Trees and forests for nonequilibrium purposes: an introduction to graphical representations

Abstract

Using local detailed balance we rewrite the Kirchhoff formula for stationary distribution of Markov jump processes in terms of a physically interpretable tree-ensemble. We use that arborification of path-space integration to derive a McLennan-tree characterization close to equilibrium, as well as to obtain response formula for the stationary distribution in the asymptotic regime of large driving. Graphical expressions of currents and of traffic are obtained, allowing the study of various asymptotic regimes. Finally, we present how the matrix-forest theorem gives a representation of quasi-potentials, as used e.g. for computing excess work and heat in nonequilibrium thermal physics. A variety of examples illustrate and explain the graph elements and constructions.

arXiv comment28 pages,19 figures

References in corpus (21)

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• Frenesy: time-symmetric dynamical activity in nonequilibria
• Local detailed balance
• Steady State Thermodynamics for Heat Conduction -- Microscopic Derivation
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• Heat bounds and the blowtorch theorem
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• Low temperature behavior of nonequilibrium multilevel systems
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Cited by in corpus (14)

• Thermodynamic Bounds on Symmetry Breaking in Linear and Catalytic Biochemical Systems
• A Nernst heat theorem for nonequilibrium jump processes
• Topologically-constrained fluctuations and thermodynamics regulate nonequilibrium response
• Exact computation of heat capacities for active particles on a graph
• The vanishing of excess heat for nonequilibrium processes reaching zero ambient temperature
• On the Poisson equation for nonreversible Markov jump processes
• Affine relationships between steady currents
• Frenetic steering in a nonequilibrium graph
• Mutual Multilinearity of Nonequilibrium Network Currents
• Drazin-Inverse and heat capacity for driven random walks on the ring
• Close-to-equilibrium heat capacity
• The generic temperature response of large biochemical networks
• Markov generators as non-hermitian supersymmetric quantum Hamiltonians: spectral properties via bi-orthogonal basis and Singular Value Decompositions
• Diagrammatic expressions for steady-state distribution and static responses in population dynamics