Building on Kirchhoff's treatment of electrical circuits, Hill and Schnakenberg – among others – proposed a celebrated theory for the thermodynamics of Markov processes and linear biochemical networks that exploited tools from graph theory to build fundamental nonequilibrium observables. However, such simple geometrical interpretation does not carry through for arbitrary chemical reaction networks because reactions can be many-to-many and are thus represented by a hypergraph, rather than a graph. We propose a generalization of the geometric intuitions behind the Hill–Schnakenberg approach to arbitrary reaction networks. In particular, we give simple procedures to build bases of cycles (encoding stationary nonequilibrium behavior) and cocycles (encoding relaxation), that we interpret in terms of circulations and gradients. Such tools allow one to properly project nonequilibrium observables onto the relevant subspaces. We develop the theory for non-equilibrium reaction networks endowed with mass-action kinetics and enrich the description with insights from the corresponding stochastic models.
Joint work with Sara Dal Cengio (LIPhy, Grenoble) and Matteo Polettini (University of Luxembourg).
Reference: arXiv:2208.01290 [cond-mat.stat-mech]