Model simplification and loss of irreversibility

In this paper, we reveal a general relationship between model simplification and irreversibility based on the model of continuous-time Markov chains with time-scale separation. According to the topological structure of the fast process, we divide the states of the chain into the transient states and the recurrent states. We show that a two-time-scale chain can be simplified to a reduced chain in two different ways: removal of the transient states and aggregation of the recurrent states. Both the two operations will lead to a decrease in the entropy production rate and its adiabatic part and will keep its nonadiabatic part the same. This suggests that although model simplification can retain almost all the dynamic information of the chain, it will lose some thermodynamic information as a trade-off.

Figure 1

Simplification of two-time-scale Markov chains. (a) The canonical decomposition of the state space. According to the topological structure of the fast process, the state space $S$ can be decomposed into the union of multiple recurrent classes $A_1,...,A_m$ and the collection $B$ of all transient states. (b) The removal of the transient states of the fast process. (c) The aggregation of the recurrent states of the fast process.