Free Energy of Petri Nets

One can often extract valuable properties of Markov chains (MCs) by “compressing” them using various notions of bisimulation (exact or approximate, strong or weak, etc) [10,3,2,4]. Typically a bisimulation will lead from a concrete and perhaps overly detailed system to a simpler and more abstract one [7]. In this paper, we will go the opposite way! We will show that for the subclass of continuous-time MCs (ctMCs) corresponding to thermodynamically consistent stochastic mass action Petri nets (the standard model for chemical reactions for fast diffusing chemicals), one can construct in a systematic fashion concrete versions of the dynamics. These concrete MCs are functionally bisimilar to their abstract counterpart and admit a simpler description of their invariant probability (equivalently, of their energy function). This can sometimes reveal interesting equilibrium properties of the original chain.

To see how the benefits of the construction come about, and fix a few notations which we will re-use in the main development, we start with a simple class of ‘urn models’. These are traditionally introduced as discrete models of diffusion.