Large rare fluctuations in systems with delayed dissipation

Abstract

We study the probability distribution and the escape rate in systems with delayed dissipation that comes from the coupling to a thermal bath. To logarithmic accuracy in the fluctuation intensity, the problem is reduced to a variational problem. It describes the most probable fluctuational paths, which are given by acausal equations due to the delay. In thermal equilibrium, the most probable path passing through a remote state has time-reversal symmetry, even though one cannot uniquely define a path that starts from a state with given system coordinate and momentum. The corrections to the distribution and the escape activation energy for small delay and small noise correlation time are obtained in explicit form.

Noise-induced switching and extinction in systems with delay

“…In particular, second-order corrections to the switching rate were obtained earlier for an underdamped system with distributed delay [38], where the first-order correction is zero.…”

“…This includes the possibility of developing a perturbation theory in the delay. It turns out that, because of the difference in the character of the delay, the predictions of the perturbation theory we develop here are qualitatively different from those in systems studied earlier [38]. We also consider the problem of the extinction rate in the presence of delay, which requires a significant extension of both the analytical theory and the Monte Carlo simulation technique, as it is necessary to consider large rare fluctuations induced by a singular multiplicative noise.…”

“…The analysis of the probability distribution far away from an attractor and of the switching rates in systems with hard delay and multiplicative noise can be done by extending the analysis for systems driven by additive noise in the absence of delay or with the delay due to linear coupling to a thermal reservoir [38,41]. The theorems about the stationary states facilitate this analysis in the case of small hard delay.…”

“…Previously we discussed the rates of rare events for a one-dimensional particle with inertia and a retarded friction force described by an integral of the velocity over time with a weighting factor [38]. We formulated a variational problem for the most probable path followed in switching between stable states in this case.…”

Capturing pattern bi-stability dynamics in delay-coupled swarms

“…More recently, however, the complex situation of several and random time delays has been researched [35][36][37]. Another important case is that of distributed time delays, when the dynamics of the system depends on a continuous p-1 interval in its past instead of on a discrete instant [38][39][40].…”