Barrier crossing in the presence of multi-exponential memory functions with unequal friction amplitudes and memory times

Abstract

We study the non-Markovian Langevin dynamics of a massive particle in a onedimensional double-well potential in the presence of multi-exponential memory by simulations. We consider memory functions as the sum of two or three exponentials with different friction amplitudes γi and different memory times τi and confirm the validity of a previously suggested heuristic formula for the mean first-passage time τMFP. Based on the heuristic formula, we derive a general scaling diagram that features a Markovian regime for short memory times, an asymptotic long-memory-time regime where barrier crossing is slowed down and τMFP grows quadratically with the memory time, and a non-Markovian intermediate regime where barrier crossing is slightly accelerated or slightly slowed down, depending primarily on the particle mass. The relative weight of different exponential memory contributions is described by the scaling variable γi/τ 2 i , i.e., memory contributions with long memory times or small amplitudes are negligible compared to other memory contributions.