Stochastic dynamics

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Stochastic dynamics

Collaborator: A. Bovier, B. Gentz

Cooperation with: G. Ben Arous (Courant Institute, New York, USA), N. Berglund (Centre de Physique Théorique, Marseille, and Université de Toulon, France), V. Gayrard (Centre de Physique Théorique, Marseille, France, and Université de Montréal, Canada), F. den Hollander (EURANDOM, Eindhoven, The Netherlands), M. Klein (Universität Potsdam), F. Nardi (Università di Roma ``La Sapienza'', Italy)

Supported by: DFG: Dutch-German Bilateral Research Group ``Mathematics of random spatial models from physics and biology''

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. For example, 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 a 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 a collaboration with F. den Hollander and F. Nardi we are currently applying these methods to the problem of nucleation in a model of conservative dynamics of a lattice gas (``Kawasaki dynamics''). The issue is to obtain precise information on the time it takes to form a supercritical droplet, and thus to initiate a vapor-liquid phase transition, in a super-saturated gas in some finite volume at low temperatures. This problem has been analyzed rigorously in the last few years by den Hollander, Olivieri, and Scoppola [3] and den Hollander, Olivieri, Scoppola, and Nardi [4] in dimensions two and three, respectively, using the conventional large deviation-type methods. By a detailed analysis of the energy landscape of the model, they obtained the logarithmic asymptotics of the nucleation time in the limit as the temperature tends to zero. In a forthcoming paper ([5]), we slightly refine the analysis of the energy landscape in the vicinity of the saddle points (critical droplets) and apply the machinery developed in [2]. As a result we obtain, as expected, striking improvements of all the estimates, and are able to compute (at least for large $\Lambda$ ) essentially the precise values of the pre-factors of the exponential rates. Interestingly, the variational problems arising in the computation of the relevant capacities are seen to be closely related to classical capacity estimates involving the free diffusion of a single particle. While in this paper we remain in the regime of very low temperatures, we see a perspective to move to the physically more interesting regimes of moderately low temperatures, and we intend to pursue this line actively in the coming years.

The investigation of more complex systems with an infinity of metastable states leading to the phenomenon of ``ageing'' ([1]) has continued rather intensely. One of the main goals here is to extend the analysis performed in [6], [7] for the random energy model to the much more complicated generalized random energy models. The issue is to reduce the dynamics on the space of spin configurations to the dynamics of an effective trap model. Work has continued in collaboration with G. Ben Arous and V. Gayrard, and first significant results can be expected soon. Another line of work concerns the analysis of trap models as such. Here, J. Cerný, in collaboration with G. Ben Arous and T. Mountford ([8]), has been able to precisely analyze various autocorrelation functions for Bouchaud's trap model on the two-dimensional lattice $\Bbb$ Z². In an attempt to better understand the signature of ageing in terms of spectral properties, we have returned to the analysis of Bouchaud's REM-like trap model. It turns out that in this case the generator can be diagonalized explicitly, and precise expressions for eigenvalues and eigenfunctions can be obtained, which allows to recover all dynamical properties purely from this spectral information. One also realizes a clear connection between aging exponents and the singularity in the limiting spectral density of the model.

Over the last years, in collaboration with Nils Berglund, we developed a new approach to random perturbations of dynamical systems, evolving on two well-separated time scales. After detailed studies of noise-induced phenomena in one-dimensional slowly time-dependent systems, [10], [11], [12], first results on fully coupled multidimensional slow-fast systems were obtained last year. These results provide estimates on the fluctuations of the fast variables near slow or centre manifolds of the corresponding deterministic system, thus allowing to study a reduced system ([13]).

This year we turned our attention towards truly multidimensional effects in non-gradient systems. As a first step we studied the random dynamics near periodic orbits of the deterministic dynamics, in particular the first-passage time through an unstable periodic orbit. Passage through an unstable periodic orbit plays a key role in many applications, in particular in those showing stochastic resonance or synchronization. Examples include climate models, phase slips in noisy systems of coupled oscillators, and stochastic resonance in lasers.

While on the exponential scale, accessible to the classical Wentzell-Freidlin theory, all points on an unstable orbit are equally likely to occur as first-exit points, the subexponential asymptotics of the distribution of first-exit points or times reflects the fact that the unstable orbit is generally not uniformly repelling. As discovered by Day ([15]), `` cycling'' occurs: The distribution of first-exit points rotates around the unstable orbit, periodically in the logarithm of the noise intensity, and thus does not converge in the zero-noise limit. In [14], we study the distribution of first-exit times for a class of model equations. A rich picture emerges, showing cycling most prominently in a metastable time regime, but also during the transitional initial phase and in the asymptotic regime. Our results cover a large range of possible values for the period of the unstable orbit, thus providing insight into the transition from the stochastic-resonance regime into the synchronization regime.

References:

A. BOVIER, Metastability and ageing in stochastic dynamics, to appear in: Proceedings of the II. Workshop on Dynamics and Randomness, Santiago de Chile, Dec. 9-13, 2002, A. Maas, S. Martínez, J. San Martin, eds., Kluwer Acad. Publishers, 2003.
A. BOVIER, M. ECKHOFF, V. GAYRARD, M. KLEIN , Metastability and low-lying spectra in reversible Markov chains, Commun. Math. Phys., 228 (2002), pp. 219-255.
F. DEN HOLLANDER, E. OLIVIERI, E. SCOPPOLA, Metastability and nucleation for conservative dynamics. Probabilistic techniques in equilibrium and nonequilibrium statistical physics, J. Math. Phys., 41 (2000), pp. 1424-1498.
F. DEN HOLLANDER, F. NARDI, E. OLIVIERI, E. SCOPPOLA, Droplet growth for three-dimensional Kawasaki dynamics, Probab. Theory Related Fields, 125 (2003), pp. 153-194.
A. BOVIER, F. DEN HOLLANDER, F. NARDI, Sharp asymptotics for Kawasaki dynamics on a finite box with open boundary, in preparation.
G. BEN AROUS, A. BOVIER, V. GAYRARD, Glauber dynamics of the random energy model. 1. Metastable motion on the extreme states, Commun. Math. Phys., 235 (2003), pp. 379-425.
, Glauber dynamics of the random energy model. 2. Aging below the critical temperature, Commun. Math. Phys., 236 (2003), pp. 1-54.
G. BEN AROUS, J. CERNÝ, T. MOUNTFORD, Aging in two-dimensional Bouchaud's model, WIAS Preprint no. 887, 2003.
A. BOVIER, V. GAYRARD, A. FAGGIONATO, in preparation.
N. BERGLUND, B. GENTZ, Pathwise description of dynamic pitchfork bifurcations with additive noise, Probab. Theory Related Fields, 122 (2002), pp. 341-388.
, A sample-paths approach to noise-induced synchronization: Stochastic resonance in a double-well potential, Ann. Appl. Probab., 12 (2002), pp. 1419-1470.
, The effect of additive noise on dynamical hysteresis, Nonlinearity, 15 (2002), pp. 605-632.
, Geometric singular perturbation theory for stochastic differential equations, J. Differential Equations, 191 (2003), pp. 1-54.
, On the noise-induced passage through an unstable periodic orbit I: Two-level model, J. Statist. Phys., 114 (2004), pp. 1577-1618.
M.V. DAY, Conditional exits for small noise diffusions with characteristic boundary, Ann. Probab., 20 (1992), pp. 1385-1419.