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17
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 53 (14 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...
Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise. Stochastic Process
 Appl
"... The ergodic properties of SDEs, and various time discretizations for SDEs, are studied. The ergodicity of SDEs is established by using techniques from the theory of Markov chains on general state spaces. Application of these Markov chain results leads to straightforward proofs of ergodicity for a va ..."
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Cited by 50 (14 self)
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The ergodic properties of SDEs, and various time discretizations for SDEs, are studied. The ergodicity of SDEs is established by using techniques from the theory of Markov chains on general state spaces. Application of these Markov chain results leads to straightforward proofs of ergodicity for a variety of SDEs, in particular for problems with degenerate noise and for problems with locally Lipschitz vector fields. The key points which need to be verified are the existence of a Lyapunov function inducing returns to a compact set, a uniformly reachable point from within that set, and some smoothness of the probability densities; the last two points imply a minorization condition. Together the minorization condition and Lyapunov structure give geometric ergodicity. Applications include the Langevin equation, the Lorenz equation with degenerate noise and gradient systems. The ergodic theorems proved are strong, yielding exponential convergence of expectations for classes of measurable functions restricted only by the condition that they grow no faster than the Lyapunov function. The same Markov chain theory is then used to study timediscrete approximations of these SDEs. It is shown that the minorization condition is robust under approximation. For globally Lipschitz vector fields this is also true of the Lyapunov condition. However in the locally Lipschitz case the Lyapunov
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 48 (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.
Spectral Theory of Thermal Relaxation
 J. Math. Phys
, 1997
"... . We review some results obtained in a recent series of papers on thermal relaxation in classical and quantum dissipative systems. We consider models where a small system S , with a finite number of degrees of freedom, interacts with a large environment R in thermal equilibrium at positive temperatu ..."
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Cited by 9 (1 self)
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. We review some results obtained in a recent series of papers on thermal relaxation in classical and quantum dissipative systems. We consider models where a small system S , with a finite number of degrees of freedom, interacts with a large environment R in thermal equilibrium at positive temperature T . The zeroth law of thermodynamics postulates that, independently of its initial configuration, the system S approaches a unique stationary state as t !1. By definition, this limiting state is the equilibrium state of S at temperature T . Statistical mechanics further identifies this state with the Gibbs canonical ensemble associated with S . For simple models we prove that the above picture is correct, provided the equilibrium state of the environment R is itself given by its canonical ensemble. In the quantum case we also obtain an exact formula for the thermal relaxation time. Spectral Theory of Thermal Relaxation 2 1. Introduction This paper is an informal outline of the results...
Berman: Resonance theory of decoherence and thermalization
 Ann. Physics
"... We present a rigorous analysis of the phenomenon of decoherence for general N−level systems coupled to reservoirs. The latter are described by free massless bosonic fields. We apply our general results to the specific cases of the qubit and the quantum register. We compare our results with the expli ..."
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Cited by 7 (5 self)
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We present a rigorous analysis of the phenomenon of decoherence for general N−level systems coupled to reservoirs. The latter are described by free massless bosonic fields. We apply our general results to the specific cases of the qubit and the quantum register. We compare our results with the explicitly solvable case of systems whose interaction with the environment does not allow for energy exchange (nondemolition, or energy conserving interactions). We suggest a new approach which applies to a wide variety of systems which are not explicitly solvable. 1
Extracting Macroscopic Stochastic Dynamics: Model Problems
, 2003
"... The purpose of this work is to shed light on an algorithm designed to extract effective macroscopic models from detailed microscopic simulations. The particular algorithm we study is a recently developed transfer operator approach due to Schtte et al. [20]. The investigations involve the formulat ..."
