Introduction General framework Illustration on simple models Conclusions Entropy-based characterizations of the observable-dependence of the fluctuation-dissipation temperature Michel Droz 1 Kirsten Martens 1 , 2 , Eric Bertin 3 1 Department of Theoretical Physics, University of Geneva, Switzerland 2 Université de Lyon; Université Lyon 1 Laboratoire de Physique de la Matière Condensée et des Nanostructures, France 3 Université de Lyon, Laboratoire de Physique, ENS Lyon, France MPI Dresden, 2011 Michel Droz
Introduction General framework Illustration on simple models Conclusions Outline Introduction 1 What is the problem? Fluctuation-dissipation relation General framework 2 Evaluation of the response function Properties of the phase-space distribution Illustration on simple models 3 A. Simple energy transfert model on a ring B. Fully connected model driven by two heat baths C. Slow relaxation model Conclusions 4 Michel Droz
Introduction General framework What is the problem? Illustration on simple models Fluctuation-dissipation relation Conclusions Outline Introduction 1 What is the problem? Fluctuation-dissipation relation General framework 2 Evaluation of the response function Properties of the phase-space distribution Illustration on simple models 3 A. Simple energy transfert model on a ring B. Fully connected model driven by two heat baths C. Slow relaxation model Conclusions 4 Michel Droz
Introduction General framework What is the problem? Illustration on simple models Fluctuation-dissipation relation Conclusions The definition of macroscopic quantities that can characterize nonequilibrium systems is a challenging question. In particular, the possibility to define an effective temperature in nonequilibrium systems has been studied in different frameworks. The introduction of effective temperatures in nonequilibrium systems through generalized fluctuation-dissipation relations (FDR) has played a major role. Michel Droz
Introduction General framework What is the problem? Illustration on simple models Fluctuation-dissipation relation Conclusions Outline Introduction 1 What is the problem? Fluctuation-dissipation relation General framework 2 Evaluation of the response function Properties of the phase-space distribution Illustration on simple models 3 A. Simple energy transfert model on a ring B. Fully connected model driven by two heat baths C. Slow relaxation model Conclusions 4 Michel Droz
Introduction General framework What is the problem? Illustration on simple models Fluctuation-dissipation relation Conclusions In equilibrium, the linear response function of an observable to a small perturbation is proportional to the nonperturbed correlation function of the corresponding fluctuations (Fluctuation Dissipation Relation). This is a very strong property, because it is independent of the details of the microscopic dynamics and of the observable considered. This relation gives rise to a universal proportionality factor, precisely given by the equilibrium temperature. Michel Droz
Introduction General framework What is the problem? Illustration on simple models Fluctuation-dissipation relation Conclusions In equilibrium, the linear response function of an observable to a small perturbation is proportional to the nonperturbed correlation function of the corresponding fluctuations (Fluctuation Dissipation Relation). This is a very strong property, because it is independent of the details of the microscopic dynamics and of the observable considered. This relation gives rise to a universal proportionality factor, precisely given by the equilibrium temperature. Michel Droz
Introduction General framework What is the problem? Illustration on simple models Fluctuation-dissipation relation Conclusions In equilibrium, the linear response function of an observable to a small perturbation is proportional to the nonperturbed correlation function of the corresponding fluctuations (Fluctuation Dissipation Relation). This is a very strong property, because it is independent of the details of the microscopic dynamics and of the observable considered. This relation gives rise to a universal proportionality factor, precisely given by the equilibrium temperature. Michel Droz
Introduction General framework What is the problem? Illustration on simple models Fluctuation-dissipation relation Conclusions However, in a nonequilibrium system this FDR relation is a priori not valid. Even if FRD is valid, it remains the question of the observable-dependence of the corresponding effective temperature. Michel Droz
Introduction General framework What is the problem? Illustration on simple models Fluctuation-dissipation relation Conclusions However, in a nonequilibrium system this FDR relation is a priori not valid. Even if FRD is valid, it remains the question of the observable-dependence of the corresponding effective temperature. Michel Droz
Introduction General framework What is the problem? Illustration on simple models Fluctuation-dissipation relation Conclusions In this work, we study how the characteristics of a nonequilibrium distribution of the microstates influence the possibility to define an observable-independent temperature in the system. Michel Droz
Introduction General framework What is the problem? Illustration on simple models Fluctuation-dissipation relation Conclusions Our main result is that for a class of systems: Close to equilibrium With stochastic markovian dynamics For which the degrees of freedom are statistically independent Michel Droz
Introduction General framework What is the problem? Illustration on simple models Fluctuation-dissipation relation Conclusions We relate the observable-dependence of the FDR-temperature to a fundamental characteristic of the nonequilibrium system, namely the Shanon entropy difference with respect to the equilibrium state with the same energy. This difference (or the “lack of entropy” ∆ S ), is defined as ∆ S = S eq ( β ∗ ) − S neq (1) where β ∗ is such that the average energy of both systems are the same. When ∆ S > 0, the effective temperature depends on the observable. This dependence generically occurs when the phase space probability distribution is nonuniform on constant energy shells. Michel Droz
Introduction General framework Evaluation of the response function Illustration on simple models Properties of the phase-space distribution Conclusions Outline Introduction 1 What is the problem? Fluctuation-dissipation relation General framework 2 Evaluation of the response function Properties of the phase-space distribution Illustration on simple models 3 A. Simple energy transfert model on a ring B. Fully connected model driven by two heat baths C. Slow relaxation model Conclusions 4 Michel Droz
Introduction General framework Evaluation of the response function Illustration on simple models Properties of the phase-space distribution Conclusions We shall consider a generic system that is described by a set of N variables x i , i = 1 , . . . , N . We introduce a family of observables B p indexed by an integer p . A small extenal field h , conjugated to the observable M puts the system in a nonequlibriun steady-state. This response will then be related to the fluctuations in the system in the absence of perturbation. Michel Droz
Introduction General framework Evaluation of the response function Illustration on simple models Properties of the phase-space distribution Conclusions The following protocol allows for the definition of the linear response of the observable B p to the external probe field. The field h takes a constant and small non-zero value until time t s , and it is then suddenly switched off. Michel Droz
Introduction General framework Evaluation of the response function Illustration on simple models Properties of the phase-space distribution Conclusions The subsequent evolution of the observable B p then provides the linear response to the probe field. The two-time linear response χ p ( t , t s ) is defined, for t > t s , as � χ p ( t , t s ) = ∂ � B p ( t , t s ) � � � � , (2) ∂ h � h = 0 � where � �· · · � � denotes an average over the dynamics corresponding to the field protocol described above. Michel Droz
Introduction General framework Evaluation of the response function Illustration on simple models Properties of the phase-space distribution Conclusions The basic idea of the FDR is to relate the linear response function χ p ( t , t s ) to the correlation function (computed in the absence of field) C p ( t , t s ) = � ( B p ( t ) − � B p ( t ) � ) ( M ( t s ) − � M ( t s ) � ) � . (3) In general, such a relation is not necessarily linear. However, in cases when it is linear, a FDR is said to hold, namely 1 χ p ( t , t s ) = T p ( t s ) C p ( t , t s ) , t > t s . (4) The proportionality factor is the inverse of the effective temperature T p . Michel Droz
Introduction General framework Evaluation of the response function Illustration on simple models Properties of the phase-space distribution Conclusions In the specific case of nonequilibrium steady state, the above FDR simplifies to, setting t s = 0, χ p ( t ) = 1 C p ( t ) , (5) T p In the following we will consider situations such that a fluctuation-dissipation relation exists, and we shall focus on the possible dependence of T p on the choice of the observable B p . Michel Droz
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