Differentiability of spatially homogeneous solutions of the Boltzmann equation in the non maxwellian case

The non linear Boltzmann equation is studied and differentiable solutions are shown to exist if the initial datum is suitably chosen     

The homogeneous Boltzmann hierarchy and statistical solutions to the homogeneous Boltzmann equation

An existence and uniqueness result for the homogeneous Boltzmann hierarchy is proven, by exploiting the statistical solutions to the homogeneous Boltzmann equation.     

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.     

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.     

Exact solutions to the time-dependent Lorentz gas Boltzmann Equation: The approach to hydrodynamics

New exact solutions to the time-dependent Lorentz gas Boltzmann equation are presented for two classes of nonequilibrium initial value problems: thedecay of localized disturbances and theresponse to applied electric fields. These exact results are used to gain some insight into the crossover of the nonequilibrium state from the early-timekinetic regime to the late-timehydrodynamic regime.     

非线性漂移的Fokker-Planck方程的近似非定态解

研究了由高斯白噪声和色噪声作用下的非线性动力学系统的不稳定态演化问题. 在弱噪声极限下, 运用本征值本征矢理论得到了非定态解ρ(x, t)的近似表达式; 分析了色噪声自关联时间τ, 强度α对ρ(x, t)以及对一、二阶矩的影响. 数值模拟发现: 1)t在一定范围内, ρ(x, t)是变量x和t的单调函数, 且随τ的增大而增大, 反之, 随α的增大而减小; 2)一阶矩是τ和α的单调函数, 但二阶矩却是非单调函数, 在参数影响下发生了相变现象. 关键词： 奥恩斯坦-乌伦贝克过程 本征值 本征矢 非定态解     

Compensation of the optically induced Lorentz force in a homogeneous optical medium

It is shown on the basis of different approaches that the density of optically induced forces applied to a homogeneous optical medium embedded in a simplest 1D structure in a form of a plane optical resonator is equal to zero. In particular, the density forces calculated on the base of the energetic approach, where no assumptions about physical nature of optically induced forces are used, are also equal to zero. At the same time the same forces calculated by means of the approach based on the Lorentz force are different from zero. A conclusion is derived that there is an additional type of optically induced force which compensates the Lorentz density forces. Thus, the Lorentz force approach used for calculation of the density of optically induced force is inconsistent.     

Propagation of sound waves through a spatially homogeneous but smoothly time-dependent medium

The propagation of sound through a spatially homogeneous but non-stationary medium is investigated within the framework of fluid dynamics. For a non-vortical fluid, especially, a generalized wave equation is derived for the (scalar) potential of the fluid velocity distribution in dependence of the equilibrium mass density of the fluid and the sound wave velocity. A solution of this equation for a finite   transition period τ$\tau$ is determined in terms of the hypergeometric function for a phenomenologically realistic, sigmoidal change of the mass density and sound wave velocity. Using this solution, it is shown that the energy flux of the sound wave is not conserved but increases always   for the propagation through a non-stationary medium, independent of whether the equilibrium mass density is increased or decreased. It is found, moreover, that this amplification of the transmitted wave arises from an energy exchange with the medium and that its flux is equal to the (total) flux of the incident and the reflected wave. An interpretation of the reflected wave as a propagation of sound backward in time is given in close analogy to Feynman and Stueckelberg for the propagation of anti-particles. The reflection and transmission coefficients of sound propagating through a non-stationary medium is analyzed in more detail for hypersonic waves with transition periods τ$\tau$ between 15 and 200 ps as well as the transformation of infrasound waves in non-stationary oceans.     

Exact solution of the Boltzmann equation in the homogeneous color conductivity problem

An exact solution of the Boltzmann equation for a binary mixture of colored Maxwell molecules is found. The solution corresponds to a nonequilibrium homogeneous steady state created by a nonconservative external force. Explicit expressions for the moments of the distribution function are obtained. By using information theory, an approximate velocity distribution function is constructed, which is exact in the limits of small and large field strengths. Comparison is made between the exact energy flux and the one obtained from the information theory distribution.     

Moment inequalities for the boltzmann equation and applications to spatially homogeneous problems

Some inequalities for the Boltzmann collision integral are proved. These inequalities can be considered as a generalization of the well-known Povzner inequality. The inequalities are used to obtain estimates of moments of the solution to the spatially homogeneous Boltzmann equation for a wide class of intermolecular forces. We obtain simple necessary and sufficient conditions (on the potential) for the uniform boundedness of all moments. For potentials with compact support the following statement is proved: if all moments of the initial distribution function are bounded by the corresponding moments of the MaxwellianA exp(−Bv ²), then all moments of the solution are bounded by the corresponding moments of the other MaxwellianA ₁ exp[−B ₁(t)v ²] for anyt > 0; moreoverB(t) = const for hard spheres. An estimate for a collision frequency is also obtained.     