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Cited by 5 (2 self)
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The purpose of this work is to shed light on an algorithm designed to extract effective macroscopic models from detailed microscopic simulations. The particular algorithm we study is a recently developed transfer operator approach due to Schtte et al. [20]. The investigations involve the formulation, and subsequent numerical study, of a class of model problems. The model problems are ordinary differential equations constructed to have the property that, when projected onto a lowdimensional subspace, the dynamics is approximately that of a stochastic differential equation exhibiting a finitestatespace Markov chain structure. The numerical studies show that the transfer operator approach can accurately extract finitestate Markov chain behavior embedded within highdimensional ordinary differential equations. In so doing the studies lend considerable weight to existing applications of the algorithm to the complex systems arising in applications such as molecular dynamics. The algorithm is predicated on the assumption of Markovian input data; further numerical studies probe the role of memory effects. Although preliminary, these studies of memory indicate interesting avenues for further development of the transfer operator methodology.
The moment map: Nonlinear dynamics of density evolution via a few moments, submitted to
 SIAM Journal on Applied Dynamical Systems
, 2005
"... We explore situations in which certain stochastic and highdimensional deterministic systems behave effectively as lowdimensional dynamical systems. We define and study moment maps, maps on spaces of loworder moments of evolving distributions, as a means of understanding equationsfree multiscale ..."
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Cited by 5 (1 self)
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We explore situations in which certain stochastic and highdimensional deterministic systems behave effectively as lowdimensional dynamical systems. We define and study moment maps, maps on spaces of loworder moments of evolving distributions, as a means of understanding equationsfree multiscale algorithms for these systems. We demonstrate how nonlinearity arises in these maps and how this results in the stabilization of metastable states. Examples are shown for a hierarchy of models, ranging from simple stochastic differential equations to molecular dynamics simulations of a particle in contact with a heat bath. Key words. Equationfree, multiscale, moment map, metastable states. AMS subject classifications. 1. Introduction. An
Stiff Oscillatory Systems, Delta Jumps and White Noise
 FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
, 2001
"... Two model problems for stiff oscillatory systems are introduced. Both comprise a linear superposition of N ≫ 1 harmonic oscillators used as a forcing term for a scalar ODE. In the first case the initial conditions are chosen so that the forcing term approximates a delta function as N →∞and in the s ..."
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Cited by 5 (2 self)
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Two model problems for stiff oscillatory systems are introduced. Both comprise a linear superposition of N ≫ 1 harmonic oscillators used as a forcing term for a scalar ODE. In the first case the initial conditions are chosen so that the forcing term approximates a delta function as N →∞and in the second case so that it approximates white noise. In both cases the fastest natural frequency of the oscillators is O(N). The model problems are integrated numerically in the stiff regime where the timestep �t satisfies N�t = O(1). The convergence of the algorithms is studied in this case in the limit N →∞and �t → 0. For the white noise problem both strong and weak convergence are considered. Order reduction phenomena are observed numerically and proved theoretically.
Analysis and Experiments for a Computational Model of a Heat Bath
, 1998
"... A question of some interest in computational statistical mechanics is whether macroscopic quantities can be accurately computed without detailed resolution of the fastest scales in the problem. To address this question a simple model for a distinguished particle immersed in a heat bath is studied (d ..."
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Cited by 3 (1 self)
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A question of some interest in computational statistical mechanics is whether macroscopic quantities can be accurately computed without detailed resolution of the fastest scales in the problem. To address this question a simple model for a distinguished particle immersed in a heat bath is studied (due to Ford and Kac). The model yields a Hamiltonian system of dimension 2N + 2 for the distinguished particle and the degrees of freedom describing the bath. It is proven that, in the limit of an innite number of particles in the heat bath (N ! 1), the motion of the distinguished particle is governed by a stochastic dierential equation (SDE) of dimension 2. Numerical experiments are then conducted on the Hamiltonian system of dimension 2N + 2 (N 1) to investigate whether the motion of the distinguished particle is accurately computed (ie. whether it is close to the solution of the SDE) when the timestep is small relative to the natural timescale of the distinguished particle, but the ...