Mutual Multilinearity of Nonequilibrium Network Currents

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.

24 pages, 4 figures

References in corpus (36)

• Entropy production along a stochastic trajectory and an integral fluctuation theorem
• Thermodynamic uncertainty relation for biomolecular processes
• Fluctuation-Dissipation: Response Theory in Statistical Physics
• Dissipation bounds all steady-state current fluctuations
• Three detailed fluctuation theorems
• Universal bounds on current fluctuations
• Fluctuation-Dissipation Theorem in Nonequilibrium Steady States
• Fluctuation theorem for currents and Schnakenberg network theory
• Irreversible thermodynamics of open chemical networks I: Emergent cycles and broken conservation laws
• Learning without neurons in physical systems
• Effective thermodynamics for a marginal observer
• Forest matrices around the Laplacian matrix
• The limits of multifunctionality in tunable networks
• Fluctuation-dissipation relations far from equilibrium
• Stochastic Thermodynamics of Non-Linear Electronic Circuits: A Realistic Framework for Computing around kT
• Thermodynamic Bound on the Asymmetry of Cross-Correlations
• Universal thermodynamic bounds on nonequilibrium response with biochemical applications
• Effective fluctuation and response theory
• Topologically-constrained fluctuations and thermodynamics regulate nonequilibrium response
• Fundamental bounds on precision and response for quantum trajectory observables
• Nonequilibrium Fluctuation-Response Relations: From Identities to Bounds
• Beyond Thermodynamic Uncertainty Relations: nonlinear response, error-dissipation trade-offs, and speed limits
• Shortcuts in stochastic systems and control of biophysical processes
• Methods and Conversations in (Post)Modern Thermodynamics
• Dissipation bounds precision of current response to kinetic perturbations
• Trees and forests for nonequilibrium purposes: an introduction to graphical representations
• Trade-offs between number fluctuations and response in nonequilibrium chemical reaction networks
• Dynamical activity universally bounds precision of response in Markovian nonequilibrium systems
• Universal Response Inequalities Beyond Steady States via Trajectory Information Geometry
• Affine relationships between steady currents
• Nonequilibrium fluctuation-response relations for state observables
• Markov generators as non-hermitian supersymmetric quantum Hamiltonians: spectral properties via bi-orthogonal basis and Singular Value Decompositions
• Fluctuation-response inequalities for kinetic and entropic perturbations
• Nonlinear Response Identities and Bounds for Nonequilibrium Steady States
• Coplanarity of rooted spanning-tree vectors
• Microscopic imprints of learned solutions in adaptive resistor networks

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• Nonequilibrium fluctuation-response relations for state-current correlations
• Nonequilibrium fluctuation-response relations for state observables
• Macroscopic fluctuation-response theory and its use for gene regulatory networks
• Mutual Linearity is a Generic Property of Steady-State Markov Networks
• Thermodynamic Geometric Constraint on the Spectrum of Markov Rate Matrices