WIAS Preprint No. 2826, (2021)

Variational structures beyond gradient flows: A macroscopic fluctuation-theory perspective


  • Patterson, Robert I. A.
  • Renger, D. R. Michiel
    ORCID: 0000-0003-3557-3485
  • Sharma, Upanshu

2020 Mathematics Subject Classification

  • 82C35 49S05 35Q8249J40 60F10 82C22 35Q84

2010 Physics and Astronomy Classification Scheme

  • 05.70.Ln, 05.40.-a


  • Non-equilibrium thermodynamics, dissipative and non-dissipative, decomposition of large-deviation rate functional




Macroscopic equations arising out of stochastic particle systems in detailed balance (called dissipative systems or gradient flows) have a natural variational structure, which can be derived from the large-deviation rate functional for the density of the particle system. While large deviations can be studied in considerable generality, these variational structures are often restricted to systems in detailed balance. Using insights from macroscopic fluctuation theory, in this work we aim to generalise this variational connection beyond dissipative systems by augmenting densities with fluxes, which encode non-dissipative effects. Our main contribution is an abstract framework, which for a given flux-density cost and a quasipotential, provides a decomposition into dissipative and non-dissipative components and a generalised orthogonality relation between them. We then apply this abstract theory to various stochastic particle systems -- independent copies of jump processes, zero-range processes, chemical-reaction networks in complex balance and lattice-gas models.

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