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49
Fourier’s law for a harmonic crystal with selfconsistent stochastic reservoirs
, 2004
"... We consider a ddimensional 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 TL and TR, respectively, while the temperatures of the ‘‘interior’ ’ baths are chosen selfconsisten ..."
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Cited by 25 (3 self)
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We consider a ddimensional 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 TL and TR, respectively, while the temperatures of the ‘‘interior’ ’ baths are chosen selfconsistently 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 (add) −1 where ad is the coupling to the intermediate reservoirs. We also analyze the effect of having a nonuniform distribution of the heat bath couplings. These results are proven rigorously by controlling the behavior of the correlations in the thermodynamic limit. KEY WORDS: Fourier’s law; harmonic crystal; nonequilibrium systems; thermodynamic limit; Green–Kubo formula.
Non equilibrium current fluctuations in stochastic lattice gases
 J. Stat. Phys
, 2006
"... We study current fluctuations in lattice gases in the macroscopic limit extending the dynamic approach for density fluctuations developed in previous articles. More precisely, we establish a large deviation principle for a spacetime fluctuation j of the empirical current with a rate functional I(j) ..."
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Cited by 14 (2 self)
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We study current fluctuations in lattice gases in the macroscopic limit extending the dynamic approach for density fluctuations developed in previous articles. More precisely, we establish a large deviation principle for a spacetime fluctuation j of the empirical current with a rate functional I(j). We then estimate the probability of a fluctuation of the average current over a large time interval; this probability can be obtained by solving a variational problem for the functional I. We discuss several possible scenarios, interpreted as dynamical phase transitions, for this variational problem. They actually occur in specific models. We finally discuss the time reversal properties of I and derive a fluctuation relationship akin to the GallavottiCohen theorem for the entropy production.
The asymmetric simple exclusion process : an integrable model for nonequilibrium statistical mechanics
 J. Phys. A: Math. Gen
, 2006
"... The Asymmetric Simple Exclusion Process (ASEP) plays the role of a paradigm in NonEquilibrium Statistical Mechanics. We review exact results for the ASEP obtained by Bethe Ansatz and put emphasis on the algebraic properties of this model. The Bethe equations for the eigenvalues of the Markov Matrix ..."
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Cited by 12 (3 self)
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The Asymmetric Simple Exclusion Process (ASEP) plays the role of a paradigm in NonEquilibrium Statistical Mechanics. We review exact results for the ASEP obtained by Bethe Ansatz and put emphasis on the algebraic properties of this model. The Bethe equations for the eigenvalues of the Markov Matrix of the ASEP are derived from the algebraic Bethe Ansatz. Using these equations we explain how to calculate the spectral gap of the model and how global spectral properties such as the existence of multiplets can be predicted. An extension of the Bethe Ansatz leads to an analytic expression for the large deviation function of the current in the ASEP that satisfies the GallavottiCohen relation. Finally, we describe some variants of the ASEP that are also solvable by Bethe Ansatz. PACS numbers: 0540.a;0560.k
A local fluctuation theorem
 Journal of Physics A
, 1998
"... a simple consequence of a time reversal symmetry; it deals with motions which are chaotic in the strong mathematical sense of being hyperbolic and transitive (ie are generated by smooth hyperbolic evolutions on a smooth compact surface (the “phase space”) and with a dense trajectory, also called Ano ..."
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Cited by 10 (2 self)
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a simple consequence of a time reversal symmetry; it deals with motions which are chaotic in the strong mathematical sense of being hyperbolic and transitive (ie are generated by smooth hyperbolic evolutions on a smooth compact surface (the “phase space”) and with a dense trajectory, also called Anosov systems) and “furthermore are time reversible”. In such systems any initial data, with the exception of a set of zero volume in phase space, have the same statistical properties in the sense that all smooth observables admit a time average independent of the initial data and expressed as an integral with respect to a probability distribution on phase space, called the ”natural stationary state”, or simply the “stationary state”. The theorem provides, asymptotically in the observation time, a quantitative and parameter free relation between the stationary state probability of observing a value of the average entropy production rate and its opposite. Although there are quite a few examples of mechanical systems which are hyperbolic and transitive in the above mathematical sense, the fluctuation theorem acquires
Nonequilibrium fluctuations in small systems: From physics to biology
 Advances in Chemical Physics
, 2006
"... In this paper I am presenting an overview on several topics related to nonequilibrium fluctuations in small systems. I start with a general discussion about fluctuation theorems and applications to physical examples extracted from physics and biology: a bead in an optical trap and single molecule fo ..."
