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Extracting macroscopic dynamics: model problems and algorithms
 NONLINEARITY
, 2004
"... In many applications, the primary objective of numerical simulation of timeevolving systems is the prediction of macroscopic, or coarsegrained, quantities. A representative example is the prediction of biomolecular conformations from molecular dynamics. In recent years a number of new algorithmic ..."
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Cited by 111 (8 self)
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In many applications, the primary objective of numerical simulation of timeevolving systems is the prediction of macroscopic, or coarsegrained, quantities. A representative example is the prediction of biomolecular conformations from molecular dynamics. In recent years a number of new algorithmic approaches have been introduced to extract effective, lowerdimensional, models for the macroscopic dynamics; the starting point is the full, detailed, evolution equations. In many cases the effective lowdimensional dynamics may be stochastic, even when the original starting point is deterministic. This review surveys a number of these new approaches to the problem of extracting effective dynamics, highlighting similarities and differences between them. The importance of model problems for the evaluation of these new approaches is stressed, and a number of model problems are described. When the macroscopic dynamics is stochastic, these model problems are either obtained through a clear separation of timescales, leading to a stochastic effect of the fast dynamics on the slow dynamics, or by considering high dimensional ordinary differential equations which, when projected onto a low dimensional subspace, exhibit stochastic behaviour through the presence of a broad frequency spectrum. Models whose stochastic microscopic behaviour leads to deterministic macroscopic dynamics are also introduced. The algorithms we overview include SVDbased methods for nonlinear problems, model reduction for linear control systems, optimal prediction techniques, asymptoticsbased mode elimination, coarse timestepping methods and transferoperator based methodologies.
NonEquilibrium Statistical Mechanics of Anharmonic Chains Coupled to Two Heat Baths at Different Temperatures
, 1999
"... . We study the statistical mechanics of a finitedimensional nonlinear Hamiltonian system (a chain of anharmonic oscillators) coupled to two heat baths (described by wave equations). Assuming that the initial conditions of the heat baths are distributed according to the Gibbs measures at two differ ..."
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Cited by 93 (22 self)
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. We study the statistical mechanics of a finitedimensional nonlinear Hamiltonian system (a chain of anharmonic oscillators) coupled to two heat baths (described by wave equations). Assuming that the initial conditions of the heat baths are distributed according to the Gibbs measures at two different temperatures we study the dynamics of the oscillators. Under suitable assumptions on the potential and on the coupling between the chain and the heat baths, we prove the existence of an invariant measure for any temperature difference, i.e., we prove the existence of steady states. Furthermore, if the temperature difference is sufficiently small, we prove that the invariant measure is unique and mixing. In particular, we develop new techniques for proving the existence of invariant measures for random processes on a noncompact phase space. These techniques are based on an extension of the commutator method of H ormander used in the study of hypoelliptic differential operators. 1. Intr...
Pautrat: “From repeated to continuous quantum interactions”, Annales Henri Poincaré (Physique Théorique
"... We consider the general physical situation of a quantum system H0 interacting with a chain of exterior systems ⊗IN ∗H, one after the other, during a small interval of time h and following some Hamiltonian H on H0 ⊗ H. We discuss the passage to the limit to continuous interactions (h → 0) in a setup ..."
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Cited by 62 (7 self)
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We consider the general physical situation of a quantum system H0 interacting with a chain of exterior systems ⊗IN ∗H, one after the other, during a small interval of time h and following some Hamiltonian H on H0 ⊗ H. We discuss the passage to the limit to continuous interactions (h → 0) in a setup which allows to compute the limit of this Hamiltonian evolution in a single state space: a continuous field of exterior systems ⊗ IR +H. Surprisingly, the passage to the limit shows the necessity for 3 different time scales in H. The limit evolution equation is shown to spontaneously produce quantum noises terms: we obtain a quantum Langevin equation as limit of the Hamiltonian evolution. For the very first time, these quantum Langevin equations are obtained as the effective limit from repeated to continuous interactions and not only as a model. These results justify the usual quantum Langevin equations considered in continual quantum measurement or in quantum optics. We show that the three time scales correspond to the normal regime, the weak coupling limit and the low density limit. Our approach allows to consider these two physical limits altogether for the first time. Their combination produces an effective Hamiltonian on the small system, which had never been described before. We apply these results to give an Hamiltonian description of the von Neumann measurement. We also consider the approximation of continuous time quantum master equations by discrete time ones. In particular we show how any Lindblad generator is obtained as the limit of completely positive maps.
Exponential Convergence to NonEquilibrium Stationary States in Classical Statistical Mechanics
 Comm. Math. Phys
, 2001
"... We continue the study of a model for heat conduction [6] consisting of a chain of nonlinear oscillators coupled to two Hamiltonian heat reservoirs at dierent temperatures. We establish existence of a Liapunov function for the chain dynamics and use it to show exponentially fast convergence of the d ..."
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Cited by 57 (12 self)
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We continue the study of a model for heat conduction [6] consisting of a chain of nonlinear oscillators coupled to two Hamiltonian heat reservoirs at dierent temperatures. We establish existence of a Liapunov function for the chain dynamics and use it to show exponentially fast convergence of the dynamics to a unique stationary state. Ingredients of the proof are the reduction of the innite dimensional dynamics to a nitedimensional stochastic process as well as a bound on the propagation of energy in chains of anharmonic oscillators. 1 Introduction In its present state, nonequilibrium statistical mechanics is lacking the rm theoretical foundations that equilibrium statistical mechanics has. This is due, perhaps, to the extremely great variety of physical phenomena that nonequilibrium statistical mechanics describes. We will concentrate here on a system which is maintained, by suitable forces, in a state far from equilibrium. In such an idealization, the nonequilibrium phenome...
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 41 (7 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.
C.A.: Mathematical theory of the WignerWeisskopf atom
 Lecture Notes in Physics 695, 145
, 2006
"... 2 Nonperturbative theory.................................. 4 ..."
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Cited by 24 (8 self)
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2 Nonperturbative theory.................................. 4
Fractional kinetics in KacZwanzig heat bath models
 J. Stat. Phys
"... We study a variant of the Kac–Zwanzig model of a particle in a heat bath. The heat bath consists of n particles which interact with a distinguished particle via springs and have random initial data. As nQ. the trajectories of the distinguished particle weakly converge to the solution of a stochasti ..."
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Cited by 19 (0 self)
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We study a variant of the Kac–Zwanzig model of a particle in a heat bath. The heat bath consists of n particles which interact with a distinguished particle via springs and have random initial data. As nQ. the trajectories of the distinguished particle weakly converge to the solution of a stochastic integrodifferential equation—a generalized Langevin equation (GLE) with powerlaw memory kernel and driven by 1/fanoise. The limiting process exhibits fractional subdiffusive behaviour. We further consider the approximation of nonMarkovian processes by higherdimensional Markovian processes via the introduction of auxiliary variables and use this method to approximate the limiting GLE. In contrast, we show the inadequacy of a socalled fractional Fokker–Planck equation in the present context. All results are supported by direct numerical experiments.
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 17 (8 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.