Topical Review - Statistical and Nonlinear Physics
Arrival time for the fastest among N switching stochastic particles
Group of Data Modeling, Computational Biology, IBENS, Ecole Normale Superieure-PSL, Paris, France
Accepted: 13 June 2022
Published online: 20 July 2022
The first arrivals among N Brownian particles is ubiquitous in the life sciences, as it often triggers cellular processes from the molecular level. We study here the case where stochastic particles, which represent proteins or molecules can switch between two states inside the non-negative real line. The switching process is modeled as a two-state Markov chain and particles can only escape in state 1. We estimate the fastest arrival time by solving asymptotically the Fokker–Planck equations for three different initial distributions: Dirac-delta, uniformly distributed and long-tail decay. The derived formulas reveal that the fastest particle avoids switching when the switching rates are much smaller than the diffusion time scale, but switches twice when the diffusion in state 2 is much faster than in state 1. The present results are compared to stochastic simulations revealing the range of validity of the derived formulas.
© The Author(s), under exclusive licence to EDP Sciences, SIF and Springer-Verlag GmbH Germany, part of Springer Nature 2022