We start from a recent theorem of microscopic thermodynamics, pictorially called thermodynamic uncertainty relation (TUR), which is part of the more general framework of thermodynamic inequalities. More specifically, the above-mentioned theorem states that the ratio between the average squared of an out-of-equilibrium current and its variance is bound from above by the entropy production. However, it was shown that this relation can be obtained in a very general way using the Ito-Dechant formalism and the Kullback-Leibler divergence that, for some particular cases, can be linked to entropy production. We will hence use the latter to obtain a new non-equilibrium inequality for systems modelled by continuous time Markov chains, i.e. performing a simple perturbation on the system's dynamic (namely a linear rescaling of the transition rates) we calculate the Kullback-Leibler divergence that arises from this procedure and show that it is proportional to the frenesy (or dynamical activity) of the system.
This final result has been called Kinetic Uncertainty Relation (KUR). Finally, we investigate the differences between the TUR and the KUR and analyse the regimes in which they give tighter constraints.
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