Peter Talkner (Augsburg):

Driven stochastic processes with metastable states: Fokker-Planck versus master equations

Abstract:

Processes occurring either in living systems on the cellular or molecular level or in artificial nanoscopic systems are most often strongly influenced by ambient thermal noise. On the other hand, intrinsic nonlinearities of the dynamical laws of these systems typically lead to multistability. The influences of the combined effects of noise and nonlinear intrinsic dynamics can often be described by a Fokker-Planck equation. As a consequence of the interaction of nonlinearities and noise the multistability is changed into metastability , i.e., on sufficiently long time scales transitions occur between the locally stable states of the noiseless dynamics. In a coarse grained description these transitions can be characterized by rates which enter a master equation, the states of which correspond to the metastable states of the system. We will demonstrate that such a description continues to hold even if the parameters of the system change with time either due to some external forcing of the system, for example by periodic or ramp-like forcing. As a prominent effect occurring in the presence of periodic forcing we mention stochastic resonance. Steadily increasing forces are often applied to molecules in order to determine their transition states and activation energies. In such forced systems the rates that determine the master equation change in time. In this talk we will discuss methods to calculate these time dependent rates. These methods also allow one to reconstruct the long time dynamics of the continuous Fokker-Planck process from the master equation. An analytic approximations for the rates will be discussed in the limit of slow driving. We will exemplify this method by a periodically driven bistable Brownian oscillator and demonstrate the use of the master equation in order to determine the point process of subsequent transition times.

download document (pdf, 2,33 MB)