Microscopic theory of anomalous diffusion based on particle interactions

We present a master equation formulation based on a Markovian random walk model that exhibits subdiffusion, classical diffusion, and superdiffusion as a function of a single parameter. The nonclassical diffusive behavior is generated by allowing for interactions between a population of walkers. At the macroscopic level, this gives rise to a nonlinear Fokker-Planck equation. The diffusive behavior is reflected not only in the mean squared displacement [$\langle r^2\left(t\right)\rangle \sim t^{\gamma }$ with $0<\gamma \le 1.5$] but also in the existence of self-similar scaling solutions of the Fokker-Planck equation. We give a physical interpretation of sub- and superdiffusion in terms of the attractive and repulsive interactions between the diffusing particles and we discuss analytically the limiting values of the exponent $\gamma$. Simulations based on the master equation are shown to be in agreement with the analytical solutions of the nonlinear Fokker-Planck equation in all three diffusion regimes.

Figure 1

The mean squared displacement as a function of time obtained from the solutions of the master equation for the cases of sub-, classical, and superdiffusion with a Gaussian initial condition. Other details of the simulation are described in Sec. 4. Only a single parameter varied is varied, $\alpha$ in Eq. (19), and the scaling exponents are predicted from the generalized diffusion equation (6), to be $\gamma =2/(\alpha +1)$. The line segments are curves of the form $\langle r^2\rangle =a{t}^{\gamma }$ and verify the diffusive scaling with the predicted exponents at long times. In this figure the mean squared displacement is given in units of lattice spacings $\delta r$ and the time in units of the fixed jump time $\delta t$.

Figure 2

The spatial distribution as a function of time as determined by direct numerical solution of the master equation (symbols) and the analytic distribution, Eq. (30) for the case $\alpha =0.5$ corresponding to a scaling exponent of $\gamma =4/3$. All quantities are dimensionless (i.e., $\delta r=\delta t=1$). In both cases, the initial distribution was the predicted scaling solution as described in the text: for the figure on the left, the initial width was $w=10$ while for the figure on the right it was $w=100$.

Figure 3

The scaling exponent $\gamma$ as determined from fits of the numerical solution of the master equation. The fits were to the predicted functional form $\langle r^2\rangle =a_0{(a_1+t)}^{\gamma }$ and were performed independently with successive windows of ${10}^6$ time steps. The figure also shows results for different initial widths $w$ (in units of lattice spacings) as described in the text. The lines are drawn at the predicted values $\gamma =2/(1+\alpha )$.

Figure 4

$\gamma$ as a function of the inverse of the initial width (in lattice units). The lines are fits to $\gamma \left(w\right)=a_0+a_1{(1/w)}^{a_2}$. For $\alpha =0.5$, the fit gives $\gamma (w\to \infty )=1.339$ and for $\alpha =0.65$, $\gamma (w\to \infty )=1.221$. The expected values of $\gamma$ based on the nonlinear Fokker-Planck equation are $4/3=1.333$ and $40/33=1.212$, respectively.