  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.
