Results 1  10
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20
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
Phase transitions and metastability in Markovian and molecular systems
, 2002
"... Diffusion models arising in analysis of large biochemical models and other complex systems are typically far too complex for exact solution, or even meaningful simulation. The purpose of this paper is to develop foundations for model reduction, and new modeling techniques for diffusion models. These ..."
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Cited by 21 (12 self)
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Diffusion models arising in analysis of large biochemical models and other complex systems are typically far too complex for exact solution, or even meaningful simulation. The purpose of this paper is to develop foundations for model reduction, and new modeling techniques for diffusion models. These foundations are all based upon recent spectral theory of Markov processes. The main assumption imposed is Vuniform ergodicity of the process. This is equivalent to any common formulation of exponential ergodicity, and is known to be far weaker than the DonskerVaradahn conditions in large deviations theory. Under this assumption it is shown that the associated semigroup admits a spectral gap in a weighted L∞norm, and real eigenfunctions provide a decomposition of the state space into ‘almost’absorbing subsets. It is shown that the process mixes rapidly in each of these subsets prior to exiting, and that the conditional distributions of exit times are approximately exponential. These results represent a significant expansion of the classical Wentzell–Freidlin theory. In particular, the results require no special structure beyond geometric ergodicity; reversibility is not assumed; and meaningful conclusions can be drawn even for models with significant variability.
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}.
Large deviations asymptotics and the spectral theory of multiplicatively regular Markov processes
 Electron. J. Probab
"... In this paper we continue the investigation of the spectral theory and exponential asymptotics of primarily discretetime Markov processes, following Kontoyiannis and Meyn [32]. We introduce a new family of nonlinear Lyapunov drift criteria, which characterize distinct subclasses of geometrically er ..."
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Cited by 16 (6 self)
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In this paper we continue the investigation of the spectral theory and exponential asymptotics of primarily discretetime Markov processes, following Kontoyiannis and Meyn [32]. We introduce a new family of nonlinear Lyapunov drift criteria, which characterize distinct subclasses of geometrically ergodic Markov processes in terms of simple inequalities for the nonlinear generator. We concentrate primarily on the class of multiplicatively regular Markov processes, which are characterized via simple conditions similar to (but weaker than) those of DonskerVaradhan. For any such process Φ = {Φ(t)} with transition kernel P on a general state space X, the following are obtained. Spectral Theory: For a large class of (possibly unbounded) functionals F: X → C, the kernel ̂ P (x, dy) = e F (x) P (x, dy) has a discrete spectrum in an appropriately defined Banach space. It follows that there exists a “maximal ” solution (λ, ˇ f) to the multiplicative Poisson equation, defined as the eigenvalue problem ̂ P ˇ f = λ ˇ f. The functional Λ(F) = log(λ) is convex, smooth, and its convex dual Λ ∗ is convex, with compact sublevel sets.
Metastability in Interacting Nonlinear Stochastic Differential Equations II: LargeN Behaviour
, 2006
"... We consider the dynamics of a periodic chain of N coupled overdamped particles under the influence of noise, in the limit of large N. Each particle is subjected to a bistable local potential, to a linear coupling with its nearest neighbours, and to an independent source of white noise. For strong co ..."
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Cited by 6 (3 self)
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We consider the dynamics of a periodic chain of N coupled overdamped particles under the influence of noise, in the limit of large N. Each particle is subjected to a bistable local potential, to a linear coupling with its nearest neighbours, and to an independent source of white noise. For strong coupling (of the order N 2), the system synchronises, in the sense that all oscillators assume almost the same position in their respective local potential most of the time. In a previous paper, we showed that the transition from strong to weak coupling involves a sequence of symmetrybreaking bifurcations of the system’s stationary configurations, and analysed in particular the behaviour for coupling intensities slightly below the synchronisation threshold, for arbitrary N. Here we describe the behaviour for any positive coupling intensity γ of order N 2, provided the particle number N is sufficiently large (as a function of γ/N 2). In particular, we determine the transition time between synchronised states, as well as the shape of the “critical droplet ” to leading order in 1/N. Our techniques involve the control of the exact number of periodic orbits of a nearintegrable twist map, allowing us to give a detailed description of the system’s potential landscape, in which the metastable behaviour is encoded.
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.
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.
On Complex Spectra and Metastability of Markov Models
, 2008
"... Abstract — The purpose of this paper is to develop methods for model reduction for diffusion processes that exhibit cyclic behavior. For this purpose we extend techniques based on the spectral theory of Markov processes to the case of complex spectra. The main idea is to augment the state process fo ..."
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Cited by 2 (2 self)
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Abstract — The purpose of this paper is to develop methods for model reduction for diffusion processes that exhibit cyclic behavior. For this purpose we extend techniques based on the spectral theory of Markov processes to the case of complex spectra. The main idea is to augment the state process for the diffusion with a clock process. For each complex eigenvalue for the original diffusion there exists a real eigenvalue for the augmented process. Results concerning metastability (or quasistationarity) are then applied to the augmented process. For the special case of a linear diffusion in two dimensions, this is analogous to analyzing the process in a rotating coordinate frame. The results are illustrated through a linear diffusion, and an empirical model of combustion dynamics. I.
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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Cited by 1 (0 self)
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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.