  Boltzmann's H-function H(t) holds a venerable place in the history of
science. However, it seems never to have been evaluated for particles with
orientational degrees of freedom. We generalize Boltzmann's H-function to a gas
of molecules that can both rotate and translate and on collision exchange both
momentum and angular momentum, obeying conservation laws. We evaluate the time
(t) evolution of single-particle joint probability distribution f (p, L, t) for
linear (p) and angular (L) momenta from an initial nonequilibrium state by
molecular dynamics simulations. We consider both prolate and oblate-shaped
particles, interacting by well-known Gay-Berne potential that depends both on
position and orientation vectors. We calculate the relaxation of the
generalized molecular H(t) from several initial (t=0) nonequilibrium states. In
the long-time limit, the H function saturates to its exact equilibrium value,
which is the sum of translational and rotational contributions to the
respective entropies. Both the translational and rotational components of H(t)
decay nearly exponentially with time; the rotational component is more
sensitive to the molecular shape that enters through the aspect ratio. A
remarkable rapid decrease in the rotational relaxation time is observed as the
spherical limit is approached, in a way tantalizingly reminiscent of Hu-Zwanzig
hydrodynamic prediction with slip boundary condition. Additionally, we obtain
H(t) analytically by solving the appropriate translational and rotational
Fokker-Planck equation and obtain a modest agreement with simulations. We
observe a remarkable signature of translation-rotation coupling as a function
of molecular shape, captured through a physically meaningful differential term
that quantifies the magnitude of translation-rotation coupling.
