2025/09/01 by Baruch Meerson, Meerson, Baruch · 1 citation
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Fractional Differential Equations Solutions #Mathematical Physics (math-ph) #Probability (math.PR) #Statistical Mechanics (cond-mat.stat-mech) #Statistical Mechanics and Entropy #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.2509.01252
openalex publication_date 2025/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the dynamics of an overdamped Brownian particle in a repulsive scale-invariant potential V(x) ∼ -xn+1. For n > 1, a particle starting at position x reaches infinity in a finite, randomly distributed time. We focus on the short-time tail T → 0 of the probability distribution P(T, x, n) of the blowup time T for integer n > 1. Krapivsky and Meerson [Phys. Rev. E 112, 024128 (2025)] recently evaluated the leading-order asymptotics of this tail, which exhibits an n-dependent essential singularity at T = 0. Here we provide a more accurate description of the T → 0 tail by calculating, for all n = 2, 3, …, the previously unknown large pre-exponential factor of the blowup-time probability distribution. To this end, we apply a WKB approximation -- at both leading and subleading orders -- to the Laplace-transformed backward Fokker--Planck equation governing P(T, x, n). For even n, the WKB solution alone suffices. For odd n, however, the WKB solution breaks down in a narrow boundary layer around x = 0. In this case, it must be supplemented by an ``internal'' solution and a matching procedure between the two solutions in their common region of validity.