TY - JOUR

T1 - Local Thermal Equilibrium for Certain Stochastic Models of Heat Transport

AU - Li, Yao

AU - Nándori, Péter

AU - Young, Lai Sang

N1 - Funding Information:
LSY was supported in part by NSF Grant DMS-1363161.

PY - 2016/4/1

Y1 - 2016/4/1

N2 - This paper is about nonequilibrium steady states (NESS) of a class of stochastic models in which particles exchange energy with their “local environments” rather than directly with one another. The physical domain of the system can be a bounded region of (Formula presented.) for any (Formula presented.). We assume that the temperature at the boundary of the domain is prescribed and is nonconstant, so that the system is forced out of equilibrium. Our main result is local thermal equilibrium in the infinite volume limit. In the Hamiltonian context, this would mean that at any location x in the domain, local marginal distributions of NESS tend to a probability with density (Formula presented.) , permitting one to define the local temperature at x to be (Formula presented.). We prove also that in the infinite volume limit, the mean energy profile of NESS satisfies Laplace’s equation for the prescribed boundary condition. Our method of proof is duality: by reversing the sample paths of particle movements, we convert the problem of studying local marginal energy distributions at x to that of joint hitting distributions of certain random walks starting from x, and prove that the walks in question become increasingly independent as system size tends to infinity.

AB - This paper is about nonequilibrium steady states (NESS) of a class of stochastic models in which particles exchange energy with their “local environments” rather than directly with one another. The physical domain of the system can be a bounded region of (Formula presented.) for any (Formula presented.). We assume that the temperature at the boundary of the domain is prescribed and is nonconstant, so that the system is forced out of equilibrium. Our main result is local thermal equilibrium in the infinite volume limit. In the Hamiltonian context, this would mean that at any location x in the domain, local marginal distributions of NESS tend to a probability with density (Formula presented.) , permitting one to define the local temperature at x to be (Formula presented.). We prove also that in the infinite volume limit, the mean energy profile of NESS satisfies Laplace’s equation for the prescribed boundary condition. Our method of proof is duality: by reversing the sample paths of particle movements, we convert the problem of studying local marginal energy distributions at x to that of joint hitting distributions of certain random walks starting from x, and prove that the walks in question become increasingly independent as system size tends to infinity.

KW - Heat transport

KW - Local thermodynamic equilibrium

KW - Nonequilibrium steady states

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U2 - 10.1007/s10955-016-1466-3

DO - 10.1007/s10955-016-1466-3

M3 - Article

AN - SCOPUS:84960382923

VL - 163

SP - 61

EP - 91

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1

ER -