From Distribution of Extremes With Degradation to the Stress Redundancy Principle

Abstract

How a degradation source affects the statistics of the first among many identical independent stochastic particles to successfully reach a narrow target remains unclear. Here, we investigate the impact of degradation on the distributions of first-passage times and exit locations by exploiting the time-dependent fluxes associated with the corresponding Fokker-Planck equation. We also compute the escape probability by solving a mixed boundary value problem. In the large initial population $N_0$ limit, we derive an asymptotic formula for the mean first passage time associated with the fastest particle, revealing how the killing field modifies the classical $1/\log (N_0)$ scaling. However, when considering the population size at the time of the first exit, we obtain a specific dependency with respect to degradation, a phenomenon that we call the stress redundancy principle. For a degradation field restricted to a narrow subregion, we use an optimal control approach to derive the optimal trajectory where the fastest escape particles will be concentrated: interestingly, we report here that the optimal trajectories avoid the degradation area depending on killing intensity. Finally, we obtain analytical expressions for the distribution of first exit points along a narrow absorbing arc in two-dimensional regular domains. In addition, for a killing subregion tangent to the target boundary, our computations reveal how geometrical parameters influence the exit distribution. In summary, a killing source can significantly affect the statistics of the fastest diffusing particles, relevant for defining cellular and molecular time scales.

arXiv comment49 pages, 8 figures, 1 table