Fractional Hydrodynamic Memory and Superdiffusion in Tilted Washboard Potentials

Diffusion in tilted washboard potentials can paradoxically exceed free normal diffusion. The effect becomes much stronger in the underdamped case due to inertial effects. What happens upon inclusion of usually neglected fractional hydrodynamics memory effects (Basset-Boussinesq frictional force), which result in a heavy algebraic tail of the velocity autocorrelation function of the potential-free diffusion making it transiently superdiffusive? Will a giant enhancement of diffusion become even stronger, and the transient superdiffusion last even longer? These are the questions that we answer in this Letter based on an accurate numerical investigation. We show that a resonancelike enhancement of normal diffusion becomes indeed much stronger and sharper. Moreover, a long-lasting transient regime of superdiffusion, including Richardson-like diffusion, $⟨\delta x^2(t)⟩\propto t^3$ and ballistic supertransport, $⟨\delta x(t)⟩\propto t^2$, is revealed.

Figure 1

Bistable fluctuations of the velocity of a single particle with the parameters given in plot (central panel). The left panel depicts probability density $p(v)$ derived from a single trajectory and the right panel shows the corresponding velocity pseudopotential $V(v)=-k_{B}T\ln p(v)$. The levels $v_1=0$ and $f/{\eta }_0$ are shown by dashed lines in the left and right panels, $⟨v⟩\approx 1.59$ and $⟨\delta v^2⟩\approx 1.87$.

Figure 2

Ratio of driven diffusion coefficient $D(U_0,f)$ to the potential-free $D_0(T)$, vs. applied force $f$ for different values of ${\gamma }_{\alpha }$ shown in the plot and $T=0.5$. Numerical critical values $f_c^{(3)}\approx 0.1270$ for ${\gamma }_{\alpha }=0$, $f_c^{(3)}\approx 0.1549$ for ${\gamma }_{\alpha }=0.1$, $f_c^{(3)}\approx 0.1900$ for ${\gamma }_{\alpha }=0.3$, and $f_c^{(3)}\approx 0.2413$ for ${\gamma }_{\alpha }=0.6623$, at $T=0$ are shown by the arrows. They indicate the onset of a diffusion enhancement over $D_0$. Notice that $f_c^{(3)}$ essentially increases with ${\gamma }_{\alpha }$. The critical value $f_c^{(2)}$ corresponding to the maximum of $D(T,f)$ slightly increases.

Figure 3

Ensemble-averaged mean-square displacement (a) and mean displacement (b) of Brownian particles vs time in log-log scaling (major plot) and linear scaling (inset) for $U_0=1$, $T=0.5$, ${\gamma }_0=0.1$, $f=0.285$, and four different values of ${\gamma }_{\alpha }$: ${\gamma }_{\alpha }=0$ (full black line), ${\gamma }_{\alpha }=0.1$ (dash-dotted red line), ${\gamma }_{\alpha }=0.3$ (dashed green line), and ${\gamma }_{\alpha }=0.6623$ (full blue line). The potential-free results, $U_0=0$, are also shown by full red lines for the case ${\gamma }_{\alpha }=0.6623$. Some fits and theoretical predictions based on $p_2(t)$ relaxation kinetics in Fig. 4 are also shown in the plot; see the main text.

Figure 4

The increase of population $p_2(t)$ of particles in the running state for two values of ${\gamma }_{\alpha }$ in Fig. 3, ${\gamma }_{\alpha }=0$ and ${\gamma }_{\alpha }=0.6623$. Two different procedures were used to find $p_2(t)$, as detailed in the text. Also, the corresponding exponential (${\gamma }_{\alpha }=0$) and power-law (${\gamma }_{\alpha }=0.6623$) fits are shown. They allow explaining the results on intermediate superdiffusion and supertransport in Fig. 3.