Lévy noise drives an exponential acceleration in transition rates within metastable systems

TL;DR

This work addresses how non-Gaussian Lévy noise alters activated transitions between metastable states beyond classical Gaussian (Kramers) theory. It develops a unified framework with finite-intensity Lévy noise and a Martin-Siggia-Rose path-integral formulation, identifying a unique weak-noise scaling $\mu=\nu=1$ that preserves all cumulants. A key result is that non-Gaussian fluctuations reduce the effective barrier, yielding $S_{\min}<\tfrac{2\Delta V}{D+\lambda h^2}$ and exponential acceleration of escape rates, with transitions potentially occurring via discontinuous jumps rather than smooth diffusion. The findings distinguish thermal and athermal fluctuations in escape dynamics and suggest noise-engineering strategies to optimize switching in metastable systems across physics, chemistry, and biology.

Abstract

Lévy noise influences diverse non-equilibrium systems across scales, including quantum devices, active biological matter, and financial markets. While such noise is pervasive, its overall impact on activated transitions between metastable states remains unclear, despite prior studies of specific noise forms and scaling limits. In this work, we introduce a unified framework for Lévy noise defined by its finite intensity and independent stationary increments. By identifying the most probable transition paths as minimizers of a stochastic action functional, we derive analytical scaling laws for escape rates under weak noise, thereby extending the classical Arrhenius law. Our results demonstrate that Lévy noise universally enhances escape efficiency by reducing the effective potential barrier compared to Gaussian noise with equivalent intensity. Strikingly, even vanishingly weak Lévy noise can exponentially increase escape rates across a broad range of amplitude distributions. This phenomenon arises from discontinuous most probable transition paths, where escape occurs via finite jumps. We validate these paths through the cumulant-generating function, a path integral representation, the mean first passage time and numerical simulations. Our findings reveal fundamental distinctions in escape dynamics under thermal and athermal fluctuations, suggesting new strategies to optimize switching processes in metastable systems through engineering noise properties.

Figures (6)

• Figure 1: The bistable potential $V(u)=\frac{u^{4}}{4}-\frac{u^{2}}{2}$ possesses two symmetric metastable states at $u=\pm1$ and an energy barrier at $u=0$.
• Figure 2: (a) The minimum action $S_{\text{min}}$ over a parameter grid of $h$ and $\lambda$; (b) The minimum action $S_{\text{min}}$ as a function of $h$ for a fixed $\lambda=0.01$; (c) The minimum action $S_{\text{min}}$ as a function of $\lambda$ for a fixed $h=5$.
• Figure 3: (a) In this regime ($\mu=0.5$, $\nu=0.5$, $\epsilon= 0.01$), escape time dynamics are dominated by non-Gaussian noise. The diffusion coefficient $D_0=D\epsilon=0.01$, jump rate $\lambda_0=\lambda/\epsilon^{\mu}=10$ and jump size $h_0=h\epsilon^{\nu}=0.1$ are defined such that the variance $\lambda_0h_0^{2}=\epsilon^{0.5}=0.1$ exceeds the Gaussian variance ($\sim\epsilon$); (b) The regime ($\mu=0.8$, $\nu=0.3$) enhances non-Gaussian dominance, leading to faster and more predictable transitions because $\lambda_0h_0^{2}\approx2.5$. The escape time histogram shows a heavy-tailed distribution.
• Figure 4: (a) In the regime ($\mu=2$, $\nu=1$), the jump rate $\lambda_0=\lambda/\epsilon^{2}=10^{4}$ is extremely high, and the jump size $h_0=h\epsilon=0.01$ is small. The resulting variance $\lambda_0h_0^2=1$ dominates the Gaussian noise; (b) For $\mu=1.5$ and $\nu=1$, the jump rate remains $\lambda_0=\lambda/\epsilon^{1.5}=1000$ with small increments $h_0=h\epsilon=0.01$ ), yielding a variance of $\lambda_0h_0^2=0.1$. The escape time distribution shows a sharp peak near $t=0.15$, with Gaussian noise adding a slight dispersion visible as a narrow spread around it; (c) The regime ($\mu=2$, $\nu=1.2$) probes the transition near the critical line $\nu=(\mu+1)/2$. It features an extremely high jump rate $\lambda_0=\lambda/\epsilon^{2}=10^4$ and microscopic increments $h_0=h\epsilon^{1.2}\approx0.004$, with a theoretical variance of $\lambda_0h_0^{2}=0.16$. This produces a narrow peak near $t\approx0.36$.
• Figure 5: (a) A Gaussian limit is achieved with $\mu=2$ and $\nu=2$, where the extremely high jump rate $\lambda_0=\lambda/\epsilon^{2}=10^{4}$ and negligible jump size $h_0=h\epsilon^{2}=0.0001$ result in a variance $\lambda_0h_0^{2}=0.0001$ that is subdominant to Gaussian noise; (b) Non-Gaussian effects are most suppressed in the Gaussian dominance regime ($\mu=3$, $\nu=3$). The aggressive scaling produces an extreme frequency $\lambda_0=\lambda/\epsilon^{3}=10^{6}$ and microscopic increments $h_0=h\epsilon^{3}=10^{-6}$, leading to a variance $\lambda_0h_0^{2}=10^{-6}$ that is insignificant compared to the Gaussian component.