First-passage time for many-particle diffusion in space-time random environments

The first-passage time for a single diffusing particle has been studied extensively, but the first-passage time of a system of many diffusing particles, as is often the case in physical systems, has received little attention until recently. We consider two models for many-particle diffusion—one treats each particle as independent simple random walkers while the other treats them as coupled to a common space-time random forcing field that biases particles nearby in space and time in similar ways. The first-passage time of a single diffusing particle under both models shows the same statistics and scaling behavior. However, for many-particle diffusions, the first-passage time among all particles (the extreme first-passage time) is very different between the two models, effected in the latter case by the randomness of the common forcing field. We develop an asymptotic (in the number of particles and location where first passage is being probed) theoretical framework to separate the impact of the random environment with that of the sampling trajectories within it. We identify a power law describing the impact of the environment on the variance of the extreme first-passage time. Through numerical simulations, we verify that the predictions from this asymptotic theory hold even for systems with widely varying numbers of particles, all the way down to 100 particles. This shows that measurements of the extreme first-passage time for many-particle diffusions provide an indirect measurement of the underlying environment in which the diffusion is occurring.

Figure 1

A system of $N={10}^5$ particles evolving according to our RWRE model. The green dashed lines denote a barrier at $-L$ and $L$. The extreme first-passage time, ${\mathrm{Min}}_L^N$, is identified by the red circle and is the first time when one of these $N$ walkers crosses this barrier. The inset illustrates the dynamics of particles in the first three time steps (not all ${10}^5$ particles are shown to avoid cluttering). The color of each box in the inset corresponds to the bias of the location (blue means upward bias and red means downward bias)

Figure 2

For the RWRE model with $N={10}^{12}$ particles, we plot the numerically measured (solid) and asymptotic theory (dashed) variance of the extreme first-passage time $\mathrm{Var}\left({\mathrm{Min}}_L^N\right)$ (red), the variance due to the environment $\mathrm{Var}\left({\mathrm{Env}}_L^N\right)$ (blue), and the variance due to sampling $\mathrm{Var}\left({\mathrm{Sam}}_L^N\right)$ (green). For the SSRW model, we plot $\mathrm{Var}\left({\widetilde{\mathrm{Min}}}_L^N\right)$, where ${\widetilde{\mathrm{Min}}}_L^N$ is the extreme first-passage time for the SSRW model (purple). Note, our asymptotic theory matches our numerics to such precision that they become indistinguishable on this scale.

Figure 3

We plot ${\mathrm{Var}}^{\mathrm{Num}}\left({\mathrm{Min}}_L^N\right)-{\mathrm{Var}}^{\mathrm{Num}}\left({\mathrm{Sam}}_L^N\right)$ for $N={10}^2,{10}^5,{10}^{12}$, and ${10}^{28}$ averaged into bins with logarithmically spaced bin edges to achieve four bins per decade of $L$. The dots represent positive values of this difference whereas the $\times$ represents the magnitude of negative values of this difference. The dashed line is ${\mathrm{Var}}^{\mathrm{Asy}}\left({\mathrm{Env}}_L^N\right)$ in Eq. (31) for each $N$ corresponding to its respective color. The lower plot shows the ratio $({\mathrm{Var}}^{\mathrm{Num}}\left({\mathrm{Min}}_L^N\right)-\mathrm{Var}\left({\mathrm{Sam}}_L^N\right))/{\mathrm{Var}}^{\mathrm{Num}}\left({\mathrm{Env}}_L^N\right)$, confirming the close fit.

Figure 4

We plot the environmental variance, ${\mathrm{Var}}^{\mathrm{Num}}\left({\mathrm{Env}}_L^N\right)$, for $N={10}^{28}$ particles (blue); the asymptotic variance $V_1$ for the short time regime given in Eqs. (22) (orange dashed line); the asymptotic variance $V_2$ for the long time regime given in Eqs. (30) (purple dashed line); the interpolation ${\mathrm{Var}}^{\mathrm{Asy}}\left({\mathrm{Env}}_L^N\right)$ between these regimes using Eq. (31) (black dashed line); and the power-law asymptotics of $V_1$ and $V_2$ (black dotted lines)

Figure 5

We plot the numerically measured (solid) and asymptotic theory (dashed) variance of the extreme first-passage time, $\mathrm{Var}\left({\mathrm{Min}}_L^N\right)$, for $N={10}^2,{10}^5,{10}^{12}$, and ${10}^{28}$ (each labeled with a different color). The asymptotic theory variance, ${\mathrm{Var}}^{\mathrm{Asy}}\left({\mathrm{Min}}_L^N\right)$, comes from Eq. (43). Note, our asymptotic theory matches our numerics to such precision that they are nearly indistinguishable on this scale. The inset shows the same data uncollapsed to better distinguish different $N$. The lower plot shows the ratio ${\mathrm{Var}}^{\mathrm{Num}}\left({\mathrm{Min}}_L^N\right)/{\mathrm{Var}}^{\mathrm{Asy}}\left({\mathrm{Min}}_L^N\right)$.

Figure 6

We plot the numerically measured (solid) and asymptotic theory (dashed) environmental variance, $\mathrm{Var}\left({\mathrm{Env}}_L^N\right)$, for $N={10}^2,{10}^5,{10}^{12}$, and ${10}^{28}$ (each labeled with a different color). The asymptotic theory variance, ${\mathrm{Var}}^{\mathrm{Asy}}\left({\mathrm{Env}}_L^N\right)$, comes from Eq. (31). Note, our asymptotic theory matches our numerics to such precision that they are nearly indistinguishable on this scale. The inset shows the same data uncollapsed to better distinguish different $N$. The lower plot shows the ratio ${\mathrm{Var}}^{\mathrm{Num}}\left({\mathrm{Env}}_L^N\right)/{\mathrm{Var}}^{\mathrm{Asy}}\left({\mathrm{Env}}_L^N\right)$.

Figure 7

We plot the mean of the extreme first-passage time, ${\mathbb{E}}^{\mathrm{Num}}\left[{\mathrm{Min}}_L^N\right]$, using solid lines of varying colors for $N={10}^2,{10}^5,{10}^{12}$, and ${10}^{28}$ particles. The dashed lines are the asymptotic curves ${\mathbb{E}}^{\mathrm{Asy}}\left[{\mathrm{Min}}_L^N\right]$ from Eq. (44). Note, our asymptotic theory matches our numerics to such precision that they are nearly indistinguishable on this scale. The inset shows the uncollapsed data to better distinguish different values of $N$. The collapse occurs by scaling both the $x$ axis and $y$ axis by $ln\left(N\right)$. The lower plot shows the ratio of ${\mathbb{E}}^{\mathrm{Num}}\left[{\mathrm{Min}}_L^N\right]/{\mathbb{E}}^{\mathrm{Asy}}\left[{\mathrm{Min}}_L^N\right]$, confirming the close fit.

Figure 8

We plot ${\mathrm{Var}}^{\mathrm{Num}}\left({\mathrm{Min}}_L^N\right)-{\mathrm{Var}}^{\mathrm{Asy}}\left({\widetilde{\mathrm{Min}}}_L^N\right)$ for $N={10}^2,{10}^5,{10}^{12}$ and ${10}^{28}$ averaged into bins with logarithmically spaced bin edges to achieve 4 bins per decade of $L$. The dots represent positive values of this difference whereas the $\times$ represents the magnitude of negative values of this difference. The dashed line is ${\mathrm{Var}}^{\mathrm{Asy}}\left({\mathrm{Env}}_L^N\right)$ in Eq. (31) for each $N$ corresponding to its respective color.