Classical uncertainty relations and entropy production in non-equilibrium statistical mechanics
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917069810827264 |
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| author | Muratore-Ginanneschi, Paolo Peliti, Luca |
| author_facet | Muratore-Ginanneschi, Paolo Peliti, Luca |
| contents | We analyze Fürth's 1933 classical uncertainty relations in the modern language of stochastic differential equations. Our interest is motivated by applications to non-equilibrium classical statistical mechanics. We show that Fürth's uncertainty relations are a property enjoyed by martingales under the measure of a diffusion process. This result implies a lower bound on fluctuations in current velocities of entropic quantifiers of transitions in stochastic thermodynamics. In cases of particular interest, we recover an inequality well known in optimal mass transport relating the mean kinetic energy of the current velocity and the squared quadratic Wasserstein distance between the probability distributions of the entropy. In performing our analysis, we also avail us of an unpublished argument due to Krzysztof Gawȩdzki to derive a lower bound to the entropy production by transition described by Langevin-Kramers process in terms of the squared quadratic Wasserstein distance between the initial and final states of the transition. Finally, we illustrate how Fürth's relations admit a straightforward extension to piecewise deterministic processes. We thus show that the results in the paper concern properties enjoyed by general Markov processes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2302_08290 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Classical uncertainty relations and entropy production in non-equilibrium statistical mechanics Muratore-Ginanneschi, Paolo Peliti, Luca Statistical Mechanics Mathematical Physics We analyze Fürth's 1933 classical uncertainty relations in the modern language of stochastic differential equations. Our interest is motivated by applications to non-equilibrium classical statistical mechanics. We show that Fürth's uncertainty relations are a property enjoyed by martingales under the measure of a diffusion process. This result implies a lower bound on fluctuations in current velocities of entropic quantifiers of transitions in stochastic thermodynamics. In cases of particular interest, we recover an inequality well known in optimal mass transport relating the mean kinetic energy of the current velocity and the squared quadratic Wasserstein distance between the probability distributions of the entropy. In performing our analysis, we also avail us of an unpublished argument due to Krzysztof Gawȩdzki to derive a lower bound to the entropy production by transition described by Langevin-Kramers process in terms of the squared quadratic Wasserstein distance between the initial and final states of the transition. Finally, we illustrate how Fürth's relations admit a straightforward extension to piecewise deterministic processes. We thus show that the results in the paper concern properties enjoyed by general Markov processes. |
| title | Classical uncertainty relations and entropy production in non-equilibrium statistical mechanics |
| topic | Statistical Mechanics Mathematical Physics |
| url | https://arxiv.org/abs/2302.08290 |