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Large deviations in stochastic heat-conduction processes provide a gradient-flow structure for heat conduction

### Abstract

We consider three one-dimensional continuous-time Markov processes on a lattice, each of which models the conduction of heat: the family of Brownian Energy Processes with parameter m (BEP(m)), a Generalized Brownian Energy Process, and the Kipnis-Marchioro-Presutti (KMP) process. The hydrodynamic limit of each of these three processes is a parabolic equation, the linear heat equation in the case of the BEP(m) and the KMP, and a nonlinear heat equation for the Generalized Brownian Energy Process with parameter a (GBEP(a)). We prove the hydrodynamic limit rigorously for the BEP(m), and give a formal derivation for the GBEP(a). We then formally derive the pathwise large-deviation rate functional for the empirical measure of the three processes. These rate functionals imply gradient-flow structures for the limiting linear and nonlinear heat equations. We contrast these gradient-flow structures with those for processes describing the diffusion of mass, most importantly the class of Wasserstein gradient-flow systems. The linear and nonlinear heat-equation gradient-flow structures are each driven by entropy terms of the form −log ρ; they involve dissipation or mobility terms of order ρ^{2} for the linear heat equation, and a nonlinear function of ρ for the nonlinear heat equation.

© 2014 AIP Publishing LLC

Received 20 March 2014
Accepted 15 August 2014
Published online 03 September 2014

Acknowledgments:
The authors are grateful for many interesting discussions with Jin Feng. M.A.P. and K.V. acknowledge the support of NWO VICI Grant No. 639.033.008.

Article outline:

I. THE HEAT EQUATION
II. THE THREE STOCHASTIC PROCESSES: BEP(*m*), GBEP(*a*), AND KMP
A. The Brownian momentum process
B. The Brownian energy process with parameter *m*
C. A generalized Brownian energy process with parameter *a*
D. The Kipnis-Marchioro-Presutti process
E. Macroscopic quantities
III. EQUILIBRIUM PROPERTIES OF THE BEP(*m*)
IV. EQUILIBRIUM LARGE DEVIATIONS OF THE BEP
V. HYDRODYNAMIC LIMIT OF THE TIME EVOLUTION
VI. THE WEAKLY ASYMMETRIC BEP(*m*)
VII. THE REPLACEMENT LEMMA
VIII. LARGE DEVIATIONS FROM THE HYDRODYNAMIC LIMIT
IX. THE GBEP(*a*) AND KMP PROCESSES
X. GRADIENT-FLOW STRUCTURES
XI. CONCLUSIONS AND DISCUSSION
A. The driving functional *E*
B. The Onsager operator *K*
C. Comparison with diffusion
D. No metric space
E. Generalizations to other models

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2014-09-03

2016-09-26

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