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

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 $$N$$ 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 $$N^{ - 1/2}$$ N - 1 / 2 and increases with time as a stretched exponential $$\approx \exp \left[ {const_{d,\,1} t^{{d/\left( {d + 2} \right)}} } \right]$$ ≈ exp c o n s t d , 1 t d / d + 2 . Therefore, the survival probability is self-averaging in parameter $$N$$ N and is strongly non-self-averaging over time $$t$$ t . To measure the concentration with the required accuracy at the required time of observation $$t_{0}$$ t 0 , the initial number of particles $$N_{0}$$ N 0 must be exponentially large in $$t_{0}$$ t 0 . At later times $$t > t_{0}$$ 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 $$N$$ N particles. The solution in 1D is exact. In higher dimensions, the leading exponential term of the solution is exact. Graphical abstract