Periodically driven stochastic systems

Abstract

Activation processes in classical metastable systems in the presence of periodic driving have recently become subject of growing research activity, stimulated by exciting new phenomena such as stochastic resonance. A theory of such systems has to deal with non-stationary stochastic processes. Thus, in chapters 2 and 3, we report on general properties and the most important notions of stochastic processes in the presence of periodic forcing. In addition to a Floquet-type formalism we introduce an equivalent description, based on the embedding of non-stationary processes in higher dimensional stationary stochastic processes. In chapter 4, the formalism is demonstrated for the exactly solvable periodically driven Ornstein-Uhlenbeck process. In chapter 5, we elaborate on the extension of the definition of escape rates in order to be applicable to non-stationary systems. In chapter 6, we consider the influence of periodic forcing and noise on bistable systems. As an archetype system the Landau-type quartic bistable potential has been chosen. Explicit numerical results for escape rates, probability distributions and spectral densities are compared with approximate theories. In chapter 7, we report on the history and recent developments of an exciting phenomenon, termed stochastic resonance. Stochastic resonance is a cooperative effect of noise, bistability and periodic forcing. It allows for the resonant enhancement of the signal/noise ratio in bistable systems as a function of the strength of the input noise background. Recently published approaches are reviewed, contrasted against each other and compared with data obtained from the numerical evaluation of a general expression for the nonlinear response of the system on periodic forcing. Transport properties and escape rates in multiwell potentials in the presence of fluctuations and periodic fields are the topic of chapter 8. In the first part of chapter 8 we discuss the connection between the escape rates, periodic orbits, mobilities and the band structure of the Fokker-Planck equation in the overdamped limit. In the second part, we focus on inertia effects in periodically driven rate processes, a field which has been termed resonance activation.