## DIMACS TR: 2001-31

## Large Deviation of the Density Profile in the Symmetric Simple Exclusion Process

### Authors: B. Derrida, J. L. Lebowitz and E. R. Speer

**
ABSTRACT
**

We consider an open one dimensional lattice gas on sites $i=1,\dots,N$, with
particles jumping independently with rate $1$ to neighboring interior empty
sites, the {\it 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 $N \to
\infty$. The probability of microscopic configurations
corresponding to some other profile $\rho(x)$, $x = i/N$,
has the asymptotic form
$\exp[-N {\cal F}(\{\rho\})]$; $\cal F$ is the {\it large deviation
functional}. In contrast to equilibrium systems, for which
${\cal F}_{eq}(\{\rho\})$ is just
the integral of the appropriately normalized
local free energy density,
the $\cal F$ we find here
for the nonequilibrium system is a nonlocal function of
$\rho$. This gives rise to the long range
correlations in the SNS predicted by fluctuating hydrodynamics
and suggests similar non-local
behavior of $\cal F$ in general SNS, where the long range correlations have
been observed experimentally.

Paper Available at:
ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/2001/2001-31.ps.gz

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