Current-current relationship in non-equilibrium networks: linearity and applications

The paradigm of irreversible thermodynamics largely revolves around pairs of conjugated observables (current and force) and linear relations among them, close to equilibrium. This study takes a different path, by examining how currents respond to changes in other currents, without directly addressing driving forces. Using continuous-time Markov chains, we establish a linear-affine relationship between any two network currents, valid even far from equilibrium, and derive a formula for the current-to-current susceptibility via spanning-tree ensembles. Finally, we discuss how inherent linearity among network currents can be exploited in complex metabolic networks, in order to map the internal reaction currents of a cell, from the knowledge of input/output currents.