Exact Large Deviation Functional of a Stationary Open Driven Diffusive System: The Asymmetric Exclusion Process

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\rho (x):{\text{ }}P_N (\{ \rho (x)\} ) \sim \exp [ - L\mathcal{F}_{[a,b]} (\{ \rho (x)\} ;\rho _a ,\rho _b )]$$ \end{document}, so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{F}$$ \end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{F}$$ \end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{F}(\{ \rho (x)\} )$$ \end{document} is not convex and others for which \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{F}(\{ \rho (x)\} )$$ \end{document} has discontinuities in its second derivatives at ρ(x)=¯ρ(x). In the latter ranges the fluctuations of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$1/\sqrt N $$ \end{document} in the density profile near ¯ρ(x) are then non-Gaussian and cannot be calculated from the large deviation function.