Periodically driven Markov processes are studied in the adiabatic limit. In order to construct the Floquet eigenfunctions and eigenvalues we use the instantaneous eigenvalues and eigenfunctions of the corresponding Master operator. In contrast to quantum mechanics where level crossings of a (hermitian) Hamiltonian are generically avoided a nonsymmetric master operator can show structurally stable encounters of eigenvalues when a parameter slowly changes. The influence of these encounters on the Floquet eigenvalues will be discussed.

Thermally activated escape over a potential barrier in the presence of periodic driving is considered. By means of novel time-dependent path-integral methods we derive asymptotically exact weak-noise expressions for both the instantaneous and the time-averaged escape rate. The agreement with accurate numerical results is excellent over a wide range of driving strengths and driving fields.

Fluctuational transitions between the stationary states of periodically-driven nonlinear oscillators are investigated by means of numerical simulations and analogue experiments for over-damped, weakly damped and chaotic motion. It is shown that transitions take place along distinct most probable escape paths in all three cases. The transition probabilities are compared with predictions based on the theory of the logarithmic susceptibility . It is found that, in agreement with theoretical predictions, the log of the transition probability displays an exponentially sharp dependence on the field frequency for the case of over-damped motion. Resonant activation, rectification of fluctuations, and current reversal, are investigated numerically for a weakly damped Brownian particle in an asymmetric periodic potential. It is shown that the dependence of the activation energy on the field amplitude is linear in both over-damped and under-damped cases. It is demonstrated that the analysis of the escape process from the quasi-attractor of a periodically-driven underdamped oscillator can be reduced to the analysis of transitions between a few saddle limit cycles of low period, thus raising the possibility of an analytic description based on an extension of the theory of logarithmic susceptibility.

I study the challenging problem whether a particle, -- held initially at rest (!) -- does exhibit a finite (stationary) current even though NO dissipation is acting (Hamiltonian dynamics). In particular, for a a driven Hamiltonian dynamics that obeys in addition TIME-REVERSAL symmetry such a finite current brings alive a "Maxwell-Loschmidt-Demon". The crucial role of overaging over random phases is also addressed.

Two loosing games can yield, when played alternately, a steady increase of capital. These games have been inspired by the functioning of flashing ratchets. They point out that result of the alternation of stochastic dynamics can be unexpected. In the seminar, I will present and discuss he original paradox and some of its variants.

Non-rational non-cooperative games are introduced and explained. When the agents are rational, this problem has been recently formulated as a disordered mean-field model. In the case of non-rational agents and using the theory of stochastic processes it could be shown that the game dynamic is a diffusion process over the space of agent strategies.

The finite element method (FEM) is one of the most popular way of solving partial differential equations. It is widely used in various branches of applied science, starting from a modeling of the electrical activity in the heart, through flow problems, ending with weather forecast. Here, its application to Fokker-Planck equations will be presented. The attention, in most of cases, will be focused on ratchet-like problems. In such cases, the efficient handling of periodic boundary conditions the evaluation of the local and global flows is of prime importance. Stationary as well as transient problems will be discussed by following examples:

Apart to already obtained results, the future applications of FEM will be also presented.

We discuss noise induced dynamics in the framework of random dynamical systems. That is, dynamical quantities are defined "pathwise" for (almost) every realization of the underlying noise process.

A random attractor is a compact set valued random variable, which attracts all trajectories as time tends to infinity. This approach allows in some sense to separate "deterministic" dynamics (which is responsible for the structure of the attractor) from the influence of the noise (which is responsible for movements of the attractor in the phase space under time evolution).

We will present a numerical algorithm, which produces a box covering of random attractors. The algorithm is applied to the stochastically forced Duffing (Kramers) oscilltator. A bifurcation (qualitative change of the structure of the attractor) is observed when the top Lyapunov exponent crosses zero. After the bifurcation the numerics shows a chaotic attractor supporting a random Sinai-Ruelle-Bowen measure, whereas before the bifurcation the "natural" invariant measure is supported by a single random point. This qualitative change of the dynamics cannot be seen in terms of the invariant Markov density.

I will explain an efficient way of implementing pseudospectral methods and predictor-corrector algorithms for integrating numerically stochastic partial differential equations with additive or multiplicative noise. Several examples, including the typical growth models of Kardar-Parisi-Zhang and related ones, will illustrate the performance (and limitations) of the algorithms.

Kinks at finite temperature are nonlinear coherent structures in a one-dimensional stochastic partial differential equation. At equilibrium there is a dynamic balance between nucleation and annihilation; thermodynamic quantities can be calculated exactly using the transfer integral and used to test numerical schemes. Because kinks are always nucleated as part of a kink-antikink pair, the nucleation rate is proportional to the square of the equilibrium density.

We study numerically and analytically the properties of the energy diffusion in systems of oscillators, which are simultaneously non linear and discrete. These systems display anomalous diffusion, due to the existence of so-called "breather-modes", non linear localised excitations. The particular characteristics of the energy transport are studied by means of several techniques, persistance distributions and projection operators essentially, which are powerfull to extract information of these complicated processes.

In the last few years numerous studies have been devoted to intrinsic localized modes in nonlinear lattices because they provide examples of localization that does not require disorder. The properties of ``discrete breathers'' as exact solutions of these nonlinear lattices are now well understood, but this is not the case for the properties of nonlinear localization and energy relaxation in thermalized lattices. In biological molecules such as DNA, where large amplitude nonlinear motions are essential for function, temporary deviations from energy equipartition could play an important role.

