Duality and exact correlations for a model of heat conduction

Giardinà, Cristian; Kurchan, Jorge; Redig, Frank

doi:10.1063/1.2711373

Cited by 85 publications

(144 citation statements)

References 20 publications

(36 reference statements)

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“…Unfortunately, the integrability of the underlying dynamics leads to several unphysical features (e.g., a vanishing temperature gradient) [2]. Later, purely stochastic models, where energy is assumed to diffuse between neighboring boxes (the oscillators), have been considered [4,5]. More recently, systems of harmonic oscillators exchanging energy with ''conservative'' noise have been proven to admit a unique stationary state with a constant heat flux and a linear temperature profile [6].…”

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confidence: 99%

Nonequilibrium Invariant Measure under Heat Flow

et al. 2008

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We provide an explicit representation of the nonequilibrium invariant measure for a chain of harmonic oscillators with conservative noise in the presence of stationary heat flow. By first determining the covariance matrix, we are able to express the measure as the product of Gaussian distributions aligned along some collective modes that are spatially localized with power-law tails. Numerical studies show that such a representation applies also to a purely deterministic model, the quartic Fermi-Pasta-Ulam chain.

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Nonequilibrium Invariant Measure under Heat Flow

et al. 2008

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“…In the case of energy transport, i.e. interacting particle systems with a continuous dynamical variable (the energy) connected at their boundaries to thermal reservoirs working at different temperatures, duality has been constructed for the KipnisMarchioro-Presutti (KMP) model [8] for heat conduction and also for other models [6]. Consequences of duality include the possibility to express the n-point energy correlation functions in terms of n (interacting) random walkers.…”

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confidence: 99%

Duality and Hidden Symmetries in Interacting Particle Systems

Kurchan²,

et al. 2009

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In the context of Markov processes, both in discrete and continuous setting, we show a general relation between duality functions and symmetries of the generator. If the generator can be written in the form of a Hamiltonian of a quantum spin system, then the "hidden" symmetries are easily derived. We illustrate our approach in processes of symmetric exclusion type, in which the symmetry is of SU(2) type, as well as for the KipnisMarchioro-Presutti (KMP) model for which we unveil its SU(1, 1) symmetry. The KMP model is in turn an instantaneous thermalization limit of the energy process associated to a large family of models of interacting diffusions, which we call Brownian energy process (BEP) and which all possess the SU(1, 1) symmetry. We treat in details the case where the system is in contact with reservoirs and the dual process becomes absorbing.

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“…We describe this procedure here somewhat intuitively. The hydrodynamic limit for the BEP(m) predicts that π N (Z(t)) evolves "typically" as ρ(t, x)dx where ρ(t, x) solves the diffusion equation (19). This behavior is a manifestation of the law of large numbers and therefore one expects corresponding exponentially small (in N) large-deviation probabilities, i.e., we expect, in the sense of large deviations…”

Section: Large Deviations From the Hydrodynamic Limitmentioning

confidence: 99%

“…Also note that for the definition of the process z i , m need not be integer-any m > 0 is admissible, although the connection to an underlying Momentum Process of course only exists for integer m. The Brownian Momentum Process was introduced in Ref. 19 with imposed-temperature boundary conditions, and further studied in Refs. 20-22.…”

Section: And This Is Again a Markov Process With Generatormentioning

confidence: 99%

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

Peletier

Redig

Vafayi

2014

Journal of Mathematical Physics

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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, a Generalized Brownian Energy Process, and the Kipnis-Marchioro-Presutti 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 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.

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Duality and exact correlations for a model of heat conduction

Cited by 85 publications

References 20 publications

Nonequilibrium Invariant Measure under Heat Flow

Nonequilibrium Invariant Measure under Heat Flow

Duality and Hidden Symmetries in Interacting Particle Systems

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

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