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

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 ρ² for the linear heat equation, and a nonlinear function of ρ for the nonlinear heat equation.
Authors:
 [1] ;  [2] ;  [3]
  1. Department of Mathematics and Computer Science and Institute for Complex Molecular Systems, Technische Universiteit Eindhoven, Postbus 513, 5600 MB Eindhoven (Netherlands)
  2. Delft Institute of Applied Mathematics, Technische Universiteit Delft, Mekelweg 4, 2628 CD Delft (Netherlands)
  3. Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, Postbus 513, 5600 MB Eindhoven (Netherlands)
Publication Date:
OSTI Identifier:
22306040
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 9; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Publisher:
American Institute of Physics (AIP)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DIFFUSION; FOURIER HEAT EQUATION; FUNCTIONALS; MARKOV PROCESS; NONLINEAR PROBLEMS; THERMAL CONDUCTION