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Cited by 9 (1 self)
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In this paper I am presenting an overview on several topics related to nonequilibrium fluctuations in small systems. I start with a general discussion about fluctuation theorems and applications to physical examples extracted from physics and biology: a bead in an optical trap and single molecule force experiments. Next I present a general discussion on path thermodynamics and consider distributions of work/heat fluctuations as large deviation functions. Then I address the topic of glassy dynamics from the perspective of nonequilibrium fluctuations due to small cooperatively rearranging regions. Finally, I
Fluctuation relations for diffusion process
 Commun. Math. Phys
"... The paper presents a unified approach to different fluctuation relations for classical nonequilibrium dynamics described by diffusion processes. Such relations compare the statistics of fluctuations of the entropy production or work in the original process to the similar statistics in the timerever ..."
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Cited by 7 (3 self)
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The paper presents a unified approach to different fluctuation relations for classical nonequilibrium dynamics described by diffusion processes. Such relations compare the statistics of fluctuations of the entropy production or work in the original process to the similar statistics in the timereversed process. The origin of a variety of fluctuation relations is traced to the use of different time reversals. It is also shown how the application of the presented approach to the tangent process describing the joint evolution of infinitesimally close trajectories of the original process leads to a multiplicative extension of the fluctuation relations. 1
K.: Fluctuation relations in simple examples of nonequilibrium steady
, 2008
"... We discuss fluctuation relations in simple cases of nonequilibrium Langevin dynamics. In particular, we show that close to nonequilibrium steady states with nonvanishing probability currents some of these relations reduce to a modified version of the fluctuationdissipation theorem. The latter ma ..."
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Cited by 7 (3 self)
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We discuss fluctuation relations in simple cases of nonequilibrium Langevin dynamics. In particular, we show that close to nonequilibrium steady states with nonvanishing probability currents some of these relations reduce to a modified version of the fluctuationdissipation theorem. The latter may be interpreted as the equilibriumlike relation in the reference frame moving with the mean local velocity determined by the probability current. 1
From Global to Local Fluctuation Theorems
 Moscow Mathematical Journal
, 2001
"... The GallavottiCohen uctuation theorem suggests a general symmetry in the fluctuations of the entropy production, a basic concept in the theory of irreversible processes, based on results in the theory of strongly chaotic maps. We study this symmetry for some standard models of nonequilibrium steady ..."
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Cited by 7 (4 self)
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The GallavottiCohen uctuation theorem suggests a general symmetry in the fluctuations of the entropy production, a basic concept in the theory of irreversible processes, based on results in the theory of strongly chaotic maps. We study this symmetry for some standard models of nonequilibrium steady states. We give a general strategy to derive a local uctuation theorem exploiting the Gibbsian features of the stationary spacetime distribution. This is applied to spin ip processes and to the asymmetric exclusion process.
Heat and Fluctuations from Order to Chaos
, 2008
"... The Heat theorem reveals the second law of equilibrium Thermodynamics (i.e.existence of Entropy) as a manifestation of a general property of Hamiltonian Mechanics and of the Ergodic Hypothesis, valid for 1 as well as 10 23 degrees of freedom systems, i.e. for simple as well as very complex systems, ..."
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Cited by 6 (5 self)
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The Heat theorem reveals the second law of equilibrium Thermodynamics (i.e.existence of Entropy) as a manifestation of a general property of Hamiltonian Mechanics and of the Ergodic Hypothesis, valid for 1 as well as 10 23 degrees of freedom systems, i.e. for simple as well as very complex systems, and reflecting the Hamiltonian nature of the microscopic motion. In Nonequilibrium Thermodynamics theorems of comparable generality do not seem to be available. Yet it is possible to find general, model independent, properties valid even for simple chaotic systems (i.e. the hyperbolic ones), which acquire special interest for large systems: the Chaotic Hypothesis leads to the Fluctuation Theorem which provides general properties of certain very large fluctuations and reflects the timereversal symmetry. Implications on Fluids and Quantum systems are briefly hinted. The physical meaning of the Chaotic Hypothesis, of SRB distributions and of the Fluctuation Theorem
Properties of stationary nonequilibrium states in the thermostatted periodic Lorentz gas III: The many colliding particles system, in preparation
"... We study numerically and analytically the properties of the stationary state of a particle moving under the influence of an electric field E in a two dimensional periodic Lorentz gas with the energy kept constant by a Gaussian thermostat. Numerically the current appears to be a continuous function o ..."
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Cited by 6 (0 self)
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We study numerically and analytically the properties of the stationary state of a particle moving under the influence of an electric field E in a two dimensional periodic Lorentz gas with the energy kept constant by a Gaussian thermostat. Numerically the current appears to be a continuous function of E whose derivative varies very irregularly, possibly in a discontinuous manner. We argue for the non differentiability of the current as a function of E utilizing a symbolic description of the dynamics based on the discontinuities of the collision map. The decay of correlations and the behavior of the diffusion constant are also investigated. KEY WORDS: Thermostatted Lorentz gas; steady state current; smoothness; regularity; symbolic dynamics.