Séminaire du LPTMS: Chikashi Arita


11:00 - 11:30

LPTMS, salle 201, 2ème étage, Bât 100, Campus d'Orsay
15 Rue Georges Clemenceau, Orsay, 91405
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Variational calculation of diffusion coefficients in stochastic lattice gases

Chikashi Arita (Universität des Saarlandes, Saarbrücken)

Deriving macroscopic behaviors from microscopic dynamics of particles   is a fundamental problem. In stochastic lattice gases one tries to  demonstrate this hydrodynamic limit. The 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  when a lattice gas additionally satisfies the « gradient condition »,  e.g. the diffusion coefficients of the simple exclusion process and   non-interacting random walks are exactly identical to their hopping  rates. We develop a procedure to obtain systematic analytical approximations for the diffusion coefficient in non-gradient lattice  gases with known equilibrium. The method relies on a variational  formula found by Varadhan and Spohn. Restriction on test functions to  finite-dimensional sub-spaces allows one to perform the minimization  and gives upper bounds for the diffusion coefficient. We apply the   procedure to 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] C. Arita, P. L. Krapivsky and K. Mallick, Variational calculation of transport coefficients in diffusive lattice gases, Phys. Rev. E 95, 032121 (2017)
  • [2] C. Arita, P. L. Krapivsky and K. Mallick, Bulk diffusion in a kinetically constrained lattice gas, preprint cond-mat arXiv:1711.10616


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