Non-Markovian stochastic Liouville equation and its Markovian representation: Extensions of the continuous-time random-walk approach

Waiting time distribution for continuous stochastic systems

The waiting time distribution (WTD) is a common tool for analyzing discrete stochastic processes in classical and quantum systems. However, there are many physical examples where the dynamics is continuous and only approximately discrete, or where it is favourable to discuss the dynamics on a discretized and a continuous level in parallel. An example is the hindered motion of particles through potential landscapes with barriers. In the present paper we propose a consistent generalization of the WTD from the discrete case to situations where the particles perform continuous barrier crossing characterized by a finite duration. To this end, we introduce a recipe to calculate the WTD from the Fokker-Planck (Smoluchowski) equation. In contrast to the closely related first passage time distribution (FPTD), which is frequently used to describe continuous processes, the WTD contains information about the direction of motion. As an application, we consider the paradigmatic example of an overdamped particle diffusing through a washboard potential. To verify the approach and to elucidate its numerical implications, we compare the WTD defined via the Smoluchowski equation with data from direct simulation of the underlying Langevin equation and find full consistency provided that the jumps in the Langevin approach are defined properly. Moreover, for sufficiently large energy barriers, the WTD defined via the Smoluchowski equation becomes consistent with that resulting from the analytical solution of a (two-state) master equation model for the short-time dynamics developed previously by us [Phys. Rev. E 86, 061135 (2012)]. Thus, our approach "interpolates" between these two types of stochastic motion. We illustrate our approach for both symmetric systems and systems under constant force.

Effect of a time-dependent field on subdiffusing particles

We analyze the effect of a time-dependent external field on non-Markovian migration described by the continuous time random walk (CTRW) approach. The rigorous method of treating the problem is proposed which is based on the Markovian representations of the CTRW approach and field modulation. With the use of this method we derive the non-Markovian stochastic Liouville equation (SLE), that describes the effect of this field, and thoroughly analyze the relation of the derived SLE with equations proposed earlier. This SLE is applied to the case of subdiffusive migration in which the exact formulas for the first and second moments of spatial distribution are obtained. In the case of oscillating external field they predict unusual dependence of the first moment on oscillation phase and anomalous time behavior of field dependent contribution to the dispersion which agree with results of earlier works. Anomalous time dependence is also found in the case of a fluctuating field. The specific features of this time dependence are analyzed in detail.