The minimum entropy production principle is an approximate characterization of certain near-equilibrium stationary states as minimizers of entropy production. It is not a universal minimization law far from equilibrium (Maes and Netočný 2007).
For a continuous-time Markov process on a finite state space, let 
be the transition rate from state 
to state 
and let 
be the state probabilities.
Assume that each permitted transition has a permitted reverse transition. In dimensionless
entropy units the entropy production rate is
 |
where the sum includes only permitted transitions. Each paired term is nonnegative because 
for 
.
A stationary distribution 
satisfying detailed balance,
defined by 
, has 
. For smooth perturbations of the rates and distributions
around an irreducible process satisfying detailed
balance, the stationary distribution
minimizes the leading quadratic term of entropy production. The full entropy production
need not be minimized outside this approximation (Maes and Netočný 2007).
In macroscopic formulations, the imposed constraints likewise matter (Polettini 2011).
