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

We analyze Fürth’s 1933 classical uncertainty relations in the modern language of stochastic differential equations. Our interest is motivated by their application to non-equilibrium classical statistical mechanics. We show that Fürth’s uncertainty relations are a property inherent in martingales within the framework of a diffusion process. This result implies a lower bound on the fluctuations in current velocities of entropic quantifiers associated with transitions in stochastic thermodynamics. In cases of particular interest, we recover a well-known inequality for 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. We take advantage in particular of an unpublished suggestion by Krzysztof Gawȩdzki to derive a lower bound to the entropy production by a transition described by a Langevin–Kramers process in terms of the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 show that the results presented in this paper pertain to the characteristics exhibited by general Markov processes.