Fractional diffusion in a periodic potential: Overdamped and inertia corrected solutions for the spectrum of the velocity correlation function

Anomalous diffusion of a particle in a cosine periodic potential is treated using fractional diffusion equations in both phase and configuration space. Exact solutions of two distinct forms of the fractional Klein-Kramers (Fokker-Planck) equation for the distribution function in phase space are obtained via matrix continued fractions yielding the average velocity, the velocity autocorrelation function, its spectrum, etc. In the overdamped limit, the results yielded by both equations agree with those from a fractional probability density diffusion equation in configuration space. A simple analytic solution for the spectrum of the velocity correlation function is also given using the effective eigenvalue approximation. The results represent generalizations of the conventional solutions for the normal diffusion of a Brownian particle in a cosine potential to fractional dynamics (giving rise to anomalous diffusion).

Figure 1

$\mathrm{Re}\left[{\tilde{C}}_v\left(\omega \right)\right]$ and $\mathrm{Im}\left[{\tilde{C}}_v\left(\omega \right)\right]$ vs $\omega \tau$ for various of σ and $b$ $=$ 5: comparison of the continued fraction solution Eq. (16) and approximate Eq. (20) (symbols).

Figure 2

$\mathrm{Re}\left[{\tilde{C}}_v\left(\omega \right)\right]$ and $\mathrm{Im}\left[{\tilde{C}}_v\left(\omega \right)\right]$ vs $\omega \eta$ for ${\beta }^{\prime }=0.1$, $\sigma =0.8$, and various barrier heights $b$ for the Barkai-Silbey (solid lines) and Metzler-Klafter (dashed lines) models.

Figure 3

$\mathrm{Re}\left[{\tilde{C}}_v\left(\omega \right)\right]$ and $\mathrm{Im}\left[{\tilde{C}}_v\left(\omega \right)\right]$ vs $\omega \tau$ for ${\beta }^{\prime }=1$, $b$ $=$ 5, and various $\sigma$ for the Barkai-Silbey (solid lines) and Metzler-Klafter (dashed lines) models.

Figure 4

$\mathrm{Re}\left[{\tilde{C}}_v\left(\omega \right)\right]$ and $\mathrm{Im}\left[{\tilde{C}}_v\left(\omega \right)\right]$ vs $\omega \eta$ for $b$ $=$ 5, $\sigma =0.8$, and various damping ${\beta }^{\prime }$ for the Barkai-Silbey (solid lines) and Metzler-Klafter (dashed lines) models. (Symbols) The effective eigenvalue Eq. (42) which is a good approximation for the low-frequency behavior for ${\beta }^{\prime }\ge 1$.

Figure 5

Comparison of the matrix continued fraction solution Eq. (B2) and the approximate Eq. (42) (symbols) in the high damping limit and low frequencies for the Barkai-Silbey and Metzler-Klafter FKKEs (for high damping, both models yield results coinciding to graphical accuracy).