Brownian particles on rough substrates: Relation between intermediate subdiffusion and asymptotic long-time diffusion

Brownian particles in random potentials show an extended regime of subdiffusive dynamics at intermediate times. The asymptotic diffusive behavior is often established at very long times and thus cannot be accessed in experiments or simulations. For the case of one-dimensional random potentials with Gaussian distributed energies, we present a detailed analysis of experimental and simulation data. It is shown that the asymptotic long-time diffusion coefficient can be related to the behavior at intermediate times, namely, the minimum of the exponent that characterizes subdiffusion and hence corresponds to the maximum degree of subdiffusion. As a consequence, investigating only the dynamics at intermediate times is sufficient to predict the order of magnitude of the long-time diffusion coefficient and the time scale at which the crossover from subdiffusion to diffusion occurs, i.e., when the long-time diffusive regime and hence thermal equilibrium is established. Inversely, theoretical predictions derived for the asymptotic long-time behavior can be used to quantitatively characterize the intermediate behavior, which hardly has been studied so far.

Figure 1

Time dependence of the (a) and (b) mean square displacement $\langle x^2\left(t\right)\rangle$ in units of the radius $R$, (c) and (d) diffusion coefficient $D$ in units of the free diffusion coefficient $D_0=R^2/2{\tau }_{\mathrm{B}}$, and (e) and (f) exponent $\nu \left(t\right)$ of the mean square displacement as in $\langle x^2\left(t\right)\rangle \propto t^{\nu \left(t\right)}$ for different degrees of roughness of the potential $\epsilon /k_{\mathrm{B}}T$ (as indicated). Time $t$ is in units of the Brownian time ${\tau }_{\mathrm{B}}$. Simulation results are represented by lines, experiments by symbols. Simulation results are not averaged over time. The experimental data shown in (a), (c), and (e) are not time averaged, while the experimental data presented in (b), (d), and (f) are obtained after an average over time (see the text for details). Fits to $D\left(t\right)/D_0$ from simulations are represented by thin gray lines. For comparison, free diffusion is indicated by black solid lines with (a) and (b) $\langle x^2\left(t\right)\rangle =2D_{0}t$, (c) and (d) $D/D_0=1$, and (e) and (f) $\nu =1$.

Figure 2

Negative logarithm of the normalized asymptotic diffusion coefficient $D_{\infty }/D_0$ as a function of the degree of roughness of the potential $\epsilon /k_{\mathrm{B}}T$. Here $D_{\infty }/D_0$ is obtained by theoretical predictions (5) (black solid line) and fits to $D\left(t\right)/D_0$ [Fig. 1], which were determined in simulations (+) and experiments ($⊡$). The logarithm of the normalized time scale ${\tau }_{\infty }/{\tau }_{\mathrm{B}}$, associated with the crossover from subdiffusion to asymptotic diffusion, is shown for the simulation results $\left(\begin{array}{c}*\end{array}\right)$. The negative logarithm of the minimal value of the exponent ${\nu }_{\min }$ is obtained from $\nu \left(t\right)$ [Fig. 1], determined in simulations (×) and experiments ($\odot$). Straight lines are fitted to ${log}_{10}\left[{\nu }_{\min }\left(\epsilon \right)\right]$, determined in simulations (blue dashed line) and experiments (red dotted line). The inset on the right shows $-{log}_{10}\left({\nu }_{\min }\right)$ from simulations together with a linear fit as in the main figure (blue dashed line) and a power law fit with exponent $0.83$ (cyan solid line). The inset on the left shows ${log}_{10}({\tau }_{\infty }/{\tau }_{\mathrm{B}})$ as a function of $-{log}_{10}(D_{\infty }/D_0)$. The green dashed line indicates a linear dependence ${\tau }_{\infty }/{\tau }_{\mathrm{B}}=f{D}_{\infty }/D_0$ [Eq. (6)], where $f\approx 1.33$ is obtained by fitting.

Figure 3

Negative logarithm of the normalized asymptotic diffusion coefficient $D_{\infty }/D_0$, which is related to the logarithm of the time characterizing the crossover to the asymptotic regime ${\tau }_{\infty }$, as a function of the negative logarithm of the minimum exponent ${\nu }_{\min }$ as obtained by simulations (main figure, $\times$) and experiments (inset, $⊡$). Lines are quadratic fits to all data (black solid line) or only the first few data points (broken lines). The gray area indicates the time scale that needs to be explored in experiments or simulations in order to predict the behavior shown in the entire figure.