Additional energy-information relations in thermodynamics of small systems

The Clausius inequality (CI) form of the second law of thermodynamics relates information changes (entropy) to changes in the first moment of the energy (heat and indirectly also work). Are there similar relations between other moments of the energy distribution, and other information measures, or is the Clausius inequality a one of a kind instance of the energy-information paradigm? If there are additional relations, can they be used to make predictions on measurable quantities? Changes in the energy distribution beyond the first moment (average heat or work) are especially important in small systems which are often very far from thermal equilibrium. The generalized Clausius inequalities (GCI's), here derived, provide positive answers to the two questions above and add another layer to the fundamental connection between energy and information. To illustrate the utility of the new GCI's, we find scenarios where the GCI's yield tighter constraints on performance (e.g. in thermal machines) compared to the second law. To obtain the GCI's we use the Bregman divergence - a mathematical tool found to be highly suitable for energy-information studies. The quantum version of the GCI's provides a thermodynamic meaning to various quantum coherence measures. It is intriguing to fully map the regime of validity of the GCI's and extend the present results to more general scenarios including continuous systems and particles exchange with the baths.