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Heat conduction and entropy production in anharmonic crystals with selfconsistent stochastic reservoirs
, 2008
"... Abstract. We investigate a class of anharmonic crystals in d dimensions, d ≥ 1, coupled to both external and internal heat baths of the OrnsteinUhlenbeck type. The external heat baths, applied at the boundaries in the 1direction, are at specified, unequal, temperatures Tl and Tr. The temperatures ..."
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Cited by 6 (2 self)
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Abstract. We investigate a class of anharmonic crystals in d dimensions, d ≥ 1, coupled to both external and internal heat baths of the OrnsteinUhlenbeck type. The external heat baths, applied at the boundaries in the 1direction, are at specified, unequal, temperatures Tl and Tr. The temperatures of the internal baths are determined in a selfconsistent way by the requirement that there be no net energy exchange with the system in the nonequilibrium stationary state (NESS). We prove the existence of such a stationary selfconsistent profile of temperatures for a finite system and show it minimizes the entropy production to leading order in (Tl − Tr). In the NESS the heat conductivity κ is defined as the heat flux per unit area divided by the length of the system and (Tl − Tr). In the limit when the temperatures of the external reservoirs goes to the same temperature T, κ(T) is given by the GreenKubo formula, evaluated in an equilibrium system coupled to reservoirs all having the temperature T. This κ(T) remains bounded as the size of the system goes to infinity. We also show that the corresponding infinite system GreenKubo formula yields a finite result. Stronger results are obtained under the assumption that the selfconsistent profile remains bounded. 1.
Nonequilibrium linear response for Markov dynamics, I: jump processes and overdamped diffusions
, 2009
"... Systems out of equilibrium, in stationary as well as in nonstationary regimes, display a linear response to energy impulses simply expressed as the sum of two specific temporal correlation functions. There is a natural interpretation of these quantities. The first term corresponds to the correlation ..."
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Cited by 5 (5 self)
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Systems out of equilibrium, in stationary as well as in nonstationary regimes, display a linear response to energy impulses simply expressed as the sum of two specific temporal correlation functions. There is a natural interpretation of these quantities. The first term corresponds to the correlation between observable and excess entropy flux yielding a relation with energy dissipation like in equilibrium. The second term comes with a new meaning: it is the correlation between the observable and the excess in dynamical activity or reactivity, playing an important role in dynamical fluctuation theory outofequilibrium. It appears as a generalized escape rate in the occupation statistics. The resulting response formula holds for all observables and allows direct numerical or experimental evaluation, for example in the discussion of effective temperatures, as it only involves the statistical averaging of explicit quantities, e.g. without needing an expression for the nonequilibrium distribution. The physical interpretation and the mathematical derivation are independent of many details of the dynamics, but in this first part they are restricted to Markov jump processes and overdamped diffusions.
Density Estimates for a Random Noise Propagating through a Chain of Di erential Equations
, 2009
"... We here provide two sided bounds for the density of the solution of a system of n di erential equations of dimension d, the rst one being forced by a nondegenerate random noise and the n − 1 other ones being degenerate. The system formed by the n equations satis es a suitable Hörmander condition: t ..."
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We here provide two sided bounds for the density of the solution of a system of n di erential equations of dimension d, the rst one being forced by a nondegenerate random noise and the n − 1 other ones being degenerate. The system formed by the n equations satis es a suitable Hörmander condition: the second equation feels the noise plugged into the rst equation, the third equation feels the noise transmitted from the rst to the second equation and so on..., so that the noise propagates one way through the system. When the coe cients of the system are Lipschitz continuous, we show that the density of the solution satis es Gaussian bounds with nondi usive time scales. The proof relies on the interpretation of the density of the solution as the value function of some optimal stochastic control problem.
JStatPhys DOI 10.1007/s1095500998528 Nonequilibrium Linear Response for Markov Dynamics, I: Jump Processes and Overdamped Diffusions
, 2009
"... Abstract Systems out of equilibrium, in stationary as well as in nonstationary regimes, display a linear response to energy impulses simply expressed as the sum of two specific temporal correlation functions. There is a natural interpretation of these quantities. The first term corresponds to the co ..."
Abstract
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Abstract Systems out of equilibrium, in stationary as well as in nonstationary regimes, display a linear response to energy impulses simply expressed as the sum of two specific temporal correlation functions. There is a natural interpretation of these quantities. The first term corresponds to the correlation between observable and excess entropy flux yielding a relation with energy dissipation like in equilibrium. The second term comes with a new meaning: it is the correlation between the observable and the excess in dynamical activity or reactivity, playing an important role in dynamical fluctuation theory outofequilibrium. It appears as a generalized escape rate in the occupation statistics. The resulting response formula holds for all observables and allows direct numerical or experimental evaluation, for example in the discussion of effective temperatures, as it only involves the statistical averaging of explicit quantities, e.g. without needing an expression for the nonequilibrium distribution. The physical interpretation and the mathematical derivation are independent of many details of the dynamics, but in this first part they are restricted to Markov jump processes and overdamped diffusions.
Progress of Theoretical Physics Supplement 1 Fluctuations and response outofequilibrium
"... We discuss some recently visited positions towards dealing with nonequilibria from the mathematical point of view of Markov networks. Statistical mechanics concentrates on deriving, correcting and extending thermodynamic treatments of multicomponent systems. Passing via different levels of mesoscopi ..."
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We discuss some recently visited positions towards dealing with nonequilibria from the mathematical point of view of Markov networks. Statistical mechanics concentrates on deriving, correcting and extending thermodynamic treatments of multicomponent systems. Passing via different levels of mesoscopic descriptions it builds bridges between more unique and fundamental microscopic laws and the plurality of macroscopic phenomena. In the absence of a well
Membres du jury:
, 2009
"... Probabilités et mécanique statistique hors équilibre Mémoire déposé en vue de l’obtention de l’habilitation à diriger des recherches. ..."
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Probabilités et mécanique statistique hors équilibre Mémoire déposé en vue de l’obtention de l’habilitation à diriger des recherches.