Universal time-dependent dispersion properties for diffusion in a one-dimensional critically tilted potential

We consider the time-dependent dispersion properties of overdamped tracer particles diffusing in a one-dimensional periodic potential under the influence of an additional constant tilting force $F$. The system is studied in the region where the force is close to the critical value $F_c$ at which the barriers separating neighboring potential wells disappear. We show that, when $F$ crosses the critical value, the shape of the mean-square displacement (MSD) curves is strongly modified. We identify a diffusive regime at intermediate-time scales with an effective diffusion coefficient which is much larger than the late-time diffusion coefficient for $F>F_c$, whereas for $F<F_c$ the late-time and intermediate-time diffusive regimes are indistinguishable. Explicit asymptotic regimes for the MSD curves are identified at all time scales.

Figure 1

(a) Sketch of the problem investigated in this paper: We calculate the dispersion properties of particles diffusing in periodic tilted potentials. When the noise is weak and the force is close to its critical value, the tracer particles spend most of their time in narrow regions, termed the slow regions, of typical size $l^{*}$ much smaller than period $L$. (b) Value of the stationary PDF $P_s\left(x\right)$ inside the slow region obtained by evaluating Eqs. (19) and (20) for $\varepsilon =-5$ ($F$ below the critical force $F_c$), $\varepsilon =0(F=F_c)$, and $\varepsilon =5(F>F_c)$. The inset: same curves on a semilogarithmic scale.

Figure 2

(a) Value of the dimensionless flux ${\tilde{J}}_s=J_s{\tau }^{*}$ for near critical forces, which is also proportional to the effective diffusion coefficient in the intermediate-time regime [see Eq. (33)]. (b) Rescaled late-time effective diffusivity $D_{\infty }$, deduced from Eq. (A9). (c) Ratio of the late-time effective diffusivity over the intermediate-time effective diffusivity.

Figure 3

MSD curves for the diffusion in a sine potential $V\left(x\right)=V_{0}sin(2\pi x/L)$ for (a) a tilting force below the critical force ($\varepsilon =-2$), (b) at the critical force ($\varepsilon =0$), and (c) above the critical force ($\varepsilon =2$). Different values of the noise amplitude are indicated in the legend. Symbols: results of the simulations of the Langevin equation (1) (see Appendix pp2 for details). Lines: results of the numerical evaluation of the Kubo formula (5). Bold lines: asymptotic results of Eq. (34) (with no free parameters). Note that the regime $\psi =2D_{0}t$ is not represented but is present at short times. Units of length and times are $L$ and $\zeta L^2/V_0$, respectively.