  We present a formalism to describe slowly decaying systems in the context of
finite Markov chains obeying detailed balance. We show that phase space can be
partitioned into approximately decoupled regions, in which one may introduce
restricted Markov chains which are close to the original process but do not
leave these regions. Within this context, we identify the conditions under
which the decaying system can be considered to be in a metastable state.
Furthermore, we show that such metastable states can be described in
thermodynamic terms and define their free energy. This is accomplished showing
that the probability distribution describing the metastable state is indeed
proportional to the equilibrium distribution, as is commonly assumed. We test
the formalism numerically in the case of the two-dimensional kinetic Ising
model, using the Wang--Landau algorithm to show this proportionality
explicitly, and confirm that the proportionality constant is as derived in the
theory. Finally, we extend the formalism to situations in which a system can
have several metastable states.
