  This is the fourth paper, the last one, on solution to the problem of absence
of detailed balance in nonequilibrium processes. It is an approach based on
another known universal dynamics: The evolutionary dynamics first conceived by
Darwin and Wallace, referring to as Darwinian dynamics in the present paper,
has been found to be universally valid in biology; The statistical mechanics
and thermodynamics, while enormously successful in physics, have been in an
awkward situation of wanting a consistent dynamical understanding; Here we
present from a formal point of view an exploration of the connection between
thermodynamics and Darwinian dynamics and a few related topics. We first show
that the stochasticity in Darwinian dynamics implies the existence temperature,
hence the canonical distribution of Boltzmann-Gibbs type. In term of relative
entropy the Second Law of thermodynamics is dynamically demonstrated without
detailed balance condition, and is valid regardless of size of the system. In
particular, the dynamical component responsible for breaking detailed balance
condition does not contribute to the change of the relative entropy. Two types
of stochastic dynamical equalities of current interest are explicitly discussed
in the present approach: One is based on Feynman-Kac formula and another is a
generalization of Einstein relation. Both are directly accessible to
experimental tests. Our demonstration indicates that Darwinian dynamics
represents logically a simple and straightforward starting point for
statistical mechanics and thermodynamics and is complementary to and consistent
with conservative dynamics that dominates the physical sciences. Present
exploration suggests the existence of a unified stochastic dynamical framework
both near and far from equilibrium.

Other Condensed Matter;Statistical Mechanics;Dynamical Systems;Adaptation and Self-Organizing Systems;Populations and Evolution