Il ruolo della varianza in sistemi stocastici classici

We characterise stochastic systems by studying their variances. In particular, we tackle this topic from two points of view, by elaborating on the subject of stochastic uncertainty relations and by discussing a novel result that we term the variance sum rule for Langevin systems. Stochastic uncertainty relations are inequalities that usually involve a signal to noise ratio g, which can be regarded as a measure of the precision associated to the observable O, and a cost function C such that g ≤ C. The thermodynamic uncertainty relation is one of the first examples of these stochastic inequalities and considers the average total entropy production in the cost function, thus refining the second law of thermodynamics. By means of an information-theoretic approach, we provide a new uncertainty relation for a system modelled by a linear generalised Langevin equation along with a novel kinetic uncertainty relation, where the upper bound to the precision is given by the mean dynamical activity, which quantifies the degree of agitation of a discrete system. We also show how the latter is often the main limiting factor for the precision in far-from-equilibrium conditions. In the second part of the thesis we introduce some variance sum rules, which can be used to infer relevant dynamical parameters also for regimes far from equilibrium. We test our method on experimental data whose model parameters are known a priori, finding very good agreement between the results of the estimation procedure and the true values of the parameters. A specific sum rule for non-Markovian systems also shows good performances in estimating the memory kernel of complex fluids. Moreover, the same approach yields a solid formula for estimating the amount of entropy production. All of this shows that the indetermination of stochastic motion is a resource that we should continue to understand and exploit for measuring physical quantities.