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59
Ergodicity for langevin processes with degenerate diffusion in momentums. Submitted; arXiv
, 2007
"... This paper presents sufficient conditions for proving ergodicity of noisedriven dynamical systems. The essential conditions are weak irreducibility and closure under second randomization of the driving noise. With these conditions one can ascertain ergodicity of Langevin processes even if the diffu ..."
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This paper presents sufficient conditions for proving ergodicity of noisedriven dynamical systems. The essential conditions are weak irreducibility and closure under second randomization of the driving noise. With these conditions one can ascertain ergodicity of Langevin processes even if the diffusion and drift matrices associated to the momentums are degenerate. The paper illustrates how to check these conditions practically in the context of a simple mechanical system governed by Langevin equations (a simple stochastic rigid body system). 1
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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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.
Multiscale expansion of invariant measures for SPDEs
, 2003
"... We derive the first two terms in an εexpansion for the invariant measure of a class of semilinear parabolic SPDEs near a change of stability, when the noise strength and the linear instability are of comparable order ε 2. This result gives insight into the stochastic bifurcation and allows to rigor ..."
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We derive the first two terms in an εexpansion for the invariant measure of a class of semilinear parabolic SPDEs near a change of stability, when the noise strength and the linear instability are of comparable order ε 2. This result gives insight into the stochastic bifurcation and allows to rigorously approximate correlation functions. The error between the approximate and the true invariant measure is bounded in both the Wasserstein and the total variation distance.
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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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.
Uniform bounds and exponential time decay . . .
, 2005
"... We consider the nonlinear VPFP system with a coulombian repulsive interaction potential and a generic confining potential in space dimension d ≥ 3. Using spectral and kinetic methods we prove the existence and uniqueness of a mild solution in weighted spaces and for small charge we find an explicit ..."
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We consider the nonlinear VPFP system with a coulombian repulsive interaction potential and a generic confining potential in space dimension d ≥ 3. Using spectral and kinetic methods we prove the existence and uniqueness of a mild solution in weighted spaces and for small charge we find an explicit exponential rate of convergence to the equilibrium in terms of the Witten Laplacian associated to the linear equation.
Exponential return to equilibrium for hypoelliptic quadratic systems
 J. Funct. Anal
"... Abstract. We study the problem of convergence to equilibrium for evolution equations associated to general quadratic operators. Quadratic operators are nonselfadjoint differential operators with complexvalued quadratic symbols. Under appropriate assumptions, a complete description of the spectrum o ..."
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Abstract. We study the problem of convergence to equilibrium for evolution equations associated to general quadratic operators. Quadratic operators are nonselfadjoint differential operators with complexvalued quadratic symbols. Under appropriate assumptions, a complete description of the spectrum of such operators is given and the exponential return to equilibrium with sharp estimates on the rate of convergence is proven. Some applications to the study of chains of oscillators and the generalized Langevin equation are given. 1.
Steady state thermodynamics
, 2004
"... We propose a thermodynamic formalism that is expected to apply to a large class of nonequilibrium steady states including a heat conducting fluid, a sheared fluid, and an electrically conducting fluid. We call our theory steady state thermodynamics (SST) after Oono and Paniconi’s original proposal. ..."
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We propose a thermodynamic formalism that is expected to apply to a large class of nonequilibrium steady states including a heat conducting fluid, a sheared fluid, and an electrically conducting fluid. We call our theory steady state thermodynamics (SST) after Oono and Paniconi’s original proposal. The construction of SST is based on a careful examination of how the basic notions in thermodynamics should be modified in nonequilibrium steady states. We define all thermodynamic quantities through operational procedures, which can be (in principle) realized experimentally. Based on SST thus constructed, we make some nontrivial predictions, including an extension of Einstein’s formula on density fluctuation, an extension of the minimum work principle, the existence of a new osmotic pressure of a purely nonequilibrium origin, and a shift of coexistence temperature. All these predictions may be checked experimentally to test SST for its quantitative validity. Contents
METROPOLIS INTEGRATION SCHEMES FOR SELFADJOINT DIFFUSIONS∗
"... Abstract. We present explicit methods for simulating diffusions whose generator is selfadjoint with respect to a known (but possibly not normalizable) density. These methods exploit this property and combine an optimized Runge–Kutta algorithm with a Metropolis–Hastings Monte Carlo scheme. The resul ..."
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Abstract. We present explicit methods for simulating diffusions whose generator is selfadjoint with respect to a known (but possibly not normalizable) density. These methods exploit this property and combine an optimized Runge–Kutta algorithm with a Metropolis–Hastings Monte Carlo scheme. The resulting numerical integration scheme is shown to be weakly accurate at finite noise and to gain higher order accuracy in the small noise limit. It also permits the user to avoid computing explicitly certain terms in the equation, such as the divergence of the mobility tensor, which can be tedious to calculate. Finally, the scheme is shown to be ergodic with respect to the exact equilibrium probability distribution of the diffusion when it exists. These results are illustrated in several examples, including a Brownian dynamics simulation of DNA in a solvent. In this example, the proposed scheme is able to accurately compute dynamics at time step sizes that are an order of magnitude (or more) larger than those permitted with commonly used explicit predictorcorrector schemes. Key words. explicit integrators, Brownian dynamics with hydrodynamic interactions,
Zygalakis. Homogenization for inertial particles in a random flow
 Commun. Math. Sci
"... Abstract. We study the problem of homogenization for inertial particles moving in a time dependent random velocity field and subject to molecular diffusion. We show that, under appropriate assumptions on the velocity field, the large–scale, long–time behavior of the inertial particles is governed by ..."
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Abstract. We study the problem of homogenization for inertial particles moving in a time dependent random velocity field and subject to molecular diffusion. We show that, under appropriate assumptions on the velocity field, the large–scale, long–time behavior of the inertial particles is governed by an effective diffusion equation for the position variable alone. This is achieved by the use of a formal multiple scales expansion in the scale parameter. The expansion relies on the hypoellipticity of the underlying diffusion. An expression for the diffusivity tensor is found and various of its properties are studied. The results of the formal multiscale analysis are justified rigorously by the use of the martingale central limit theorem. Our theoretical findings are supported by numerical investigations where we study the parametric dependence of the effective diffusivity on the various non–dimensional parameters of the problem. Key words. Homogenization theory, multiscale analysis, martingale central limit theorem, hypoelliptic diffusions, Gaussian velocity fields. subject classifications. 1. Introduction Inertial