Robustness of the Second Law of Thermodynamics under Generalizations of the Maximum Entropy Method

It is shown that the laws of thermodynamics are extremely robust under generalizations of the form of entropy. Using the Bregman-type relative entropy, the Clausius inequality is proved to be always valid. This implies that thermodynamics is highly universal and does not rule out consistent generalization of the maximum entropy method. There is a consensus of a certain kind that among the physical laws those of thermodynamics are extremely universal and will persist. This optimism might have its origin in the fact that the thermodynamic principles have remained unchanged when the shift was made from classical statistical mechanics to quantum. This is indeed astonishing if one takes into account that the formulation of thermodynamics was completed in the 19th century, before the establishment of atomism in physics. Here, by the term “statistical mechanics”, we refer to ordinary equilibrium statistical mechanics that describes normal systems, in which ergodicity is supposed to be well realized (though it still seems vague in quantum theory).