Eur. Phys. J. B (2022) 95: 88
https://doi.org/10.1140/epjb/s10051-022-00350-9

Regular Article - Statistical and Nonlinear Physics

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

Institute of Biochemical Physics, Russian Academy of Sciences, Kosygin Street 4, 119 334, Moscow, Russia

^a kpronin@mail.ru

Received: 16 October 2021
Accepted: 6 May 2022
Published online: 25 May 2022

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 $\approx exp\left[cons{t}_{d,1}{t}^{d/\left(d+2\right)}\right]$. 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.