Stochastic pump effect and geometric phases in dissipative and stochastic systems

The success of Berry phases in quantum mechanics stimulated the study of similar phenomena in other areas of physics, including the theory of living cell locomotion and motion of patterns in nonlinear media. More recently, geometric phases have been applied to systems operating in a strongly stochastic environment, such as molecular motors. We discuss such geometric effects in purely classical dissipative stochastic systems and their role in the theory of the stochastic pump effect (SPE).Comment: Review. 35 pages. J. Phys. A: Math, Theor. (in press

A step towards holistic discretisation of stochastic partial differential equations

The long term aim is to use modern dynamical systems theory to derive discretisations of noisy, dissipative partial differential equations. As a first step we here consider a small domain and apply stochastic centre manifold techniques to derive a model. The approach automatically parametrises subgrid scale processes induced by spatially distributed stochastic noise. It is important to discretise stochastic partial differential equations carefully, as we do here, because of the sometimes subtle effects of noise processes. In particular we see how stochastic resonance effectively extracts new noise processes for the model which in this example helps stabilise the zero solution.Comment: presented at the 5th ICIAM conferenc

Stochastic dynamics of correlations in quantum field theory: From Schwinger-Dyson to Boltzmann-Langevin equation

The aim of this paper is two-fold: in probing the statistical mechanical properties of interacting quantum fields, and in providing a field theoretical justification for a stochastic source term in the Boltzmann equation. We start with the formulation of quantum field theory in terms of the Schwinger - Dyson equations for the correlation functions, which we describe by a closed-time-path master ($n = \infty PI$) effective action. When the hierarchy is truncated, one obtains the ordinary closed-system of correlation functions up to a certain order, and from the nPI effective action, a set of time-reversal invariant equations of motion. But when the effect of the higher order correlation functions is included (through e.g., causal factorization-- molecular chaos -- conditions, which we call 'slaving'), in the form of a correlation noise, the dynamics of the lower order correlations shows dissipative features, as familiar in the field-theory version of Boltzmann equation. We show that fluctuation-dissipation relations exist for such effectively open systems, and use them to show that such a stochastic term, which explicitly introduces quantum fluctuations on the lower order correlation functions, necessarily accompanies the dissipative term, thus leading to a Boltzmann-Langevin equation which depicts both the dissipative and stochastic dynamics of correlation functions in quantum field theory.Comment: LATEX, 30 pages, no figure

Random Attractors for Stochastic Partly Dissipative Systems

We prove the existence of a global random attractor for a certain class of stochastic partly dissipative systems. These systems consist of a partial (PDE) and an ordinary differential equation (ODE), where both equations are coupled and perturbed by additive white noise. The deterministic counterpart of such systems and their long-time behaviour have already been considered but there is no theory that deals with the stochastic version of partly dissipative systems in their full generality. We also provide several examples for the application of the theory.Comment: 29 page