  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.
