Affine relationships between steady currents

TL;DR

This work addresses how steady currents in finite Markov jump processes respond to changes in transition rates, revealing a nonperturbative, graph‑theoretic link between currents on different edges. The authors derive a current‑current susceptibility via mean first‑passage time differences, enabling an affine relation that generalizes to multiple currents on a multigraph. The contributions include a shorter derivation, interpretation in terms of MFPTs and quasipotentials, and a Kirchhoff‑based graphical representation, with potential applications to traffic regulation and biological networks.

Abstract

Perturbing transition rates in a steady nonequilibrium system, e.g. modelled by a Markov jump process, causes a change in the local currents. Their susceptibility is usually expressed via Green-Kubo relations or their nonequilibrium extensions. However, we may also wish to directly express the mutual relation between currents. Such a nonperturbative interrelation was discovered by P.E. Harunari et al. in [1] by applying algebraic graph theory showing the mutual linearity of currents over different edges in a graph. We give a novel and shorter derivation of that current relationship where we express the current-current susceptibility as a difference in mean first-passage times. It allows an extension to multiple currents, which remains affine but the relation is not additive.

Figures (3)

• Figure 1: The system is modeled as a random walker on a multigraph (a), where each connection $x\leftrightarrow y$ may have different channels (b).
• Figure 2: The graph of Example \ref{['exthreeloops']}. The example illustrates the relationship between the current over the edge $(4,6)$ with currents over the edges $(1,2)$ and $(8,9)$ and its nonadditivity.
• Figure 3: Two loops connected via one edge.

Theorems & Definitions (2)

• Example 5.1
• Example A.1