  It has been proposed that equilibrium thermodynamics is described on Legendre
submanifolds in contact geometry. It is shown in this paper that Legendre
submanifolds embedded in a contact manifold can be expressed as attractors in
phase space for a certain class of contact Hamiltonian vector fields. By giving
a physical interpretation that points outside the Legendre submanifold can
represent nonequilibrium states of thermodynamic variables, in addition to that
points of a given Legendre submanifold can represent equilibrium states of the
variables, this class of contact Hamiltonian vector fields is physically
interpreted as a class of relaxation processes, in which thermodynamic
variables achieve an equilibrium state from a nonequilibrium state through a
time evolution, a typical nonequilibrium phenomenon. Geometric properties of
such vector fields on contact manifolds are characterized after introducing a
metric tensor field on a contact manifold. It is also shown that a contact
manifold and a strictly convex function induce a lower dimensional dually flat
space used in information geometry where a geometrization of equilibrium
statistical mechanics is constructed. Legendre duality on contact manifolds is
explicitly stated throughout.
