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Short-time blowup statistics of a Brownian particle in repulsive potentials

Baruch Meerson

TL;DR

The paper addresses the short-time tail of the blowup-time distribution for an overdamped Brownian particle in a repulsive scale-invariant potential $V(x,n)=-x^{n+1}/(n+1)$. By applying a time-independent WKB (Laplace-transformed backward Fokker–Planck) analysis to the generating function $oldsymbol{ ho}(s,x,n)$, it derives the leading and subleading contributions and, crucially, the large pre-exponential factor of the blowup-time distribution for all integer $n>1$. For even $n$, the WKB solution suffices across all $x$, yielding a compact expression for the small-$T$ tail; for odd $n$, a boundary layer near $x=0$ requires an internal-region solution and matching, producing an additional multiplicative factor and a refined tail. The resulting asymptotic form is $oldsymbol{ m P}(T o 0,n) \\simeq \mu_n \kappa_n \; T^{-(3n-1)/(2(n-1))} \exp[-\beta_n T^{-(n+1)/(n-1)}]$, with explicit $eta_n$ and $ κ_n$ in terms of Gamma functions, and $eta_n$ matches the leading exponent found by prior optimal fluctuation methods. The results improve understanding of short-time blowups and offer a method applicable to a broader class of first-passage problems.

Abstract

We study the dynamics of an overdamped Brownian particle in a repulsive scale-invariant potential $V(x) \sim -x^{n+1}$. For $n > 1$, a particle starting at position $x$ reaches infinity in a finite, randomly distributed time. We focus on the short-time tail $T \to 0$ of the probability distribution $P(T, x, n)$ of the blowup time $T$ for integer $n > 1$. Krapivsky and Meerson [Phys. Rev. E \textbf{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 $T \to 0$ tail by calculating, for all $n = 2, 3, \dots$, 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.

Short-time blowup statistics of a Brownian particle in repulsive potentials

TL;DR

The paper addresses the short-time tail of the blowup-time distribution for an overdamped Brownian particle in a repulsive scale-invariant potential . By applying a time-independent WKB (Laplace-transformed backward Fokker–Planck) analysis to the generating function , it derives the leading and subleading contributions and, crucially, the large pre-exponential factor of the blowup-time distribution for all integer . For even , the WKB solution suffices across all , yielding a compact expression for the small- tail; for odd , a boundary layer near requires an internal-region solution and matching, producing an additional multiplicative factor and a refined tail. The resulting asymptotic form is , with explicit and in terms of Gamma functions, and matches the leading exponent found by prior optimal fluctuation methods. The results improve understanding of short-time blowups and offer a method applicable to a broader class of first-passage problems.

Abstract

We study the dynamics of an overdamped Brownian particle in a repulsive scale-invariant potential . For , a particle starting at position reaches infinity in a finite, randomly distributed time. We focus on the short-time tail of the probability distribution of the blowup time for integer . Krapivsky and Meerson [Phys. Rev. E \textbf{112}, 024128 (2025)] recently evaluated the leading-order asymptotics of this tail, which exhibits an -dependent essential singularity at . Here we provide a more accurate description of the tail by calculating, for all , 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 . For even , the WKB solution alone suffices. For odd , however, the WKB solution breaks down in a narrow boundary layer around . 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.

Paper Structure

This paper contains 9 sections, 37 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Generating function of $\mathcal{P}(T)$
  3. Generating function of $\mathcal{P}(T)$
  4. WKB theory, internal region and matching
  5. WKB solution
  6. Even $n$: the WKB solution holds for all $x$
  7. Odd $n$: internal region and matching
  8. Short-time tail of $\mathcal{P}(T,n)$
  9. Summary and Discussion

Figures (4)

  • Figure 1: Solid line: numerical solution of Eq. (\ref{['Pi-eq']}) for the generating function for $n=4$ and $s=10$artrelaxation. Red dashed line: the WKB asymptotic given by Eqs. (\ref{['WKBansatz']}) and (\ref{['S']}). Blue dashed line: the asymptotic $\Pi(x\to \infty)=1$.
  • Figure 2: Solid line: the exact generating function $\Pi(x)$ as described by Eq. (\ref{['Pi-sol']}) for $n=3$ and $s=10$. Red dashed line: the WKB asymptotic of $\Pi(x)$ given by Eqs. (\ref{['WKBansatz']}) and (\ref{['S']}).
  • Figure 3: Solid line: the exact expression (\ref{['Pi-0']}) for the generating function $\Pi(s,0)$ for $n=3$. Dashed line: the large-$s$ asymptotic of $\Pi(s,0)$, given by Eq. (\ref{['const']}).
  • Figure 4: Solid line: the blowup time probability distribution $\mathcal{P}(T)$ for $n=3$ and $x=0$, obtained by a numerical inverse Laplace transform of the exact generating function (\ref{['Pi-0']}). The red dashed line shows the short-time asymptotic (\ref{['PsmallT']}). The blue dashed line shows the long-time asymptotic $\mathcal{P}(T) \simeq 2.5 \, e^{-1.3685\dots T}$ obtained in Ref. KM2025.