  Biological systems are typically highly open, non-equilibrium systems that
are very challenging to understand from a statistical mechanics perspective.
While statistical treatments of evolutionary biological systems have a long and
rich history, examination of the time-dependent non-equilibrium dynamics has
been less studied. In this paper we first derive a generalized master equation
in the genotype space for diploid organisms incorporating the processes of
selection, mutation, recombination, and reproduction. The master equation is
defined in terms of continuous time and can handle an arbitrary number of gene
loci and alleles, and can be defined in terms of an absolute population or
probabilities. We examine and analytically solve several prototypical cases
which illustrate the interplay of the various processes and discuss the
timescales of their evolution. The entropy production during the evolution
towards steady state is calculated and we find that it agrees with predictions
from non-equilibrium statistical mechanics where it is large when the
population distribution evolves towards a more viable genotype. The stability
of the non-equilibrium steady state is confirmed using the Glansdorff-Prigogine
criterion.
