Kramers’ turnover phenomenon in the spatial diffusion region

In this paper we have presented the dynamics of a Brownian particle with time-delayed feedback. It clearly suggests that the delayed feedback may introduce a dissipation-like effect. As a result of this a breakdown of the fluctuation–dissipation relation occurs and the system behaves like an open one. Therefore, the stationary distribution deviates from the Boltzmann type. It depends on the damping strength. The probability at the barrier top increases with the enhancement of the damping strength. This is a sharp contrast to closed systems (which obey the fluctuation–dissipation relation) and the usual open systems, where the noise strength does not depend on the damping strength. This special feature motivated us to calculate Kramers' rate in the presence of the delayed feedback. Our calculation shows that the activation energy decreases with an increase in the damping strength. This peculiarity introduces a noticeable observation. Kramers' turnover behaviour appears even in the spatial diffusive regime. Thus its origin is quite different from the known Kramers' turnover, which is a result of an interplay of energy and spatial diffusive regimes. It should be mentioned here that all our theoretical results are well justified by the numerical experiments.