Heat flow in an exactly solvable model

Kipnis, Claude; Marchioro, Carlo; Presutti, Errico

doi:10.1007/bf01011740

Cited by 262 publications

(411 citation statements)

References 4 publications

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“…This is in the spirit of the general work in the last two decades proving the existence of hydrodynamical laws in the appropriate scaling limits for systems evolving via stochastic dynamics. (7) We should mention here in particular the work of Kipnis, Marchioro, and Presutti (21) who proved results similar to ours for a model with purely stochastic internal dynamics. They were in fact able to consider a situation where the energy is strictly conserved in the bulk rather than just in the average as in the model considered here.…”

Section: Discussionmentioning

confidence: 99%

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Fourier's Law for a Harmonic Crystal with Self-Consistent Stochastic Reservoirs

Bonetto

Lebowitz

Lukkarinen

2004

Journal of Statistical Physics

170

237

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We consider a d-dimensional harmonic crystal in contact with a stochastic Langevin type heat bath at each site. The temperatures of the ''exterior'' left and right heat baths are at specified values T L and T R , respectively, while the temperatures of the ''interior'' baths are chosen self-consistently so that there is no average flux of energy between them and the system in the steady state. We prove that this requirement uniquely fixes the temperatures and the self consistent system has a unique steady state. For the infinite system this state is one of local thermal equilibrium. The corresponding heat current satisfies Fourier's law with a finite positive thermal conductivity which can also be computed using the Green-Kubo formula. For the harmonic chain (d=1) the conductivity agrees with the expression obtained by Bolsterli, Rich, and Visscher in 1970 who first studied this model. In the other limit, d ± 1, the stationary infinite volume heat conductivity behaves as (a d d) −1 where a d is the coupling to the intermediate reservoirs. We also analyze the effect of having a non-uniform distribution of the heat bath couplings. These results are proven rigorously by controlling the behavior of the correlations in the thermodynamic limit.

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Section: Discussionmentioning

confidence: 99%

“…(2.17). The stationary covariance is then given by 21) and the stationary measure is the Gibbs measure at temperature T.…”

Section: C(toe T)=˛ementioning

confidence: 99%

Fourier's Law for a Harmonic Crystal with Self-Consistent Stochastic Reservoirs

Bonetto

Lebowitz

Lukkarinen

2004

Journal of Statistical Physics

170

237

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“…In this case all (nearest neighbor) pairs collide with equal probability P l,p = L −1 , independently of their energy. This choice (together with α = 1 above) corresponds to the Kipnis-Marchioro-Presutti (KMP) model of energy transport [23], which can be considered as a coarse-grained description of the physics of a large class of quasi-1D real diffusive systems. For instance, it is one of the very few instances where Fourier's law can be rigorously proved [23].…”

Section: A General Class Of Nonlinear Driven Dissipative Modelsmentioning

confidence: 99%

“…This choice (together with α = 1 above) corresponds to the Kipnis-Marchioro-Presutti (KMP) model of energy transport [23], which can be considered as a coarse-grained description of the physics of a large class of quasi-1D real diffusive systems. For instance, it is one of the very few instances where Fourier's law can be rigorously proved [23]. In addition, the KMP model has been used to investigate the validity of the additivity principle for current fluctuations [8] and the GallavottiCohen fluctuation theorem [5] and its generalization in refs.…”

Section: A General Class Of Nonlinear Driven Dissipative Modelsmentioning

confidence: 99%

“…In particular, when the colliding pair is chosen completely at random, independently of the value of its energy, the Kipnis-Marchioro-Presutti (KMP) model [23] for heat conduction is recovered in the conservative case. The KMP model plays a main role in nonequilibrium statistical physics as a touchstone to test theoretical advances [5,6,8,[11][12][13][14][15]23]. Our general class of models contains the essential ingredients characterizing most dissipative media, namely: (i) diffusive dynamics, (ii) bulk dissipation, and (iii) boundary injection.…”

Section: Introductionmentioning

confidence: 99%

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Renormalized time scale for anticipating and lagging synchronization

2016

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We consider fluctuations of the dissipated energy in nonlinear driven diffusive systems subject to bulk dissipation and boundary driving. With this aim, we extend the recently-introduced macroscopic fluctuation theory to nonlinear driven dissipative media, starting from the fluctuating hydrodynamic equations describing the system mesoscopic evolution. Interestingly, the action associated to a path in mesoscopic phase-space, from which large-deviation functions for macroscopic observables can be derived, has the same simple form as in non-dissipative systems. This is a consequence of the quasi-elasticity of microscopic dynamics, required in order to have a nontrivial competition between diffusion and dissipation at the mesoscale. Euler-Lagrange equations for the optimal density and current fields that sustain an arbitrary dissipation fluctuation are also derived. A perturbative solution thereof shows that the probability distribution of small fluctuations is always gaussian, as expected from the central limit theorem. On the other hand, strong separation from the gaussian behavior is observed for large fluctuations, with a distribution which shows no negative branch, thus violating the Gallavotti-Cohen fluctuation theorem as expected from the irreversibility of the dynamics. The dissipation large-deviation function exhibits simple and general scaling forms for weakly and strongly dissipative systems, with large fluctuations favored in the former case but heavily supressed in the latter. We apply our results to a general class of diffusive lattice models for which dissipation, nonlinear diffusion and driving are the key ingredients. The theoretical predictions are compared to extensive numerical simulations of the microscopic models, and excellent agreement is found. Interestingly, the large-deviation function is in some cases non-convex beyond some dissipation. These results show that a suitable generalization of macroscopic fluctuation theory is capable of describing in detail the fluctuating behavior of nonlinear driven dissipative media.

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Stochastic Duality and Eigenfunctions

Redig

2019

Springer Proceedings in Mathematics &Amp; Statistics

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We start from the observation that, anytime two Markov generators share an eigenvalue, the function constructed from the product of the two eigenfunctions associated to this common eigenvalue is a duality function. We push further this observation and provide a full characterization of duality relations in terms of spectral decompositions of the generators for finite state space Markov processes. Moreover, we study and revisit some well-known instances of duality, such as Siegmund duality, and extract spectral information from it. Next, we use the same formalism to construct all duality functions for some solvable examples, i.e., processes for which the eigenfunctions of the generator are explicitly known.

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Heat flow in an exactly solvable model

Cited by 262 publications

References 4 publications

Fourier's Law for a Harmonic Crystal with Self-Consistent Stochastic Reservoirs

Fourier's Law for a Harmonic Crystal with Self-Consistent Stochastic Reservoirs

Renormalized time scale for anticipating and lagging synchronization

Stochastic Duality and Eigenfunctions

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