Nonequilibrium Force-Flow Relations in Networks

Abstract: In a companion paper [arXiv:2410.09277], we outlined a general theory for nonequilibrium (NEQ) network statistical physics, $\textit{Caliber Force Theory}$, (CFT). It resembles the "First and Second Laws" basis of Equilibrium (EQ) Thermodynamics, except that dynamics requires maximization of $\textit{path entropy}$ ($\mathfrak{S}\text{path}$), not $\textit{state entropy}$. Here, we elaborate the mathematics of the forces $(\text{d} \mathfrak{S}\text{path}/\text{d}{\textbf{x}})$, the fluctuation-respond relations, and the Maxwell-Onsager-like symmetry relations. And we give applications beyond thermal baths -- such as networks in ecology, biochemistry, vehicle traffic, and other force-flow networks. Beyond past work on inequalities, it gives equalities; and it gives forces, which allows for studies of optimization, conflicts, congestion and network design; and it gives a third Kirchhoff-like relation that applies to networks that have fluctuating flows.