Thermodynamic uncertainty relation

The starting point of my thesis is a recent result of microscopic thermodynamics obtained with techniques related to linear response theory. The theorem in question, pictorially called thermodynamic uncertainty principle, forms part of the more general framework of thermodynamic inequalities, which arise from the handling of microscopic processes in a non-equilibrium state. More specifically, the above-mentioned theorem puts a bound on the ratio between the average rate of the an out of equilibrium current and the generalized diffusivity (proportional to the variance of the current) times the entropy production rate. However, it was showed that this thermodynamic relation can be obtained in a more general way, namely using a mathematical object called the Kullback-Leibler divergence that in some particular cases can be linked to entropy production. Hence, in this thesis, we use the latter to obtain general non-equilibrium inequalities for systems modelled by continuous time Markov chains and for some specific Langevin systems, not necessarily aiming to recover the thermodynamic uncertainty relation. In fact, performing some simpler perturbations on the system's dynamic and using linear response theory we calculate the Kullback-Leibler divergences that arise from this procedure and we will relate the obtained results to other relevant observables of the system which will not necessarily be entropy production . This will be done in particular for general jump processes where we perform a linear perturbation of the transition rates obtaining a relation between the variance of a given observable, the time derivative of its mean and the activity of the system, namely the average total number of jumps that the system performed up to time T. We will also discuss general Brownian motion with memory effects and time dependent external force, the results we obtain involve the variance of a given observable, its susceptibility to the performed perturbation and of course the Kullback-Leibler divergence that we believe, in this particular situation, to be linked to the dissipation of the system as the perturbations we use involves the friction kernel of the Langevin equation used to model the system in question. Moreover, using position as observable in the obtained inequalities and plotting the saturation ratio for these we will get interesting informations about the dominant components (deterministic or random) of the dynamics at different times.