Ensemble averaging versus non-self-averaging: survival probability in the presence of traps-sinks

Abstract

We consider nonstationary diffusion in a medium with static random traps-sinks. We address the problem of self-averaging of the survival probability (or concentration) of the ensemble of $N$ particles in the fluctuation regime in the long-time limit. We demonstrate that the relative standard deviation of the survival probability decreases with the number of engaged particles as $N^{ - 1/2}$ and increases with time as a stretched exponential $\approx \exp \left[ {const_{d,\,1} t^{{d/\left( {d + 2} \right)}} } \right]$. Therefore, the survival probability is self-averaging in parameter $N$ and is strongly non-self-averaging over time $t$. To measure the concentration with the required accuracy at the required time of observation $t_{0}$, the initial number of particles $N_{0}$ must be exponentially large in $t_{0}$. At later times $t > t_{0}$ the relative fluctuations continue to diverge exponentially beyond the required accuracy. In the limit of high dimensions, there is no tendency to restore self-averaging over time in the ensemble of $N$ particles. The solution in 1D is exact. In higher dimensions, the leading exponential term of the solution is exact.

Graphical abstract