Effective mass approach to memory in non-Markovian systems

Recent pioneering experiments on non-Markovian dynamics done, e.g., for active matter have demonstrated that our theoretical understanding of this challenging yet hot topic is rather incomplete and there is a wealth of phenomena still awaiting discovery. It is related to the fact that typically for simplification the Markovian approximation is employed and as a consequence the memory is neglected. Therefore, methods allowing to study memory effects are extremely valuable. We demonstrate that a non-Markovian system described by the Generalized Langevin Equation (GLE) for a Brownian particle of mass $M$ can be approximated by the memoryless Langevin equation in which the memory effects are correctly reproduced solely via the effective mass $M^{*}$ of the Brownian particle which is determined only by the form of the memory kernel. Our work lays the foundation for an impactful approach which allows one to readily study memory-related corrections to Markovian dynamics.

Figure 1

The average velocity $\langle \widehat{v}\rangle$ of a Brownian particle as a function of the correlation time $\tau$ for $f=0.1$ obtained from (i) the GLE (33), (ii) the effective mass approach (35) and (iii) the standard Markovian approximation $\tau \to 0$ when $m^{*}=m$. The parameters are $\{m=1$, $a=15$, $\omega =4$, $f=0.1$, $D={10}^{-3}\}$.

Figure 2

The average velocity $\langle \widehat{v}\rangle$ of the Brownian particle as a function of the static bias $f$ for $\tau =0.025$. Other parameters are the same as in Fig. 1.

Figure 3

Time evolution of the mean velocity averaged over the external driving period $\langle \widehat{\mathsf{v}}\rangle$ for $\tau =0.025$ and $f=0.1$. Other parameters are the same as in Fig. 1.

Figure 4

The average velocity $\langle \widehat{v}\rangle$ of the Brownian particle as a function of the memory time $\tau$ for $m=149$, $a=50$, $\omega =0.2$, $f=0.1$ and $D=0.01$. The effective mass approach is correct up to $\tau \approx {\tau }_0=1$.