Sommaire:

Stochastic lattice gases (interacting particle systems) are used to model transport phenomena, where one starts from defining microscopic rules and tries to show behaviors of macroscopic quantities. The macroscopic evolution of a stochastic lattice gas with symmetric hopping rules is described by a diffusion equation with density-dependent diffusion coefficient. In practice, even when the equilibrium properties of a lattice gas are analytically known, the diffusion coefficient cannot be explicitly computed, except for a few simple examples, e.g. for the symmetric simple exclusion process. We develop a procedure to systematically obtain approximations for the diffusion coefficient in explicit forms. The method relies on a variational formula found in mathematics literature. Restriction on test functions to finite-dimensional sub-spaces allows one to perform the minimization and gives upper bounds for the diffusion coefficient. We demonstrate calculations in the following two models; one-dimensional generalized exclusion processes, where each site can accommodate at most two particles (2-GEPs) [1], and the Kob-Andersen (KA) model on the square lattice, which is classified into kinetically-constrained gas [2]. The prediction of the diffusion coefficient depends on the domain ("shape") of test functions. The smallest shapes give approximations which coincide with the mean-field theory, but the larger shapes, the more precise upper bounds we obtain. For the 2-GEPs, our analytical predictions provide upper bounds which are very close to simulation results throughout the entire density range. For the KA model, we also find improved upper bounds when the density is small. By combining the variational method with a perturbation approach, we discuss the asymptotic behavior of the diffusion coefficient in the high density limit.

[1] CA, P L Krapivsky, K Mallick: Phys. Rev. E 95, 032121 (2017)
[2] CA, P L Krapivsky, K Mallick: J. Phys. A (accepted)

Pour plus d'informations, merci de contacter Parmeggiani A.