2002/05/16 by Bernard Derrida, B. Derrida, Derrida, B. +6 · 1 citation
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Statistical Mechanics (cond-mat.stat-mech) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.stat-mech
paper · pdf · doi:10.48550/arxiv.cond-mat/0205353
Latex, one PicTeX figure in a separate file
arxiv created 2002/05/16 · openalex publication_date 2002/05/16 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the asymmetric exclusion process (ASEP) in one dimension on sites i = 1,..., N, in contact at sites i=1 and i=N with infinite particle reservoirs at densities ρa and ρb. As ρa and ρb are varied, the typical macroscopic steady state density profile ρ(x), x∈[a,b], obtained in the limit N=L(b-a)→∞, exhibits shocks and phase transitions. Here we derive an exact asymptotic expression for the probability of observing an arbitrary macroscopic profile ρ(x): PN(\ρ(x)\)∼exp[-L\cal F[a,b](\ρ(x)\);ρa,ρb], so that \cal F is the large deviation functional, a quantity similar to the free energy of equilibrium systems. We find, as in the symmetric, purely diffusive case q=1 (treated in an earlier work), that \cal F is in general a non-local functional of ρ(x). Unlike the symmetric case, however, the asymmetric case exhibits ranges of the parameters for which \cal F(\ρ(x)\) is not convex and others for which \cal F(\ρ(x)\) has discontinuities in its second derivatives at ρ(x) = ρ(x); the fluctuations near ρ(x) are then non-Gaussian and cannot be calculated from the large deviation function.