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It is a well-known fact that, in small mean free path regimes, kinetic equations can lead to diffusion equations. Besides, kinetic equations can be approached by a closed system of moments equations. In this paper, we are interested in a special closure based on an entropy minimization principle, as introduced earlier by Levermore. We investigate the behavior of the resulting nonlinear hyperbolic system in the diffusive scaling. We first establish various fundamental facts on this system, then we show that the hyperbolic system admits global smooth solutions, and is consistent with the diffusion limit. Similar features are also discussed for a simpler limited flux equation. [ABSTRACT FROM AUTHOR]

We consider a system of N particles in dimension one, interacting through a zero‐range repulsive potential whose strength is proportional to N-1. We construct the finite and the infinite Schrödinger hierarchies for the reduced density matrices of subsystems with n particles. We show that the solution of the finite hierarchy converges in a suitable sense to a solution of the infinite one. Besides, the infinite hierarchy is solved by a factorized state, built as a tensor product of many identical one‐particle wave functions which fulfil the cubic nonlinear Schrödinger equation. Therefore, choosing a factorized initial datum and assuming propagation of chaos, we provide a derivation for the cubic NLSE. The result, achieved with operator‐analysis techniques, can be considered as a first step towards a rigorous deduction of the Gross–Pitaevskii equation in dimension one. The problem of proving propagation of chaos is left untouched. [ABSTRACT FROM AUTHOR]

In this article we study a model for the dynamics of particles moving freely between two horizontal plates. Our goal is to characterize the diffusive behavior for such systems in the long‐time and large (horizontal) scale limits. Using homogenization techniques for PDEs, we obtain a diffusion equation as a limit of the original kinetic equation, appropriately scaled. This limiting diffusion can be understood intuitively by observing that in the limit of long times and vanishing vertical distance between the plates, for a given particle, the mean free path between two successive reflections at the plates also vanishes, and therefore the number of reflections grows unboundedly with the length of the time interval. The nature of a particular reflection law then efficiently randomizes the particle motion. More specifically, we study the dependence of the diffusion constant Dα on mixed boundary conditions: the case of specular reflection (based on the “Arnold cat map”) with an isotropic component (i.e., with small “accommodation” coefficient α). We find that the diffusion tensor Dα is positive definite for every α∈(0,1]. Furthermore, in the limit of vanishing isotropic component (α→0), we recover the result of Bardos, Golse and Colonna [Physica D, 104 (1997), 32–60]. [ABSTRACT FROM AUTHOR]

In the kinetic theory of gases, half‐space problems are important because they govern the behavior of correctors to the Hilbert expansion near the boundaries. This paper takes up the discussion in Coron et al., Comm. Pure Appl. Math. 41 (1988) by a different method, based on the index formula for Wiener–Hopf operators on the half‐line. This method leads naturally to a new result on analogues of Chandrasekhar’s H function, which does not seem easy to obtain by the energy method of Coron et al. (1988). [ABSTRACT FROM AUTHOR]

Deals with a study which proved a classical diffusion approximation method for determining the anomalous diffusion limit of a Knudsen gas. Description of a Knudsen gas; Asymptotic expansion of the equation; Proof of the anomalous diffusion approximation.