BEST statistics of Markovian fluxes: a tale of Eulerian tours and Fermionic ghosts

Abstract

We provide an exact expression for the statistics of the fluxes of Markov jump processes at all times, improving on asymptotic results from large deviation theory. The main ingredient is a generalization of the BEST theorem in enumeratoric graph theory to Eulerian tours with open ends. In the long-time limit we reobtain Sanov's theorem for Markov processes, which expresses the exponential suppression of fluctuations in terms of relative entropy. The finite-time power-law term, increasingly important with the system size, is a spanning-tree determinant that, by introducing Grassmann variables, can be absorbed into the effective Lagrangian of a Fermionic ghost field on a metric space, coupled to a gauge potential. With reference to concepts in nonequilibrium stochastic thermodynamics, the metric is related to the dynamical activity that measures net communication between states, and the connection is made to a previous gauge theory for diffusion processes.

22 pages

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Cited by in corpus (8)

• Tightening the uncertainty principle for stochastic currents
• Path integral methods for the dynamics of stochastic and disordered systems
• Effective fluctuation and response theory
• Nonequilibrium thermodynamic potentials for continuous-time Markov chains
• Graph-combinatorial approach for large deviations of Markov chains
• Delocalization-localization dynamical phase transition of random walks on graphs
• Large deviations at level 2.5 and for trajectories observables of diffusion processes : the missing parts with respect to their random-walks counterparts
• A supersymmetric quantum perspective on the explicit large deviations for reversible Markov jump processes, with applications to pure and random spin chains