Alessia Nota

Derivation of the Fick's Law for the Lorentz Model in a low density regime

We consider the Lorentz model in a slab with two mass reservoirs at the boundaries. We show that, in a low density regime, there exists a unique stationary solution for the microscopic dynamics, which converges to the stationary solution of the heat equation, namely to the linear profile of the density. In the same regime, the macroscopic current in the stationary state is given by the Fick’s law, with the diffusion coefficient determined by the Green–Kubo formula.

A Diffusion Limit for a Test Particle in a Random Distribution of Scatterers

We 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.

Summability of Connected Correlation Functions of Coupled Lattice Fields

We consider two nonindependent random fields ψ and φ defined on a countable set Z. For instance, Z = Z or Z = Z × I , where I denotes a finite set of possible “internal degrees of freedom” such as spin. We prove that, if the cumulants of both ψ and φ are l1-clustering up to order 2n, then all joint cumulants between ψ and φ are l2-summable up to order n, in the precise sense described in the text. We also provide explicit estimates in terms of the related l1-clustering norms, and derive a weighted l2-summation property of the joint cumulants if the fields are merely l2-clustering. One immediate application of the results is given by a stochastic process ψt(x) whose state is l1-clustering at any time t: then the above estimates can be applied with ψ = ψt and φ = ψ0 and we obtain uniform in t estimates for the summability of time-correlations of the field. The above clustering assumption is obviously satisfied by any l1-clustering stationary state of the process, and our original motivation for the control of the summability of time-correlations comes from a quest for a rigorous control of the Green-Kubo correlation function in such a system. A key role in the proof is played by the properties of non-Gaussian Wick polynomials and their connection to cumulants.

Derivation of the Linear Landau Equation and Linear Boltzmann Equation from the Lorentz Model with Magnetic Field

We consider a test particle moving in a random distribution of obstacles in the plane, under the action of a uniform magnetic field, orthogonal to the plane. We show that, in a weak coupling limit, the particle distribution behaves according to the linear Landau equation with a magnetic transport term. Moreover, we show that, in a low density regime, when each obstacle generates an inverse power law potential, the particle distribution behaves according to the linear Boltzmann equation with a magnetic transport term. We provide an explicit control of the error in the kinetic limit by estimating the contributions of the configurations which prevent the Markovianity. We compare these results with those ones obtained for a system of hard disks in Bobylev et al. (Phys Rev Lett 75:2, 1995), which show instead that the memory effects are not negligible in the Boltzmann-Grad limit.

Harmonic Chain with Velocity Flips: Thermalization and Kinetic Theory

We consider the detailed structure of correlations in harmonic chains with pinning and a bulk velocity flip noise during the heat relaxation phase which occurs on diffusive time scales, for

On the theory of Lorentz gases with long range interactions

t=O(L^2)

Diffusive Limit for the Random Lorentz Gas

t=O(L2) where L is the chain length. It has been shown earlier that for non-degenerate harmonic interactions these systems thermalize, and the dominant part of the correlations is given by local thermal equilibrium determined by a temperature profile which satisfies a linear heat equation. Here we are concerned with two new aspects about the thermalization process: the first order corrections in 1xa0/xa0L to the local equilibrium correlations and the applicability of kinetic theory to study the relaxation process. Employing previously derived explicit uniform estimates for the temperature profile, we first derive an explicit form for the first order corrections to the particle position-momentum correlations. By suitably revising the definition of the Wigner transform and the kinetic scaling limit we derive a phonon Boltzmann equation whose predictions agree with the explicit computation. Comparing the two results, the corrections can be understood as arising from two different sources: a current-related term and a correction to the position-position correlations related to spatial changes in the phonon eigenbasis.

Long time asymptotics for homoenergetic solutions of the Boltzmann equation. Collision-dominated case

AbstractIn this paper we study a class of solutions of the Boltzmann equation which have the form f (x, v, t) =xa0g (v − L (t) x, t) where L (t) =xa0 A (I +xa0tA)−1 with the matrix A describing a shear flow or a dilatation or a combination of both. These solutions are known as homoenergetic solutions. We prove the existence of homoenergetic solutions for a large class of initial data. For different choices for the matrix nA and for different homogeneities of the collision kernel, we characterize the long time asymptotics of the velocity distribution for the corresponding homoenergetic solutions. For a large class of choices of A we then prove rigorously, in the case of Maxwell molecules, the existence of self-similar solutions of the Boltzmann equation. The latter are non Maxwellian velocity distributions and describe far-from-equilibrium flows. For Maxwell molecules we obtain exact formulas for the H-function for some of these flows. These formulas show that in some cases, despite being very far from equilibrium, the relationship between density, temperature and entropy is exactly the same as in the equilibrium case. We make conjectures about the asymptotics of homoenergetic solutions that do not have self-similar profiles.