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Collaborator: A. Bovier, J. Cerný, A. Faggionato, B. Gentz
Cooperation with: G. Ben Arous (Courant Institute, New York, USA), N. Berglund (Centre de Physique Théorique, Marseille, and Université de Toulon, France), F. den Hollander (EURANDOM, Eindhoven, The Netherlands), T. Mountford (Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland), 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'', DFG Research Center MATHEON, project E1
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 microscopic 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 [1].
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.
In a collaboration with F. den Hollander and F. Nardi, [3], we have
studied 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
[5] and den Hollander, Nardi, Olivieri, and Scoppola
[4]
in dimensions two and three, respectively, using the conventional
large
deviation-type methods.
The shortcoming of these methods is the limited precision that usually allows
only to obtain the exponential rate of the nucleation times.
In [3],
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
) essentially the precise values of the prefactors 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.
A challenge for the coming years will be the extension of this analysis to finite temperatures and larger volumes.
The investigation of more complex systems with an infinity of metastable states leading to the phenomenon of ``ageing'' has continued rather intensely.
Work in the past year has been focused on the analysis of trap models.
The first objective was to prove some fine properties of the one-dimensional Bouchaud trap model, i.e. finding asymptotic properties of the behavior of ageing two-point functions on time scales that are different from the right ageing scales. These properties were observed numerically, Bouchaud and Bertin [10]. Cerný [11] extended the methods used in Fontes, Isopi, and Newman [12] and Ben Arous and Cerný [7] to give a rigorous analytic derivation of these properties. The refined methods have also allowed to partially justify the concept of the local equilibrium introduced in Rinn, Maass, and Bouchaud [13].
Another goal was to formalize the main properties of Markov chains that lead to ageing behavior. Together with G. Ben Arous, [8], we tried to isolate several conditions that should be verified in order to prove ageing. These conditions were then used to give different, and much shorter, proofs of ageing in the Random Energy Model (first proved by Ben Arous, Bovier, and Gayrard [6]) and on the cubic lattice (proved by Ben Arous, Cerný, and Mountford [9]). To verify the conditions in the REM case, we developed some new potential-theoretic methods for the simple random walk on the hypercube. The results obtained by these methods can be of independent interest.
One of the desiderata in the field is a better understanding of the relation between ageing and spectral properties of the generator of the process, as this may lead to attractive methods for the numerical analysis of such systems. As a first step we have concentrated in [14] on the Bouchaud REM-like trap model. Here the generator can be diagonalized explicitly, and precise expressions for eigenvalues and eigenfunctions were obtained. From these it was possible to derive the dynamical properties of the system on all relevant time scales. A clear connection between ageing exponents and the singularity of the limiting spectral density at the bottom of the spectrum can be traced. As a second example, we have started to analyze Sinai's random walk in a random environment. In this case, an exact diagonalization of the generator is not possible, but it is possible to compute perturbatively the eigenvalues at the bottom of the spectrum, as well as the corresponding eigenfunctions.
In another line of research, we continued our investigation of noise-induced phenomena in non-autonomous dynamical systems. Over the last years, we developed a novel approach to the mathematics of small random perturbations of singularly perturbed dynamical systems, providing a constructive method to describe typical sample-path behavior, and yielding precise estimates on atypical behavior at the same time.
One of the key applications was the phenomenon of stochastic resonance which plays an eminent rôle in many applications. Examples include climate models and neural models as well as technical applications such as ring lasers and Schmitt triggers. Studying stochastic resonance in a periodically modulated double-well potential, we had so far focused on the parameter regime in which noise-induced synchronization is observed, [15]. In this regime, with high probability, inter-well transitions are concentrated near the first instant of minimal barrier height.
One of the popular means to quantify stochastic resonance is the distribution of residence times. It can be derived from the distribution of inter-well transitions which are characterized by passage through the unstable periodic orbit separating the domains of attraction of the potential wells. While the exponential asymptotics, accessible to the classical Wentzell-Freidlin theory, is trivial, the sub-exponential asymptotics reflects the fact that the unstable orbit is generally not uniformly repelling. As first observed by Day [19], ``cycling'' occurs, i.e., a periodic dependence on the logarithm of the noise intensity. Generalizing results obtained last year for a class of model equations [16], the first-passage density is derived in [17], [18]. Remarkably, the density is close to an exponential one, modulated by a universal cycling profile, the profile depending only on the product of the period of the unstable orbit with its Lyapunov exponent. This representation holds in a large parameter regime, ranging from noise-induced synchronization to the general stochastic resonance regime.
References:
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