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Classical uncertainty relations and entropy production in non-equilibrium statistical mechanics

Paolo Muratore-Ginanneschi, Luca Peliti

TL;DR

The paper recasts Fürth's classical uncertainty relations in the language of stochastic differential equations to address non-equilibrium classical statistical mechanics. It derives a family of bounds that couple the variance of a martingale observable to the variance of its time-symmetric derivative via the carré du champ, and extends these results to adapted martingales and piecewise deterministic processes. Crucially, it connects these dynamical uncertainties to optimal transport through the squared Wasserstein distance, providing lower bounds on entropy production and entropy-flow fluctuations in Langevin–Kramers and related dynamics. The findings unify stochastic-analytic and transport-geometric perspectives, yielding universal bounds for broad classes of Markov processes with applications to stochastic thermodynamics and non-equilibrium statistical mechanics. The approach leverages Girsanov's theorem, Benamou–Brenier inequalities, and transport distances to illuminate how dissipation and irreversibility are quantified by current-velocity fluctuations.

Abstract

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 In performing our analysis, we also avail us of an unpublished argument due to Krzysztof Gawȩdzki to derive a lower bound to the entropy production by transition described by Langevin-Kramers process in terms of the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.

Classical uncertainty relations and entropy production in non-equilibrium statistical mechanics

TL;DR

The paper recasts Fürth's classical uncertainty relations in the language of stochastic differential equations to address non-equilibrium classical statistical mechanics. It derives a family of bounds that couple the variance of a martingale observable to the variance of its time-symmetric derivative via the carré du champ, and extends these results to adapted martingales and piecewise deterministic processes. Crucially, it connects these dynamical uncertainties to optimal transport through the squared Wasserstein distance, providing lower bounds on entropy production and entropy-flow fluctuations in Langevin–Kramers and related dynamics. The findings unify stochastic-analytic and transport-geometric perspectives, yielding universal bounds for broad classes of Markov processes with applications to stochastic thermodynamics and non-equilibrium statistical mechanics. The approach leverages Girsanov's theorem, Benamou–Brenier inequalities, and transport distances to illuminate how dissipation and irreversibility are quantified by current-velocity fluctuations.

Abstract

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.
Paper Structure (19 sections, 1 theorem, 125 equations)

This paper contains 19 sections, 1 theorem, 125 equations.

Table of Contents

  1. Introduction
  2. Osmotic derivative and current velocity
  3. Uncertainty relation with respect to the osmotic derivative
  4. Fürth's 1933 example
  5. Relation with multivariate inequalities
  6. Fürth's inequality for adapted martingales
  7. Applications to statistical mechanics
  8. Relation with the squared quadratic Wasserstein distance between the initial and the final distribution of the martingale
  9. Lower bound on fluctuations of the relative entropy flow
  10. Lower bound on fluctuations of the entropy production flow
  11. Relation with the squared quadratic Wasserstein distance between the initial and final probability distributions of a state in stochastic thermodynamics
  12. Uncertainty relation for piecewise deterministic processes
  13. Current and osmotic derivatives of a test function
  14. Fürth's uncertainty relation
  15. Conclusions
  16. ...and 4 more sections

Key Result

Proposition 1

For any test function $f\colon\mathbb{R}^{d}\,\times\,\mathbb{R}_{+}\mapsto\mathbb{R}$ one has

Theorems & Definitions (2)

  • Proposition
  • proof