The entropy production P for Markov chains of superior order r > 1 is introduced in connection with the corresponding entity arising in non equilibrium statistical physics. In case of circuit processes, asymptotics is studied for suitable sequences that convenge almost surely to P such that it is uniquely determined by limiting classes  , of weighted directed circuits. Finally , the consistency condition for the limiting weights of the classes  , o f being invariant under circuit-reversal , provides a reversibility criterion for superior order Markov chains in terms of the entropy production