[Paper Review] Large Deviation of the Density Profile in the Symmetric Simple Exclusion Process

Bernard Derrida, Joel L. Lebowitz+1 more|arXiv (Cornell University)|Sep 19, 2001

Stochastic processes and statistical mechanics14 citations

TL;DR

This paper studies large deviations in the symmetric simple exclusion process (SSEP) in a one-dimensional open system driven by particle reservoirs at different chemical potentials, leading to a stationary nonequilibrium state (SNS). It derives a nonlocal large deviation functional F{ρ} for the density profile ρ(x), revealing long-range spatial correlations absent in equilibrium systems, which explains fluctuating hydrodynamics predictions and suggests broader nonlocality in nonequilibrium free energy functionals.

ABSTRACT

Abstract We consider an open one dimensional lattice gas on sites i = 1,...,N, with particles jumping independently with rate 1 to neighboring interior empty sites, the simple symmetric exclusion process. The particle fluxes at the left and right boundaries, corresponding to exchanges with reservoirs at different chemical potentials, create a stationary nonequilibrium state (SNS) with a steady flux of particles through the system. The mean density profile in this state, which is linear, describes the typical behavior of a macroscopic system, i.e., this profile occurs with probability 1 when N → ∞. The probability of microscopic configurations corresponding to some other profile ρ(x), x = i/N, has the asymptotic form exp[−NF({ρ})]; F is the large deviation functional. In contrast to equilibrium systems, for which Feq({ρ}) is just the integral of the appropriately normalized local free energy density, the F we find here for the nonequilibrium system is a nonlocal function of ρ. This gives rise to the long range correlations in the SNS predicted by fluctuating hydrodynamics and suggests similar non-local behavior of F in general SNS, where the long range correlations have been observed experimentally. Key words: Large deviations, symmetric simple exclusion process, open system, stationary nonequilibrium state.

Motivation & Objective

• To understand the statistical mechanics of nonequilibrium steady states (SNS) in driven diffusive systems.
• To characterize the probability of observing atypical density profiles ρ(x) in the SSEP under open boundary conditions.
• To derive the large deviation functional F{ρ} for the density profile in the SNS, contrasting it with equilibrium free energy functionals.
• To investigate the origin and implications of long-range spatial correlations in nonequilibrium systems.

Proposed method

• Model the system as a one-dimensional lattice of N sites with symmetric exclusion dynamics and particle exchange at boundaries.
• Use the large deviation principle to express the probability of a macroscopic density profile ρ(x) as exp[−NF{ρ}].
• Derive the functional F{ρ} using hydrodynamic scaling and the generator of the stochastic process.
• Show that F{ρ} is nonlocal in ρ, meaning it depends on the global shape of the profile, not just local values.
• Compare the nonequilibrium F{ρ} to the equilibrium free energy functional, highlighting the absence of locality in the former.
• Establish that the nonlocality of F{ρ} implies long-range correlations in the SNS, consistent with fluctuating hydrodynamics.

Experimental results

Research questions

• RQ1 How does the large deviation functional F{ρ} for the density profile in the SSEP differ from equilibrium free energy functionals?
• RQ2 What is the functional form of F{ρ} in a stationary nonequilibrium state with particle flux?
• RQ3 Why does the SNS exhibit long-range spatial correlations, and how is this related to the structure of F{ρ}?
• RQ4 Is the large deviation functional in nonequilibrium systems inherently nonlocal, and what are the consequences?
• RQ5 Can the nonlocality of F{ρ} explain experimentally observed long-range correlations in driven systems?

Key findings

• The large deviation functional F{ρ} for the density profile in the SSEP is nonlocal, depending on the global shape of ρ(x), unlike in equilibrium systems where it is local.
• The nonlocality of F{ρ} directly implies the presence of long-range spatial correlations in the stationary nonequilibrium state.
• The functional F{ρ} cannot be expressed as an integral of a local free energy density, distinguishing it from equilibrium statistical mechanics.
• The steady-state density profile is linear in the thermodynamic limit, corresponding to the typical macroscopic behavior with probability one.
• The derivation confirms that fluctuating hydrodynamics predictions of long-range correlations in driven diffusive systems are consistent with the large deviation structure of the SSEP.
• The results suggest that nonlocality in the large deviation functional may be a generic feature of stationary nonequilibrium states beyond the SSEP.