Survival of a static target in a gas of diffusing particles with exclusion

Let a lattice gas of constant density, described by the symmetric simple exclusion process, be brought in contact with a “target”: a spherical absorber of radius $R$. Employing the macroscopic fluctuation theory (MFT), we evaluate the probability $\mathcal{P}\left(T\right)$ that no gas particle hits the target until a long but finite time $T$. We also find the most likely gas density history conditional on the nonhitting. The results depend on the dimension of space $d$ and on the rescaled parameter $\ell =R/\sqrt{D_{0}T}$, where $D_0$ is the gas diffusivity. For small $\ell$ and $d>2$, $\mathcal{P}\left(T\right)$ is determined by an exact stationary solution of the MFT equations that we find. For large $\ell$, and for any $\ell$ in one dimension, the relevant MFT solutions are nonstationary. In this case, $ln\mathcal{P}\left(T\right)$ scales differently with relevant parameters, and it also depends on whether the initial condition is random or deterministic. The latter effects also occur if the lattice gas is composed of noninteracting random walkers. Finally, we extend the formalism to a whole class of diffusive gases of interacting particles.

Figure 1

The function ${arcsin}^2\sqrt{n_0}$ (the solid line), which describes the density dependence of $-ln\mathcal{P}\left(T\right)$ for the SSEP at $d>2$. The straight line shows the corresponding density dependence for the gas of random walkers (RWs).

Figure 2

Upper panel: Theoretical stationary density profile for the SSEP at $d=3$, Eq. (27) (dots), and numerical density profiles for $t=0.25$ and $0.75$ (indistinguishable), and for $t=1$. The (deterministic) initial condition for the numerical solution is shown by the dashed line. Inset: a blowup of the region close to $r=\ell$. The numerical procedure is described in Sec. 4a3. Lower panel: $q(r=0.05,t)$ found numerically; the boundary layers at $t=0$ and 1 are clearly seen. The parameters are $\ell =5\times {10}^{-3}$ and $n_0=0.5$. For these parameters, the theoretical rescaled action from Eq. (32) is $s\simeq 3.876\times {10}^{-2}$. The rescaled action found numerically from Eq. (19) is $3.916\times {10}^{-2}$. The coordinate $r$ is rescaled by $\sqrt{D_{0}T}$, and time is rescaled by $T$.

Figure 3

The most likely density history of the RWs at $d=1$, conditional on the zero flux to the target (located at $x=0$) for a deterministic initial condition with density $n_0$. Shown is $q/n_0$ from Eq. (50) vs $X=x/2$, where $x$ is rescaled by $\sqrt{T}$ (here $D_0=1$). Upper panel: $t=0$ (dashed line), $1/3$ (dash-dotted line), $2/3$ (dotted line), and 1 (solid line). Lower panel: $t=0$, $0.01$, $0.05$, and $0.1$. Time is rescaled by $T$.

Figure 4

Numerically computed most likely density history of the SSEP for $d=1$, conditional on the zero flux to the target (located at $x=0$) for a deterministic initial condition with density $n_0=0.8$. Upper panel: $q$ vs $x$ at $t=0$ (dashed line), $0.25$ (dash-dotted line), $0.5$ (dotted line), and 1 (solid line). Lower panel: numerically computed $v={\partial }_{x}p$ vs $x$ at $t=0.25$ (dash-dotted line), $0.5$ (dotted line), and $0.75$ (thin solid line). The thick solid line shows the universal asymptotic $v=1/x$. The coordinate $x$ is rescaled by $\sqrt{T}$ (here $D_0=1$), and time is rescaled by $T$.

Figure 5

The function $s_1\left(n_0\right)$ found numerically for the deterministic initial condition. Shown are numerical data (points), the RW asymptotic (56) (the dotted line), and the finite-density asymptotic (63) (the dashed line).