  We study the time evolution of eleven microscopic entropy definitions (of
Boltzmann-surface, Gibbs-volume, canonical, coarse-grained-observational,
entanglement and diagonal type) and three microscopic temperature definitions
(based on Boltzmann, Gibbs or canonical entropy). This is done for the
archetypal nonequilibrium setup of two systems exchanging energy, modeled here
with random matrix theory, based on numerical integration of the Schroedinger
equation. We consider three types of pure initial states (local energy
eigenstates, decorrelated and entangled microcanonical states) and three
classes of systems: (A) two normal systems, (B) a normal and a negative
temperature system and (C) a normal and a negative heat capacity system.
  We find: (1) All types of initial states give rise to the same macroscopic
dynamics. (2) Entanglement and diagonal entropy sensitively depend on the
microstate, in contrast to all other entropies. (3) For class B and C,
Gibbs-volume entropies can violate the second law and the associated
temperature becomes meaningless. (4) For class C, Boltzmann-surface entropies
can violate the second law and the associated temperature becomes meaningless.
(5) Canonical entropy has a tendency to remain almost constant. (6) For a Haar
random initial state, entanglement or diagonal entropy behave similar or
identical to coarse-grained-observational entropy.
