TY - JOUR

T1 - Large deviations for a stochastic model of heat flow

AU - Bertini, Lorenzo

AU - Gabrielli, Davide

AU - Lebowitz, Joel L.

N1 - Funding Information:
It is a pleasure to thank C. Bernardin, A. De Sole, G. Jona–Lasinio, C. Landim, and E. Presutti for useful discussions. We also thank an anonymous referee for clarifying the mathematical details left open. L.B. and D.G. acknowledge the support COFIN MIUR 2002027798 and 2003018342. The work of J.L.L. was supported by NSF Grant DMR 01-279-26 and AFOSR Grant AF 49620-01-1-0154.

PY - 2005/12

Y1 - 2005/12

N2 - 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 τ - and τ+. Kipnis et al. (J. Statist. Phys., 27:65-74 (1982).) proved that this model satisfies Fourier's law and that in the hydrodynamical scaling limit, when N → ∞, the stationary state has a linear energy density profile θ(u), u [-1,1]. We derive the large deviation function S(θ(u)) for the probability of finding, in the stationary state, a profile θ(u) different from θ(u). The function S(θ) 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 models and find the features common in these two and other models whose S(θ) is known.

AB - 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 τ - and τ+. Kipnis et al. (J. Statist. Phys., 27:65-74 (1982).) proved that this model satisfies Fourier's law and that in the hydrodynamical scaling limit, when N → ∞, the stationary state has a linear energy density profile θ(u), u [-1,1]. We derive the large deviation function S(θ(u)) for the probability of finding, in the stationary state, a profile θ(u) different from θ(u). The function S(θ) 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 models and find the features common in these two and other models whose S(θ) is known.

KW - Boundary driven stochastic systems

KW - Large deviations

KW - Stationary nonequilibrium states

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U2 - 10.1007/s10955-005-5527-2

DO - 10.1007/s10955-005-5527-2

M3 - Article

AN - SCOPUS:29044447618

VL - 121

SP - 843

EP - 885

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 5-6

ER -