Is there memory for the memoryless? (Vol. 50, No. 4)

Inertia plays a role on the evolution of Brownian particles. Nevertheless, the interplay of inertial time-scale contributions and an overdamped dynamics with non-Markovian stochastic forces leads to contradictions that make equilibration impossible. This is due to assuming memory correlations for the dissipation, which seems to be inconsistent with the overdamped approximation, where thermal fluctuations adjust instantaneously to the state of the particle. Effectively, by taking the noise correlation time-scale to be zero (no memory) we certainly recover the expected physical behaviour of the problem, e.g., the equilibrium distribution. On the other hand, we can deal with the contradiction by inserting another source of noise, of Markovian type, and with “effective temperature” different from the non-Markovian noise. As a result, the stationary state may be regularized and the equilibrium recovered if both noises have same temperatures, even for finite memory time-scales. The additional white noise brings the system back to equilibrium, no matter how small the new noise intensity is.

E. S. Nascimento and W. A. M. Morgado, Non-Markovian effects on overdamped systems, EPL 126, 10002 (2019)
[Abstract]