Results 1  10
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30
Isotropic hypoellipticity and trend to the equilibrium for the FokkerPlanck equation with high degree potential
, 2002
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Fluctuations of the entropy production in anharmonic chains
 Ann. Henri Poincare
, 2002
"... Abstract. We prove the GallavottiCohen fluctuation theorem for a model of heat conduction through a chain of anharmonic oscillators coupled to two Hamiltonian reservoirs at different temperatures. 1 ..."
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Cited by 22 (4 self)
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Abstract. We prove the GallavottiCohen fluctuation theorem for a model of heat conduction through a chain of anharmonic oscillators coupled to two Hamiltonian reservoirs at different temperatures. 1
Spectral Properties of Hypoelliptic Operators
 Commun. Math. Phys
, 2003
"... We study hypoelliptic operators with polynomially bounded coefficients that are of the form K = i X i + X0 + f , where the X j denote first order differential operators, f is a function with at most polynomial growth, and X i denotes the formal adjoint of X i in L . For any # > 0 we show t ..."
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Cited by 18 (0 self)
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We study hypoelliptic operators with polynomially bounded coefficients that are of the form K = i X i + X0 + f , where the X j denote first order differential operators, f is a function with at most polynomial growth, and X i denotes the formal adjoint of X i in L . For any # > 0 we show that an inequality of the form C(#u#0,# + + iy)u#0,0 ) holds for suitable # and C which are independent of R, in weighted Sobolev spaces (the first index is the derivative, and the second the growth). We apply this result to the FokkerPlanck operator for an anharmonic chain of oscillators coupled to two heat baths. Using a method of H erau and Nier [HN02], we conclude that its spectrum lies in a cusp {x+ iy # y c, # (0, 1], c R}.
Ergodicity of the finite dimensional approximation of the 3d navier–stokes equations forced by a degenerate
, 2002
"... Abstract. We prove ergodicity of the finite dimensional approximations of the three dimensional NavierStokes equations, driven by a random force. The forcing noise acts only on a few modes and some algebraic conditions on the forced modes are found that imply the ergodicity. The convergence rate to ..."
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Cited by 11 (1 self)
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Abstract. We prove ergodicity of the finite dimensional approximations of the three dimensional NavierStokes equations, driven by a random force. The forcing noise acts only on a few modes and some algebraic conditions on the forced modes are found that imply the ergodicity. The convergence rate to the unique invariant measure is shown to be exponential. 1.
Large deviations of lattice Hamiltonian dynamics coupled to stochastic thermostats.
, 802
"... We discuss the DonskerVaradhan theory of large deviations in the framework of Hamiltonian systems thermostated by a Gaussian stochastic coupling. We derive a general formula for the DonskerVaradhan large deviation functional for dynamics which satisfy natural properties under time reversal. Next, ..."
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Cited by 6 (0 self)
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We discuss the DonskerVaradhan theory of large deviations in the framework of Hamiltonian systems thermostated by a Gaussian stochastic coupling. We derive a general formula for the DonskerVaradhan large deviation functional for dynamics which satisfy natural properties under time reversal. Next, we discuss the characterization of the stationary state as the solution of a variational principle and its relation to the minimum entropy production principle. Finally, we compute the large deviation functional of the current in the case of a harmonic chain thermostated by a Gaussian stochastic coupling. 1 Introduction. Attempts to study large systems out of equilibrium through fluctuation theory has received a lot of attention in recent years [2, 3, 4, 5, 16, 18, 19, 20]. In a recent series of papers [27, 8, 28, 29, 30], it has been understood that in random systems driven out of equilibrium, the theory of large deviations provides naturally a variational
Statistical Mechanics of anharmonic lattices
 In Advances in Differential Equations and Mathematical Physics, Contemporary Mathematics 327
, 2003
"... Abstract. We discuss various aspects of a series of recent works on the nonequilibrium stationary states of anharmonic crystals coupled to heat reservoirs (see also [7]). We expose some of the main ideas and techniques and also present some open problems. 1. ..."
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Cited by 5 (1 self)
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Abstract. We discuss various aspects of a series of recent works on the nonequilibrium stationary states of anharmonic crystals coupled to heat reservoirs (see also [7]). We expose some of the main ideas and techniques and also present some open problems. 1.
Exponential Mixing for a Stochastic PDE Driven by Degenerate Noise
, 2001
"... We study stochastic partial differential equations of the reactiondiffusion type. ..."
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Cited by 3 (1 self)
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We study stochastic partial differential equations of the reactiondiffusion type.
NonEquilibrium Steady States
, 2002
"... The mathematical physics of mechanical systems in thermal equilibrium is a well studied, and relatively easy, subject, because the Gibbs distribution is in general an adequate guess for the equilibrium state. On the other hand, the mathematical physics of nonequilibrium systems, such as that of a c ..."
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Cited by 3 (0 self)
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The mathematical physics of mechanical systems in thermal equilibrium is a well studied, and relatively easy, subject, because the Gibbs distribution is in general an adequate guess for the equilibrium state. On the other hand, the mathematical physics of nonequilibrium systems, such as that of a chain of masses connected with springs to two (infinite) heat reservoirs is more difficult, precisely because no such a priori guess exists. Recent work has, however, revealed that under quite general conditions, such states can not only be shown to exist, but are unique, using the H"ormander conditions and controllability. Furthermore, interesting properties, such as energy flux, exponentially fast convergence to the unique state, and fluctuations of that state have been successfully studied. Finally, the ideas used in these studies can be extended to certain stochastic PDE's using Malliavin calculus to prove regularity of the process.
Strange Heat Flux in (An)Harmonic Networks
, 2003
"... We study the heat transport in systems of coupled oscillators driven out of equilibrium by Gaussian heat baths. We illustrate with a few examples that such systems can exhibit "strange" transport phenomena. In particular, circulation of heat flux may appear in the steady state of a system of three o ..."
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Cited by 2 (1 self)
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We study the heat transport in systems of coupled oscillators driven out of equilibrium by Gaussian heat baths. We illustrate with a few examples that such systems can exhibit "strange" transport phenomena. In particular, circulation of heat flux may appear in the steady state of a system of three oscillators only. This indicates that the direction of the heat fluxes can in general not be "guessed" from the temperatures of the heat baths. Although we primarily consider harmonic couplings between the oscillators, we explain why this strange behavior persists under weak anharmonic perturbations.
How hot can a heat bath get?
, 2008
"... We study a model of two interacting Hamiltonian particles subject to a common potential in contact with two Langevin heat reservoirs: one at finite and one at infinite temperature. This is a toy model for ‘extreme ’ nonequilibrium statistical mechanics. We provide a full picture of the longtime be ..."
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We study a model of two interacting Hamiltonian particles subject to a common potential in contact with two Langevin heat reservoirs: one at finite and one at infinite temperature. This is a toy model for ‘extreme ’ nonequilibrium statistical mechanics. We provide a full picture of the longtime behaviour of such a system, including the existence / nonexistence of a nonequilibrium steady state, the precise tail behaviour of the energy in such a state, as well as the speed of convergence toward the steady state. Despite its apparent simplicity, this model exhibits a surprisingly rich variety of long time behaviours, depending on the parameter regime: if the surrounding potential is ‘too stiff’, then no stationary state can exist. In the softer regimes, the tails of the energy in the stationary state can be either algebraic, fractional exponential, or exponential. Correspondingly, the speed of convergence to the stationary state can be either algebraic, stretched exponential, or exponential. Regarding both types of claims, we obtain matching upper and lower bounds.