Entropy Production of Nonequilibrium Steady States with Irreversible   Transitions
  In nature stationary nonequilibrium systems cannot exist on their own, rather they need to be driven from outside in order to keep them away from equilibrium. While the internal mean entropy of such stationary systems is constant, the external drive will on average increase the entropy in the environment. This external entropy production is usually quantified by a simple formula, stating that each microscopic transition of the system between two configurations $c \to c'$ with rate $w_{c\to c'}$ changes the entropy in the environment by $\Delta S_{\rm env} = {\ln w_{c \to c'}}-{\ln w_{c' \to c}}$. According to this formula irreversible transitions $c \to c'$ with a vanishing backward rate $w_{c'\to c}=0$ would produce an infinite amount of entropy. However, in experiments designed to mimic such processes, a divergent entropy production, that would cause an infinite increase of heat in the environment, is not seen. The reason is that in an experimental realization the backward process can be suppressed but its rate always remains slightly positive, resulting in a finite entropy production. The paper discusses how this entropy production can be estimated and specifies a lower bound depending on the observation time. 
Somayeh Zeraati, Farhad H. Jafarpour, and Haye Hinrichsen
cond-mat.stat-mech