Mutual Multilinearity of Nonequilibrium Network Currents

TL;DR

This paper extends the linearity relations of network currents in Markov chains to multiple controlled edges, providing explicit susceptibility formulas and linking them to fluctuations and Kirchhoff's law.

Contribution

It introduces a generalized mutual multilinearity of currents controlled at multiple edges and offers two proofs and explicit susceptibility expressions.

Findings

Currents exhibit linear-affine relations when multiple edges are controlled.

Explicit formulas for current susceptibilities are derived.

Kirchhoff's law emerges as a special case of the linearity relations.

Abstract

Continuous-time Markov chains have been successful in modelling systems across numerous fields, with currents being fundamental entities that describe the flows of energy, particles, individuals, chemical species, information, or other quantities. They apply to systems described by agents transitioning between vertices along the edges of a network (at some rate in each direction). It has recently been shown by the authors that, at stationarity, a hidden linearity exists between currents that flow along edges: if one controls the current of a specific "input" edge (by tuning transition rates along it), any other current is a linear-affine function of the input current [PRL 133, 047401 (2024)]. In this paper, we extend this result to the situation where one controls the currents of several edges, and prove that other currents are in linear-affine relation with the input ones. Two proofs with distinct insights are provided: the first relies on Kirchhoff's current law and reduces the input set inductively through graph analysis, while the second utilizes the resolvent approach via a Laplace transform in time. We obtain explicit expressions for the current-to-current susceptibilities, which allow one to map current dependencies through the network. We also verify from our expression that Kirchhoff's current law is recovered as a limiting case of our mutual linearity. Last, we uncover that susceptibilities can be obtained from fluctuations when the reference system is originally at equilibrium.