A Markov Process Associated with a Boltzmann Equation Without Cutoff and for Non-Maxwell Molecules (original) (raw)
Related papers
2011
Consider the homogeneous Boltzmann equation for Maxwellian molecules. We provide a new representation for its solution in the form of expectation of a random probability measure M. We also prove that the Fourier transform of M is a conditional characteristic function of a sum of independent random variables, given a suitable sigma-algebra. These facts are then used to prove a CLT for Maxwellian molecules, that is the statement of a necessary and sufficient condition for the weak convergence of the solution of the equation. Such a condition reduces to the finiteness of the second moment of the initial distribution \mu_0. As a further application, we give a refinement of some inequalities, due to Elmroth, concerning the evolution of the moments of the solution.
Annali di Matematica Pura ed Applicata (1923 -), 2014
This paper proves the existence of weak solutions to the the spatially homogeneous Boltzmann equation for Maxwellian molecules, when the initial data are chosen from the space of all Borel probability measures on R 3 with finite second moments and the (angular) collision kernel satisfies a very weak cutoff condition, namely 1 −1 x 2 b(x)dx < +∞. Conservation of momentum and energy is also proved for these weak solutions, without resorting to any boundedness of the entropy.
Logan & Kac 1976 Fluctuations and the Boltzmann Equation I
Physical Review A, 1976
The evolution of a homogeneous dilute gas is treated as a Markov process in the complete set of K coarse-grained velocity states of all N particles. From the Siegert master equation for the process, a Fokker-Planck equation is derived which describes, in the limit N → ∞, the fluctuations in the occupation numbers ni(t), whose average behavior is governed by the (appropriately discretized) Boltzmann equation. The continuum limit K → ∞ corresponds to fluctuations in the usual molecular distribution function f(r v;t). On similar reasoning, a Fokker-Planck equation is obtained for the fluctuation process near equilibrium, where the average is governed by the linearized Boltzmann equation. The theory of linear irreversible processes, which offers a statistical description of fluctuations on a thermodynamical basis, is applied to the linearized Boltzmann equation—treated as a linear phenomenological equation—following the development given recently by Fox and Uhlenbeck. The resulting stochastic equation is seen to be equivalent to the Fokker- Planck equation obtained from the master equation, yielding a multidimensional Ornstein-Uhlenbeck process which describes the fluctuations in molecular phase space.
Journal of Statistical Physics, 1995
This paper deals with the trend to equilibrium of solutions to the spacehomogeneous Boltzmann equation for Maxwellian molecules with angular cutoff as well as with infinite-range forces. The solutions are considered as densities of probability distributions. The Tanaka functional is a metric for the space of probability distributions, which has previously been used in connection with the Boltzmann equation. Our main result is that, if the initial distribution possesses moments of order 2+ε, then the convergence to equilibrium in his metric is exponential in time. In the proof, we study the relation between several metrics for spaces of probability distributions, and relate this to the Boltzmann equation, by proving that the Fourier-transformed solutions are at least as regular as the Fourier transform of the initial data. This is also used to prove that even if the initial data only possess a second moment, then ∫∣v∣>R f(v, t) ∣v∣2dv→0 asR→∞, and this convergence is uniform in time.
Probability Metrics and Uniqueness of the Solution to the Boltzmann Equation for a Maxwell Gas
Journal of Statistical Physics, 1999
We consider a metric for probability densities with finite variance on Rd, and compare it with other metrics. We use it for several applications both in probability and in kinetic theory. The main application in kinetic theory is a uniqueness result for the solution of the spatially homogeneous Boltzmann equation for a gas of true Maxwell molecules.
Monte Carlo Methods for the Linearized Poisson-Boltzmann Equation
2003
We review efficient grid-free random walk methods for solving boundary value problems for the linearized Poisson-Boltzmann equation (LPBE). First we introduce the "Walk On Spheres" (WOS) algorithm [1] for the LPBE. Based on this WOS algorithm, another, related, Monte Carlo algorithm is presented. This modified Monte Carlo method reinterprets the weights used in the original WOS algorithm as survival probabilities for the random walker used in the computation . In addition, a Feynman-Kac path-integral implementation for solving the LPBE is given . This Feynman-Kac approach uses the WOS method to provide a technique for estimating certain Gaussian path integrals without the need for simulating Brownian trajectories in detail. We then similarly interpret the exponential weight in the Feynman-Kac formula as a survival probability. It is then shown that this method is mathematically equivalent to the previous modified WOS method for the LPBE. The effectiveness of these methods is illustrated by computing four analytically solvable problems. In all four cases, excellent agreement is shown. In particular, for the problem of calculating the electrostatic potential in an electrolyte between two infinite parallel flat plates, our modified WOS method is compared with the old WOS method and with our Feynman-Kac WOS (FK WOS) method. Our modified WOS method is the most efficient one, but FK WOS method holds the promise of extension to more complicated equations such as the time-independent Schrödinger equation. Finally, we illustrate the use of a Monte Carlo approach for the LPBE in a more complicated setting related to the computation of the electrostatic free energy of a large molecule. Here, we couple the LPBE solution in the exterior of a compact domain (molecule) with the solution of the Poisson equation inside, and with continuity boundary conditions linking these two solutions. The Monte Carlo method performs quite well in this complicated situation.
Statistical mechanics of the fluctuating lattice Boltzmann equation
Physical Review E, 2007
We propose a new formulation of the fluctuating lattice Boltzmann equation that is consistent with both equilibrium statististical mechanics and fluctuating hydrodynamics. The formalism is based on a generalized lattice-gas model, with each velocity direction occupied by many particles. We show that the most probable state of this model corresponds to the usual equilibrium distribution of the lattice Boltzmann equation. Thermal fluctuations about this equilibrium are controlled by the mean number of particles at a lattice site. Stochastic collision rules are described by a Monte Carlo process satisfying detailed balance. This allows for a straightforward derivation of discrete Langevin equations for the fluctuating modes. It is shown that all non-conserved modes should be thermalized, as first pointed out by Adhikari et al.; any other choice violates the condition of detailed balance. A Chapman-Enskog analysis is used to derive the equations of fluctuating hydrodynamics on large length and time scales; the level of fluctuations is shown to be thermodynamically consistent with the equation of state of an isothermal, ideal gas. We believe this formalism will be useful in developing new algorithms for thermal and multiphase flows.
On a simulation scheme for the Boltzmann equation
Mathematical Methods in The Applied Sciences, 1986
A scheme for the simulation of solutions of the Boltzmann equation derived by Nanbu is investigated. Rigorous results concerning questions of justification, the computation effort and the energy fluctuations are presented.