Large Deviations in Switching Diffusion: from Free Cumulants to Dynamical Transitions
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| Format: | Preprint |
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2025
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| _version_ | 1866909721297944576 |
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| author | Guéneau, Mathis Majumdar, Satya N. Schehr, Gregory |
| author_facet | Guéneau, Mathis Majumdar, Satya N. Schehr, Gregory |
| contents | 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 $10^{-2000}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_13754 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Large Deviations in Switching Diffusion: from Free Cumulants to Dynamical Transitions Guéneau, Mathis Majumdar, Satya N. Schehr, Gregory Statistical Mechanics Mathematical Physics Probability 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 $10^{-2000}$. |
| title | Large Deviations in Switching Diffusion: from Free Cumulants to Dynamical Transitions |
| topic | Statistical Mechanics Mathematical Physics Probability |
| url | https://arxiv.org/abs/2501.13754 |