Probabilistic treatment of steady states of classical overdamped noisy systems with time-delay

Abstract:

We explore possibilities to describe the dynamical behaviour of classical overdamped, noisy systems with a time-delayed feedback force (that depends on the system state at one earlier instant in time t-τ) via probability densities. Due to the non-Markovian character of such systems, there is no standard Fokker-Planck (FP) equation which corresponds to the (delayed) stochastic Langevin equation [1,2]. In my talk, I will first review earlier theoretical work on how the FP approach for delayed systems yields an infinite hierarchy of coupled differential equations that involves n-time (joint) probability densities depending on an increasing number of instances in time n -> ∞ [1]. Although these equations are not self-sufficient, they are a valuable starting point for approximation schemes. In particular, I will discuss a first order perturbation-theoretical approach [3] and its application to two exemplary systems involving a Brownian particle in a one-dimensional potential with delayed feedback. We compare the perturbation-theoretical results with those from Brownian dynamics simulations of the underlying delayed Langevin equation. Further, we discuss properties of the two-time probability density, an essential ingredient for the first member of the delayed FP equation.
[1] S. Guillouzic et al., Phys. Rev. E 59, 3970 (1999).
[2] M. L. Rosinberg et al., Phys. Rev. E 91, 042114 (2015).
[3] T. D. Frank, Phys. rev. E 71, 031106 (2005).