Superdiffusion induced by a long-correlated external random force

We consider a particle immersed in a thermal reservoir and simultaneously subjected to an external random force that drives the system to a nonequilibrium situation. Starting from a Langevin equation description, we derive exact expressions for the mean-square displacement and the velocity autocorrelation function of the diffusing particle. An effective temperature is introduced to characterize the deviation from the internal equilibrium situation. Using a power-law force autocorrelation function, the mean-square displacement and the velocity autocorrelation function are analytically obtained in terms of Mittag-Leffler functions. In this case, we show that the present model exhibits a superdiffusive regime as a consequence of the competition between passive and active processes.

Figure 1

The term $\Delta (t,0)$ given by Eq. (37) for $\alpha =$ 0.2, 0.4, 0.6, and 0.8 from bottom to top. The used parameters are ${\Lambda }_{\alpha }/m^2=1$ and ${\tau }_0=1$.

Figure 2

VACF vs time lag $\tau$ for $\alpha =$ 0.2, 0.4, 0.6, and 0.8 from top to bottom, with $\varepsilon =1$ and ${\tau }_0=1$.

Figure 3

The logarithmic derivative of the MSD as a function of the time lag for $\alpha =1/5$ and $\varepsilon =$ 0.01, 0.05, 0.1, 0.5, 1, and 5 from bottom to top, with ${\Lambda }_{\alpha }/m^2=1$ and ${\tau }_0=1$. The corresponding values for ${\tau }_{*}$ are 603.26, 80.68, 33.92, 4.54, 1.91, and 0.26, respectively. In all cases, the asymptotic value is $\beta \left(\infty \right)=1.8$.