Séminaires du mardi 20 février

Séminaire du LPTMS: Chikashi Arita


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

Séminaire du LPTMS: Sergej Moroz


Séminaire du LPTMS: Alexandre Lazarescu


On the hydrodynamic behaviour of interacting lattice gases far from equilibrium

Alexandre Lazarescu (Centre de Physique Théorique, École Polytechnique)

Lattice gases are a particularly rich playground to study the large scale emergent behaviour of microscopic models. A few things are known in general for models that are sufficiently close to equilibrium (i.e. with rates close to detailed balance, and where the dynamics is typically diffusive): in particular, the local density of particles behaves autonomously in the macroscopic limit, even at the level of large deviations, and the system can be described through a Langevin equation involving only a few quantities called transport coefficients. As demonstrated in the previous talk, obtaining those coefficients in practice can be quite challenging, but we can usually be confident that they exist. I will be talking about a situation that is quite different at first sight: systems far from equilibrium, where the dynamics is propagative, and where very little is known in general. The question is then whether one can hope to be able to describe those models with a similar hydrodynamic structure, or if that description breaks down (if, for instance, long-range correlations become relevant). I will present recent results showing that, for a broad class of 1D models with hard-core repulsion but also interactions and space-dependent rates, the answer is yes and no: all those models exhibit a dynamical phase transition between a hydrodynamic regime and a highly correlated one, which can be related to the so-called "third order phase transitions". The methods involved are quite general and likely to be applicable to many more families of models.