Concentration inequalities, dynamical activity, and trade-off relations

We present a concentration inequality in stochastic thermodynamics that provides a lower bound for the probability distribution of observables in Markov processes, where the lower bound includes the dynamical activity, a central thermodynamic quantity in trade-off relations. The derived inequality is called the thermodynamic concentration inequality. As a first corollary of the thermodynamic concentration inequality, by combining it with the Markov inequality, we derive an upper bound on the expectation of an observable given its maximum. Furthermore, as a second corollary, we obtain a generalization of the thermodynamic uncertainty relation, where the first and second moments of the original relation are replaced by the p-norm. These corollaries are valid for an arbitrary Markov process with time-dependent transition rates and for an arbitrary initial probability distribution. The thermodynamic concentration inequality provides the foundation for discovering trade-off relations.