Stochastic dynamics

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Stochastic dynamics

Collaborator: A. Bovier , B. Gentz

Cooperation with: G. Ben Arous (Ecole Polytechnique Fédérale, Lausanne, Switzerland), N. Berglund (Eidgenössische Technische Hochschule Zürich, Switzerland), V. Gayrard (Centre de Physique Théorique, Marseille, France and Ecole Polytechnique Fédérale, Lausanne, Switzerland), M. Klein, F. Manzo (Universität Potsdam)

Description: The central issue that is addressed in this project is how to adequately describe a complex system, whose dynamics is specified on a microscopic scale, on spatially coarsened macro- or mesoscopic scales in terms of an effective dynamics on different time-scales inherent to the system. The emphasis here is to be put on the fact that these effective dynamics must depend, in general, on the time-scale considered. E.g., while even in microscopically stochastic systems one expects generally deterministic limit dynamics for the spatially coarsened system on short time-scales (homogenization), on much longer time-scales stochastic effects may again become relevant and may even appear in deterministic systems as a residual effect of the integrated short-wavelength degrees of freedom.

One of the central concepts in this context is that of metastability. It applies to situations where the state space of a system can be decomposed into several (``quasi-invariant'') subsets in which the process remains for a very long time before transiting from one such set into another. Over the last years, we have developed an entirely novel approach to the analysis of both probabilistic (distribution of transition times) and spectral (eigenvalues and eigenfunctions of the generator) quantities and their relations. This approach allows in particular to obtain rigorous results that have a far greater precision than the standard exponential estimates obtained in the Wentzell-Freidlin theory. In the past year, we have been building on the methods and results of [2] in various directions.

In [4] we have shown that the results of [2] naturally apply in the context of reversible Markov chains with exponentially small transition probabilities and that, in this setting, all key quantities can be computed up to multiplicative errors that tend to one in terms of properties of the energy function. As a particularly interesting example we considered the stochastic Ising model in finite volume when the temperature is a small parameter. In this context we could compute the mean nucleation time, e.g., in the presence of a positive external field, from a metastable state of negative magnetization to the stable +-state including the precise prefactor. Following the general theory of [2], this quantity was also shown to be the inverse of the spectral gap of the generator.

A far more challenging program was the extension of the methods of [2] to disordered spin systems with an unbounded number of ``metastable states''. In this situation, on the time-scale where transitions between the involved metastable states occur, one may observe the appearance of an effective non-Markovian dynamics. These phenomena have received considerable attention in the physics literature under the name of ``aging'', in particular in materials such as glasses and spin glasses, or glassy polymers. The current state of the art in the physical literature involves the ad-hoc introduction of a simple effective dynamics on the set of metastable states that can then be analyzed exactly (so-called ``trap models''). The literature on these models as well as the observed phenomenology is abundant. However, the rigorous derivation of these effective models is missing, and the very first results were obtained last year in the context of the random energy model (REM), one of the simplest spin glass models ([5, 6, 7]). The REM is considered in the physics literature as one of the basic paradigms for the study of the phenomenon of aging. However, all work has previously been based on an ad-hoc simplification of the model that consists in effectively replacing the dynamics on the underlying spin-space by an effective Markov chain defined in terms of transitions between the ``meta-stable'' states only. The same procedure is used in much more general situations to get an idea of the long-time dynamics of highly disordered systems. In the references cited above, a mathematically rigorous derivation of the results based on the trap model ansatz could be given for the first time in the context of the REM. Let us mention that the results on the equilibrium properties of the REM obtained in [3] (see also [1]) have been instrumental for this work. Extensions of this analysis towards the more complex Generalized Random Energy Model are now under way.

A second direction of research that has made substantial progress last year is the investigation of slow explicit time dependence in the parameters of random dynamical systems. These are of major importance in applications where stochastic resonance or hysteresis are observed, such as ring lasers, electronic circuits, neurons, and climate models, for instance for the Atlantic thermohaline circulation.

We used the methods developed in [8] to study the overdamped motion of a Brownian particle in a periodically modulated Ginzburg-Landau potential in a regime of moderately low frequency of the modulation: In [9], we obtained a precise mathematical understanding of the behavior of sample paths in the case when the amplitude of the modulation is too small to allow for transitions between wells in the absence of noise. There is a threshold value for the noise intensity as a function of amplitude and frequency of the modulation. Below threshold, typical sample paths remain in the same potential well for many periods while above threshold, typical paths switch wells twice per period. The probability of atypical paths decays exponentially, with an exponent scaling with the small parameters of the problem. Inter-well transitions are concentrated in small windows around the instant of minimal barrier height and the width of these windows depends only on the noise intensity.

In [10], the area of random hysteresis cycles in the same model is investigated. There are three qualitatively different regimes. Below a critical noise intensity, the area enclosed by a sample path over one period is concentrated near the deterministic area, which is microscopic for small amplitudes and macroscopic for larger amplitudes. Above the critical noise intensity, noise causes transitions regardless of the amplitude, and the typical hysteresis area depends, to leading order, on the noise intensity only. We estimated the probability of deviations from the typical area in all three regimes.

In addition, we also discussed the effect of colored noise instead of Gaussian white noise and applications to simple climate models, see [11] and [12], respectively.

References:

A. BOVIER, Statistical mechanics of disordered systems, MaPhySto Lecture Notes 10, Aarhus, 2001.
A. BOVIER, M. ECKHOFF, V. GAYRARD, M. KLEIN, Metastability and low-lying spectra in reversible Markov chains, WIAS Preprint no. 601, 2000, to appear in: Commun. Math. Phys.
A. BOVIER, I. KURKOVA, M. LÖWE, Fluctuations of the free energy in the REM and the p-spin SK-models, WIAS Preprint no. 595, 2000, to appear in: Ann. Probab.
A. BOVIER, F. MANZO, Metastability in Glauber dynamics in the low-temperature limit: Beyond exponential asymptotics, WIAS Preprint no. 663, 2001, to appear in: J. Statist. Phys.
A. BOVIER, G. BEN AROUS, V. GAYRARD, Glauber dynamics of the random energy model. 1. Metastable motion on the extreme states, WIAS Preprint no. 690, 2001, submitted.
$\dito$ , Glauber dynamics of the random energy model. 2. Aging below the critical temperature, WIAS Preprint no. 691, 2001, submitted.
$\dito$ ,Aging in the random energy model, WIAS Preprint no. 692, 2001, to appear in: Phys. Rev. Letts.
N. BERGLUND, B. GENTZ, Pathwise description of dynamic pitchfork bifurcations with additive noise, Probab. Theory Related Fields (2001). Published online under DOI 10.1007/s004400100174.
$\dito$ , A sample-paths approach to noise-induced synchronization: Stochastic resonance in a double-well potential, WIAS Preprint no. 627, 2001, submitted.
$\dito$ , The effect of additive noise on dynamical hysteresis, WIAS Preprint no. 668, 2001, to appear in: Nonlinearity.
$\dito$ , Beyond the Fokker-Planck equation: Pathwise control of noisy bistable systems, WIAS Preprint no. 688, 2001, to appear in: J. Phys. A.
$\dito$ , Metastability in simple climate models: Pathwise analysis of slowly driven Langevin equations, WIAS Preprint no. 696, 2001.