The second law of thermodynamics from symmetry and unitarity
The second law of thermodynamics states that for a thermally isolated system entropy never decreases. Most physical processes we observe in nature involve variations of macroscopic quantities over spatial and temporal scales much larger than microscopic molecular collision scales and thus can be considered as in local equilibrium. For a many body system in local equilibrium a stronger version of the second law applies which says that the entropy production at each spacetime point should be non negative. In this paper we provide a proof of the second law for such systems and a first derivation of the local second law. For this purpose we develop a general non equilibrium effective field theory of slow degrees of freedom from integrating out fast degrees of freedom in a quantum many body system and consider its classical limit. The key elements of the proof are the presence of a $Z_2$ symmetry, which can be considered as a proxy for local equilibrium and micro time reversibility, and a classical remnant of quantum unitarity. The $Z_2$ symmetry leads to a local current from a procedure analogous to that used in the Noether theorem. Unitarity leads to a definite sign of the divergence of the current. We also discuss the origin of an arrow of time, as well as the coincidence of causal and thermodynamical arrows of time. Applied to hydrodynamics, the proof gives a first principle derivation of the phenomenological entropy current condition and provides a constructive procedure for obtaining the entropy current.