2021/07/02 by Vaibhava Srivastava, Srivastava, Vaibhava, Alexei F. Cheviakov +1
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Diffusion and Search Dynamics #FOS: Physical sciences #Mathematical Physics (math-ph) #Stochastic processes and statistical mechanics #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.2107.01233
openalex publication_date 2021/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The narrow escape problem is a first-passage problem concerned with randomly\nmoving particles in a physical domain, being trapped by absorbing surface traps\n(windows), such that the measure of traps is small compared to the domain size.\nThe expected value of time required for a particle to escape is defined as mean\nfirst passage time (MFPT), which satisfies the Poisson partial differential\nequation subject to a mixed Dirichlet-Neumann boundary condition. The primary\nobjective of this work is a direct numerical simulation of multiple particles\nundergoing Brownian motion in a three-dimensional sphere with boundary traps,\ncompute MFPT values by averaging Brownian escape times, and compare the results\nwith asymptotic results obtained by solving the Poisson PDE problem. A\ncomprehensive study of results obtained from the simulations shows that the\ndifference between Brownian and asymptotic results for the escape times mostly\nnot exceed 1 % accuracy. This comparison in some sense validates the narrow\nescape PDE problem itself as an approximation (averaging) of the multiple\nphysical Brownian motion runs. This work also predicted that how many\nsingle-particle simulations are required to match the predicted asymptotic\naveraged MFPT values. The next objective of this work is to study dynamics of\nBrownian particles near the boundary by estimating the average percentage of\ntime spent by Brownian particle near the domain boundary for both the\nanisotropic and isotropic diffusion. It is shown that the Brownian particles\nspend more in the boundary layer than predicted by the boundary layer relative\nvolume, with the effect being more pronounced in a narrow layer near the\nspherical wall. It is also shown that taking into account anisotropic diffusion\nyields larger times a particle spends near the boundary, and smaller escape\ntimes than those predicted by the isotropic diffusion model.\n