Lorenzo Bertini, Davide Gabrielli and Joel L. Lebowitz
Large deviations for a stochastic model of heat flow
(110K, Latex)
ABSTRACT. We investigate a one dimensional chain of $2N$ harmonic
oscillators in which neighboring sites have their energies
redistributed randomly. The sites $-N$ and $N$ are in contact with
thermal reservoirs at different temperature $\tau_-$ and $\tau_+$.
Kipnis, Marchioro, and Presutti \cite{KMP} proved that this model
satisfies {}Fourier's law and that in the hydrodynamical scaling
limit, when $N \to \infty$, the stationary state has a linear energy
density profile $\bar \theta(u)$, $u \in [-1,1]$. We derive the
large deviation function $S(\theta(u))$ for the probability of
finding, in the stationary state, a profile $\theta(u)$ different
from $\bar \theta(u)$. The function $S(\theta)$ has striking
similarities to, but also large differences from, the
corresponding one of the symmetric exclusion process. Like the
latter it is nonlocal and satisfies a variational equation. Unlike
the latter it is not convex and the Gaussian normal fluctuations
are enhanced rather than suppressed compared to the local
equilibrium state. We also briefly discuss more general model and find
the features common in these two and other models whose $S(\theta)$ is
known.