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arXiv · 2501.13754

Large Deviations in Switching Diffusion: from Free Cumulants to Dynamical Transitions

Mathis Guéneau·Satya N. Majumdar·Gregory Schehr
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Abstract

We study the diffusion of a particle with a time-dependent diffusion constant $D(t)$ that switches between random values drawn from a distribution $W(D)$ at a fixed rate $r$. Using a renewal approach, we compute exactly the moments of the position of the particle $\langle x^{2n}(t) \rangle$ at any finite time $t$, and for any $W(D)$ with finite moments $\langle D^n \rangle$. For $t \gg 1$, we demonstrate that the cumulants $\langle x^{2n}(t) \rangle_c$ grow linearly with $t$ and are proportional to the free cumulants of a random variable distributed according to $W(D)$. For specific forms of $W(D)$, we compute the large deviations of the position of the particle, uncovering rich behaviors and dynamical transitions of the rate function $I(y=x/t)$. Our analytical predictions are validated numerically with high precision, achieving accuracy up to $10^{-2000}$.

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BibTeXRIS

Mathis Guéneau, Satya N. Majumdar, Gregory Schehr. 2025-01-23. Large Deviations in Switching Diffusion: from Free Cumulants to Dynamical Transitions. https://doi.org/10.1103/rqsn-bzr6

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