,t=0) given. The linear operatorL operates only on the ξ variable and is non-negative, but, for the soft potentials considered here, its continuous spectrum extends to the origin. Thus one cannot expect exponential decay forf, but in this paper it is shown thatf decays likee ^?λ t ^β with β<1. This result will be used in Part II to show existence of solutions of the initial value problem for the full nonlinear, spatially dependent problem for initial data that is close to equilibrium.     

The Boltzmann equation with a soft potential

The results of Part I are extended to include linear spatially periodic problems-solutions of the initial value are shown to exist and decay like . Then the full non-linear Boltzmann equation with a soft potential is solved for initial data close to equilibrium. The non-linearity is treated as a perturbation of the linear problem, and the equation is solved by iteration.Supported by the National Science Foundation under Grant Nos. MCS78-09525 and MCS76-07039 and by the United States Army under Contract No. DAAG29-75-C-0024     

The generalized Boltzmann equation

From the BBGKY-hierarchy of the kinetic equations taking into account three possible levels of scales, connected with the mean time of particle interactions, mean time between collisions and the hydrodynamic time, the generalized Boltzmann equation is derived. The generalized H-theorem is proven.     

An exactly soluble Boltzmann equation with persistent scattering

A model Boltzmann equation with persistence of state is described and shown to lead to both exact, similarity solutions and general solutions as moment expansions. These, unlike any previously known, are parametrized over the whole range from “small-scattering” to “diffuse-scattering” extremes.     

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].     

One-dimensional Boltzmann equation with a three-body collision term

It is shown that a linearized one-dimensional Boltzmann equation with a certain simple three-body collision term is trivially soluable.     

The p-q model Boltzmann equation

The “p-q” model earlier introduced by the authors to describe persistent scattering under a scalar Boltzmann equation is here examined in detail. After deriving the scattering kernel and exhibiting its properties we obtain moment and similarity solutions and show how the model effectively parametrizes all intermediate conditions between the extremes of diffusion-like “small-scattering” and the strong-collisional limit of “diffuse-scattering” characteristic of earlier, more restrictive models. Both continuous and discrete-variable versions of the model are discussed and shown to be straightforwardly interrelated. Our derivations, carried out in natural energy-like variables, parallel those given recently by Ernst and Hendriks using transform methods.     

Quantum corrections for Boltzmann equation

We present the lowest order quantum correction to the semiclassical Boltzmann distribution function, and the equation satisfied by this correction is given. Our equation for the quantum correction is obtained from the conventional quantum Boltzmann equation by explicitly expressing the Planck constant in the gradient approximation, and the quantum Wigner distribution function is expanded in powers of Planck constant, too. The negative quantum correlation in the Wigner distribution function which is just the quantum correction terms is naturally singled out, thus obviating the need for the Husimi’s coarse grain averaging that is usually done to remove the negative quantum part of the Wigner distribution function. We also discuss the classical limit of quantum thermodynamic entropy in the above framework. Supported by the National Natural Science Foundation of China (Grant No. 10404037) and the Scientific Research Fund of GUCAS (Grant No. 055101BM03)     

Discrete Boltzmann equation for microfluidics

We propose a discrete Boltzmann model for microfluidics based on the Boltzmann equation with external forces using a single relaxation time collision model. Considering the electrostatic interactions in microfluidics systems, we introduce an equilibrium distribution function that differs from the Maxwell-Boltzmann distribution by an exponential factor to represent the action of an external force field. A statistical mechanical approach is applied to derive the equivalent external acceleration force exerting on the lattice particles based on a mean-field approximation, resulting from the electro-static potential energy and intermolecular potential energy between fluid-fluid and fluid-substrate interactions.     

A density-corrected quantum Boltzmann equation

Binary correlations are a recognized part of the pair density operator, but the influence of binary correlations on the singlet density operator is usually not emphasized. Here free motion and binary correlations are taken as independent building blocks for the structure of the nonequilibrium singlet and pair density operators. Binary correlations are assumed to arise from the collision of twofree particles. Together with the first BBGKY equation and a retention of all terms that are second order in gas density, a generalization of the Boltzmann equation is obtained. This is an equation for thefree particle density operator rather than for the (full) singlet density operator. The form for the pressure tensor calculated from this equation reduces at equilibrium to give the correct (Beth-Uhlenbeck) second virial coefficient, in contrast to a previous quantum Boltzmann equation, which gave only part of the quantum second virial coefficient. Generalizations to include higher-order correlations and collision types are indicated.     

Iterative solution of the Boltzmann equation

We define an iterative scheme to solve the nonlinear Boltzmann equation. Conservation rules are maintained at each iterative step. We apply this method to a spatially uniform and isotropic velocity distribution function on the Maxwell and very-hard-particle models. A particular example is evaluated and results are compared with the exact solution. It shows to be a very fast convergent approach.     

Two generalizations of the Boltzmann equation

We connect two different extensions of Boltzmann's kinetic theory by requiring the same stationary solution. Non-extensive statistics can be produced by either using corresponding collision rates nonlinear in the one-particle densities or equivalently by using nontrivial energy composition rules in the energy conservation constraint part. Direct transformation formulas between key functions of the two approaches are given.     

Temperature transform of the Boltzmann equation

We define an integral transform of the energy distribution function for an isotropic and homogeneous diluted gas. It may be interpreted as a linear combination of equilibrium states with variable temperatures. We show that the temporal evolution features of the distribution function are determined by the singularities of this temperature transform. We compare the relaxation to the equilibrium process for Maxwell and very hard-particle interaction models, finding many basic discrepancies. Finally, we formulate an alternative approach, which is given by anN-pole approximation with a clear physical meaning.Fellow of the Conselho Nacional de Desenvolvimento Cientifico e Tecnólogico, Brazil.     

Nonisotropic solutions of the Boltzmann equation

We consider the relaxation to equilibrium of a spatially uniform Maxwellian gas. We expand the solution of the nonlinear Boltzmann equation in a truncated series of orthogonal functions. We integrate numerically the equation for non-isotropic initial conditions. For certain simple conditions we find interesting proximity effects and other transient relaxation phenomena at thermal energies. Furthermore, we define a resummation of the orthogonal expansion which is more convenient than the original one for the numerical analysis of the relaxation process.     

Origin of the Landau-Lifshitz hydrodynamic fluctuations in nonequilibrium systems and a new method for reducing the Boltzmann equation

The Landau-Lifshitz fluctuating fluxes in fluctuating hydrodynamics are derived from the deterministic Boltzmann equation with the aid of a reduction method developed by Fujisaka and Mori. Thus it is shown that the hydrodynamic fluctuations innonequilibrium systems are generated by the reduction of variables from the-space distribution function to its five momentum moments, i.e., the hydrodynamic variables. This differs from the Bixon-Zwanzig and Fox-Uhlenbeck theories, in which the Landau-Lifshitz fluctuating fluxes are derived from the molecular fluctuating force in thestochastic Boltzmann-Langevin equation, which is, however, negligible in nonequilibrium systems. Thus the present method improves the Chapman-Enskog reduction method so as to include the hydrodynamic fluctuations generated by the reduction of variables.Supported in part by the Scientific Research Fund of the Ministry of Education.