After a brief introduction to intrinsic localized modes, the talk will address the following questions:

External fluctuations have a wide variety of constructive effects on the dynamical behavior of spatially extended systems, as described by stochastic partial differential equations. A set of paradigmatic situations, mainly noise-induced ordering transitions, exhibiting such effects are briefly reviewed and the physical mechanisms are discused in this paper, in an attempt to provide a concise but thorough introduction to this active field of research, and at the same time an overview of its current status.

Instabilites of coupled parametric oscillators with quenched random phases are investigated by a mean field theory and computer simulations. In addition to incoherent parametric resonance, we found two kinds of collective instabilites: in one instability the mean amplitude monotonically explodes whereas in another it oscillates with a diverging amplitude. The phase diagrams of these instabilities and other properties such as energy localization will be discussed.

The effect of external noise on the spatiotemporal dynamics of activator-inhibitor systems is discussed. We will concentrate on the particular case of very different time scales for each of the two species. In this situation, the system exhibits excitable properties for certain parameter values. Two different kinds of models will be presented, namely cellular automata and continuous reaction-diffusion models. In the two cases, external noise displays a rich variety of constructive effects, which will be reviewed in this communication.

The application of scaling concepts in Statistical Physics has revealed their great usefulness to obtain relevant information about static and dynamical quantities of the the systems without entering into complex calculations. These techniques have been employed in many classical branches of that discipline, such as phase transitions, hydrodynamics, polymer physics, and non-linear physics, and more recently to growth phenomena, fracture, and economy, to mention but a few. We show how scaling arguments can also be applied to analyze the dynamics of a wide class of systems exhibiting, as a main characteristic, a dynamics evolusion which is periodically modulated when the system is affected by noise. In particular we will focus on the phenonemon of Stochastic Resonance. We present a methodology based upon general scaling arguments which enables one to predict the appearance of an ordered behavior due to the presence of noise in temporal and spatial systems.

In order to improve the separation of any given chemical species from a mixture of compounds with close thermodynamic and kinetic properties, we propose a new chromatography procedure in the presence of a uniform time-periodic field. In the framework of a macroscopic reaction-diffusion model in an external field, we prove that the apparent motion of the chemical species is of diffusion type and determine an approximate analytical expression for the effective diffusion coefficient. Considering this coefficient as a function of the rate constants and maximizing it leads to specific relations between rate constants and field properties interpreted as stochastic resonances. In the case of an electric field, we show that these constraints are compatible with typical experimental values.

In this talk, we report some theoretical and numerical results which were obtained in collaboration with Anne de Bouard and Laurent Di Menza concerning the Cauchy problem and finite-time blow-up of solutions of some stochastic nonlinear Schr\"odinger equations.

The nonlinear Schr\"odinger equation is one of the basic models for nonlinear waves. It arises in various areas of physics such as hydrodynamics, nonlinear optics or plasma physics. In some circumstances, randomness has to be taken into account, often to model phenomena with characteristic time scales much smaller than the characteristic time scales of the deterministic phenomena. Hence it is natural that the time-dependence of the random terms appearing in the equation is a white noise type dependence, while the spacial correlation of these terms may take different forms. A random term may arise in the equation as an "additive term", that is as an external force, or as a ``multiplicative term", such as for example a random potential.

I shall present an overview of a research programme of experiments and simulations by the Laboratory for Crystallographic Studies in Granada. This programme is designed to further our understanding of:

It is well known that small-scale, rapidly-varying fluctuations in wind speed can contribute significantly to the variability of the planetary- scale vorticity field. What seems to be less appreciated in the meteorological community is the fact that these same fluctuations also contribute to the mean patterns themselves. Also, the fact that appropriate numerical techniques for simulating stochastic-forced geophysical processes may depend on the physics of the situation is usually undervalued. In this study, we apply the theory of stochastic differential equations and the numerical generation thereof to investigate the extent to which stochastic fluctuations in the base state can affect the mean and variance of the response to steady forcing in the context of a barotropic vorticity model of the atmosphere. Appropriate modifications to the barotropic vorticity equations are derived using the classic theory of stochastic differential equations. We show the effect of stochastic fluctua- tions in a superrotation flow on Rossby-wave propagation. For realistic values of variation in the zonal wind, we find the effect on the mean response to be small to medium, but always significant. The variability is strongly enhanced in all cases.

We discuss the principles governing the activity of "hair cells" which are responsible for sound detection in non mamalian vertebrates.We introduce in particular the concept of self tuned criticality which allows to give a natural description of not less than five oddities which characterize the function of hair cells.

We performed direct Monte-Carlo simulations and a careful finite-size analysis of the reversible diffusion-limited process A + A <-> A to study the effect of fluctuations on the interface between stable and unstable phases. The mean-field description of this process, Fisher's reaction-diffusion equation, admits stable nonlinear propagating wavefronts. We found that the mean-field description breaks down in spatial dimensions 1 and 2, while it appears qualitatively and quantitatively accurate in 4 dimensions. In particular, there is no stable steady wavefront in 2-d where we find that the interface width grows ~t^{.27}. This is joint work with Jason Riordan and Daniel ben Avraham.