On exact spatially nonuniform solutions of the Boltzmann equation in the generalized Kac model

It is shown that the Boltzmann equation in the generalized Kac model can be cast into a nonlinear differential equation of the second order for a certain class of functions f(t, x, v). Some explicit solutions of this equation are pointed out.     

Boltzmann equation with fluctuations

From the Liouville equation, by the method of multiple-time-scales, a generalized Boltzmann-equation with fluctuations is obtained on the statistical considerations of the randomness of the many-particle correlations in the macroscopic picture. These fluctuations lead to anH theorem in which theH function decreases, with fluctuations with time toward equilibrium. These fluctuations furnish a source for a random force term introduced by Fox and Uhlenbeck in the Boltzmann equation.     

Steady state convergence acceleration of the generalized lattice Boltzmann equation with forcing term through preconditioning

Several applications exist in which lattice Boltzmann methods (LBM) are used to compute stationary states of fluid motions, particularly those driven or modulated by external forces. Standard LBM, being explicit time-marching in nature, requires a long time to attain steady state convergence, particularly at low Mach numbers due to the disparity in characteristic speeds of propagation of different quantities. In this paper, we present a preconditioned generalized lattice Boltzmann equation (GLBE) with forcing term to accelerate steady state convergence to flows driven by external forces. The use of multiple relaxation times in the GLBE allows enhancement of the numerical stability. Particular focus is given in preconditioning external forces, which can be spatially and temporally dependent. In particular, correct forms of moment projections of source/forcing terms are derived such that they recover preconditioned Navier–Stokes equations with non-uniform external forces. As an illustration, we solve an extended system with a preconditioned lattice kinetic equation for magnetic induction field at low magnetic Prandtl numbers, which imposes Lorentz forces on the flow of conducting fluids. Computational studies, particularly in three-dimensions, for canonical problems show that the number of time steps needed to reach steady state is reduced by orders of magnitude with preconditioning. In addition, the preconditioning approach resulted in significantly improved stability characteristics when compared with the corresponding single relaxation time formulation.     

Exact propagator of the Fokker-Planck equation with a space-dependent diffusion coefficient and a time-dependent mean-reverting force

We have investigated the algebraic structure of the Fokker-Planck equation with a variable diffusion coefficient and a time-dependent mean-reverting force. Such a model could be useful to study the general problem of a Brownian walker with a space-dependent diffusion coefficient. We also show that this model is related to the Fokker-Planck equation with a constant diffusion coefficient and a time-dependent anharmonic potential of the form V(x, t) = ?a(t)x ² + b ln x, which has been widely applied to model different physical and biological phenomena, e.g. the study of neuron models and stochastic resonance in monostable nonlinear oscillators. Using the Lie algebraic approach we have derived the exact diffusion propagators for the Fokker-Planck equations associated with different boundary conditions, namely (i) the case of a single absorbing barrier, and (ii) the case of two absorbing barriers. These exact diffusion propagators enable us to study the time evolution of the corresponding stochastic systems. Received 23 October 2001 and Received in final form 24 December 2001     

Solutions of the Boltzmann equation for Maxwell interactions and singular angle-dependent cross sections

We consider the spatially homogeneous and isotropic Boltzmann distribution function in the case of nonisotropic, binary cross sections inversely proportional to the relative speed of the colliding particles. Further, we allow the angle dependence of the differential cross section() to be singular in the forward direction ( 0). We assume (), d < which includes the case of a Maxwellian interaction. We explicitly show how to construct the solutions of the Boltzmann equation, study their properties, and obtain for a class of solutions sufficient conditions for their existence at any positive time value. We extend the formalism to the more general case of arbitrary dimensionality. We observe an effect noticed previously by Krook, Wu, and Tjon in other models of the Boltzmann equations-namely, for special initial distributions, we find solutions which exhibit an excess of higher energy particles at later time.     

Boundary-free propagation with the time-dependent Schrodinger equation

We present two methods that allow for the efficient numerical propagation of continuum wave packets to large times. Time-dependent solutions of the Schrodinger equation that include continuum components are numerically challenging to solve because the wave packet travels, spreads, and acquires a spatial phase gradient. The methods we propose account for these kinematic effects analytically in general and numerically tractable schemes.     

Brans-Dicke cosmology with time-dependent cosmological term

Berman and Som's solution for a Brans-Dicke cosmology with time-dependent cosmological term, Robertson-Walker metric, perfect fluid, and perfect gas law of state solves the horizon, homogeneity, and isotropy problems without requiring any unnatural fine tuning in the very early universe, thus being an alternative model to inflation. The model also does not need recourse to quantum cosmology, and solves the flatness and magnetic monopole problems.     

Brans-Dicke models with time-dependent cosmological term

More general solutions than those presented by Bertolami are deduced in the Brans-Dicke cosmology, endowed with a time-dependent cosmological term, for a Robertson-Walker metric and a perfect fluid obeying the perfect gas law of state.     

On the master equation with time-dependent transition probabilities

The asymptotic behaviour of solutions of the master equation with time-dependent transition probabilities by means of the Kullback information is investigated. We assume that there are no absorbing states.