Majorization requires infinitely many second laws

Abstract

In statistical mechanics, the possibility of spontaneous transitions between the different states of an isolated system are usually characterized by the increase of entropy via the second law of thermodynamics. However, a stronger criterion that holds for systems obeying a master equation, the so-called principle of increasing mixing character, has also been proposed to describe these transitions. This principle relies on a more fundamental notion of disorder, which is described through the majorization preorder. Here, we show that, when replacing entropy by disorder, one requires a countably infinite family of entropy-like functions in order to obtain the corresponding second law of thermodynamics. Hence, regarding complexity, disorder goes well beyond the second law of thermodynamics. Furthermore, we prove an analogous result holds for the transitions of a system that is coupled to a heat bath