Diffusion in an expanding medium: Fokker-Planck equation, Green's function, and first-passage properties

We present a classical, mesoscopic derivation of the Fokker-Planck equation for diffusion in an expanding medium. To this end, we take a conveniently generalized Chapman-Kolmogorov equation as the starting point. We obtain an analytical expression for the Green's function (propagator) and investigate both analytically and numerically how this function and the associated moments behave. We also study first-passage properties in expanding hyperspherical geometries. We show that in all cases the behavior is determined to a great extent by the so-called Brownian conformal time $\tau \left(t\right)$, which we define via the relation $\dot{\tau }=1/a^2$, where $a\left(t\right)$ is the expansion scale factor. If the medium expansion is driven by a power law $[a\left(t\right)\propto t^{\gamma }$ with $\gamma >0]$, then we find interesting crossover effects in the mixing effectiveness of the diffusion process when the characteristic exponent $\gamma$ is varied. Crossover effects are also found at the level of the survival probability and of the moments of the first passage-time distribution with two different regimes separated by the critical value $\gamma =1/2$. The case of an exponential scale factor is analyzed separately both for expanding and contracting media. In the latter situation, a stationary probability distribution arises in the long-time limit.

Figure 1

Schematic picture of the random walker's motion due to the combined action of random steps (solid arrows) and the deterministic drift arising from the medium expansion (dotted arrows). Recall that ${\mathbf{y}}^{\prime }=\mathbf{y}-\mathbf{z}=\mathbf{F}(\mathbf{y}-\mathbf{\epsilon },t_n)$.

Figure 2

Simulation results for the probability $P(y,t)$ to find the random walker at position $y$ after a time $t$ given the initial value $P(y,0)=[\delta (y-y_0)+\delta (y+y_0)]/2$. We have set $y_0=10, D=1/2, t_0=100$, and $t-t_0=10,20,100,500,5000$ (symbols: solid circle, solid square, circle, square, and star, respectively). The medium expands according to the power law $a\left(t\right)={(t/t_0)}^{\gamma }$ for $\gamma =1/4$. The solid lines represent the corresponding theoretical results.

Figure 3

Simulation results for the probability $P(y,t)$ to find the random walker at position $y$ after a time $t$ for a double-peaked initial condition $P(y,0)=[\delta (y-y_0)+\delta (y+y_0)]/2$ representing a pair of diffusive pulses. We take $y_0=10, t_0=100, D=1/2$, and $t-t_0=10,20,100,500,5000$ (symbols: solid circle, solid square, circle, square, and star, respectively). The medium expands according to the power law $a\left(t\right)={(t/t_0)}^{\gamma }$ and $\gamma =2$. The solid lines represent the corresponding theoretical results. There is no complete mixing between the two diffusive pulses, since the Brownian event horizon $r_{\text{Beh}}={[2D{t}_0/(2\gamma -1)]}^{1/2}\approx 8.16$ is shorter than the semidistance (=10) between the two source points located at $y=\pm y_0$.

Figure 4

Brownian horizons ${\overline{r}}_{\text{Bh}}$ as described in the main text. The parameter values are the same as in Figs. 2 and 3 (except $\gamma$, which is specified in each subfigure). The solid (dashed) line corresponds to the Brownian pulse that is emitted at $y=10$ ($y=-10$) at time $t_0$. It is clear that in comoving coordinates both pulses become narrower as $\gamma$ grows. Note also that for $\gamma =1/4$ both pulses overlap, but they do not for $\gamma =2$, in agreement with what is shown in Figs. 2 and 3. (a) $\gamma =0$; (b) $\gamma =1/4$; (c) $\gamma =1/2$; and (d) $\gamma =2$.

Figure 5

Simulation results for the probability $P(y,t)$ to find random walkers at position $y$ after a time $t$ in an exponentially contracting medium with $P(y,0)=\delta \left(y\right), t_0=100, D=1/2$, and $t-t_0=100$ for $H=-1/50$ (circles), $t-t_0=400$ for $H=-1/200$ (squares), and $t-t_0=800$ for $H=-1/400$ (triangles). The solid lines represent the corresponding final stationary distribution $P_s\left(y\right)$ as given by Eq. (63). The corresponding theoretical distributions $P(y,t)$ are also plotted (broken lines), but they are hardly distinguishable from the stationary distribution.