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

We analyze F\"urth'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 F\"urth'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 In performing our analysis, we also avail us of an unpublished argument due to Krzysztof Gaw\c{e}dzki to derive a lower bound to the entropy production by transition described by Langevin-Kramers process in terms of the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 F\"urth'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.