1,720,971 research outputs found
A Diffusion Limit for a Test Particle in a Random Distribution of Scatterers
No full textWe consider a point particle moving in a random distribution of obstacles described by a potential barrier. We show that, in a weak-coupling regime, under a diffusion limit suggested by the potential itself, the probability distribution of the particle converges to the solution of the heat equation. The diffusion coefficient is given by the Green-Kubo formula associated to the generator of the diffusion process dictated by the linear Landau equation
A gradient flow approach to linear Boltzmann equations
No full textWe introduce a gradient flow formulation of linear Boltzmann equations. Under a diffusive scaling we derive a diffusion equation by using the machinery of gradient flows
Large Deviations for Kac-Like Walks
No full textWe introduce a Kac’s type walk whose rate of binary collisions preserves the total momentum but not the kinetic energy. In the limit of large number of particles we describe the dynamics in terms of empirical measure and flow, proving the corresponding large deviation principle. The associated rate function has an explicit expression. As a byproduct of this analysis, we provide a gradient flow formulation of the Boltzmann-Kac equatio
Large deviations for a binary collision model: energy evaporation
Full text linkWe analyze the large deviations for a discrete energy Kac-like walk. In particular, we exhibit a path, with probability exponentially small in the number of particles, that looses energy
Derivation of the Fick's law for the Lorentz model in a low density regime
Full text linkWe consider a point particle moving in a random distribution of obstacles described by a potential barrier. We show that, in a weak-coupling regime, under a diffusion limit suggested by the potential itself, the probability distribution of the particle converges to the solution of the heat equation. The diffusion coefficient is given by the Green-Kubo formula associated to the generator of the diffusion process dictated by the linear Landau equation
Asymptotic probability of energy increasing solutions to the homogeneous Boltzmann equation
No full textWeak solutions to the homogeneous Boltzmann equation with increasing energy have been constructed by Lu and Wennberg. We consider an underlying microscopic stochastic model with binary collisions (Kac's model) and show that these solutions are atypical. More precisely, we prove that the probability of observing these paths is exponentially small in the number of particles and compute the exponential rate. This result is obtained by improving the established large deviation estimates in the canonical setting. Key ingredients are the extension of Sanov's theorem to the microcanonical ensemble and large deviations for the Kac's model in the microcanonical setting
Convergence of a kinetic equation to a fractional diffusion equation
No full textABSTRACT. A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process (K(t), Y (t)) on (T × R), where T is the one-dimensional torus. K(t) is a autonomous reversible jump process, with waiting times between two jumps with finite expectation value but infinite variance. Y (t) is an additive functional of K, defined as ∫ t 0 v(K(s))ds, where |v | ∼ 1 for small k. We prove that the rescaled process N−2/3Y (Nt) converge in distribution to a symmetric Lévy process, stable with index α = 3/2. 1. INTRODUCTION. The understanding of thermal conductance in both classical and quantum me-chanical systems is one of the fundamental problems of non-equilibrium statistical mechanics. A particular aspect that has attracted much interest is the observation that autonomous translation invariant systems in dimensions one and two exhibi
Donsker-Varadhan asymptotics for degenerate jump Markov processes
Full text linkWe consider a class of continuous time Markov chains on a compact metric space that admit an invariant measure strictly positive on open sets together with absorbing states. We prove the joint large deviation principle for the empirical measure and flow. Due to the lack of uniform ergodicity, the zero level set of the rate function is not a singleton. As corollaries, we obtain the Donsker-Varadhan rate function for the empirical measure and a variational expression of the rate function for the empirical flow
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