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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.07571 |
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| _version_ | 1866912638071472128 |
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| author | Baranau, Vasili |
| author_facet | Baranau, Vasili |
| contents | What is the fastest possible "diffusion"? A trivial answer would be "a process that converts a Dirac delta-function into a uniform distribution infinitely fast". Below, we consider a more reasonable formulation: a process that maximizes differential entropy of a probability density function (pdf) $f(\vec{x}, t)$ at every time $t$, under certain restrictions. Specifically, we focus on a case when the rate of the Kullback-Leibler divergence $D_{\text{KL}}$ is fixed. If $Δ(\vec{x}, t, d{t}) = \frac{\partial f}{ \partial t} d{t}$ is the pdf change at a time step $d{t}$, we maximize the differential entropy $H[f + Δ]$ under the restriction $D_{\text{KL}}(f + Δ|| f) = A^2 d{t}^2$, $A = \text{const} > 0$. It leads to the following equation: $\frac{\partial f}{ \partial t} = - κf (\ln{f} - \int f \ln{f} d{\vec{x}})$, with $κ= \frac{A}{\sqrt{ \int f \ln^2{f} d{\vec{x}} - \left( \int f \ln{f} d{\vec{x}} \right)^2 } }$. Notably, this is a non-local equation, so the process is different from the Itô diffusion and a corresponding Fokker-Planck equation. We show that the normal and exponential distributions are solutions to this equation, on $(-\infty; \infty)$ and $[0; \infty)$, respectively, both with $\text{variance} \sim e^{2 A t}$, i.e. diffusion is highly anomalous. We numerically demonstrate for sigmoid-like functions on a segment that the entropy change rate $\frac{d H}{d t}$ produced by such an optimal "diffusion" is, as expected, higher than produced by the "classical" diffusion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07571 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | What is the most optimal diffusion? Baranau, Vasili Statistical Mechanics What is the fastest possible "diffusion"? A trivial answer would be "a process that converts a Dirac delta-function into a uniform distribution infinitely fast". Below, we consider a more reasonable formulation: a process that maximizes differential entropy of a probability density function (pdf) $f(\vec{x}, t)$ at every time $t$, under certain restrictions. Specifically, we focus on a case when the rate of the Kullback-Leibler divergence $D_{\text{KL}}$ is fixed. If $Δ(\vec{x}, t, d{t}) = \frac{\partial f}{ \partial t} d{t}$ is the pdf change at a time step $d{t}$, we maximize the differential entropy $H[f + Δ]$ under the restriction $D_{\text{KL}}(f + Δ|| f) = A^2 d{t}^2$, $A = \text{const} > 0$. It leads to the following equation: $\frac{\partial f}{ \partial t} = - κf (\ln{f} - \int f \ln{f} d{\vec{x}})$, with $κ= \frac{A}{\sqrt{ \int f \ln^2{f} d{\vec{x}} - \left( \int f \ln{f} d{\vec{x}} \right)^2 } }$. Notably, this is a non-local equation, so the process is different from the Itô diffusion and a corresponding Fokker-Planck equation. We show that the normal and exponential distributions are solutions to this equation, on $(-\infty; \infty)$ and $[0; \infty)$, respectively, both with $\text{variance} \sim e^{2 A t}$, i.e. diffusion is highly anomalous. We numerically demonstrate for sigmoid-like functions on a segment that the entropy change rate $\frac{d H}{d t}$ produced by such an optimal "diffusion" is, as expected, higher than produced by the "classical" diffusion. |
| title | What is the most optimal diffusion? |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2510.07571 |