Stochastic Hydrodynamics of Complex Fluids: Discretisation and Entropy Production

Michael E. Cates, Étienne Fodor, Tomer Markovich*, Cesare Nardini, Elsen Tjhung

*Corresponding author for this work

Research output: Contribution to journalReview articlepeer-review


Many complex fluids can be described by continuum hydrodynamic field equations, to which noise must be added in order to capture thermal fluctuations. In almost all cases, the resulting coarse-grained stochastic partial differential equations carry a short-scale cutoff, which is also reflected in numerical discretisation schemes. We draw together our recent findings concerning the construction of such schemes and the interpretation of their continuum limits, focusing, for simplicity, on models with a purely diffusive scalar field, such as ‘Model B’ which describes phase separation in binary fluid mixtures. We address the requirement that the steady-state entropy production rate (EPR) must vanish for any stochastic hydrodynamic model in a thermal equilibrium. Only if this is achieved can the given discretisation scheme be relied upon to correctly calculate the nonvanishing EPR for ‘active field theories’ in which new terms are deliberately added to the fluctuating hydrodynamic equations that break detailed balance. To compute the correct probabilities of forward and time-reversed paths (whose ratio determines the EPR), we must make a careful treatment of so-called ‘spurious drift’ and other closely related terms that depend on the discretisation scheme. We show that such subtleties can arise not only in the temporal discretisation (as is well documented for stochastic ODEs with multiplicative noise) but also from spatial discretisation, even when noise is additive, as most active field theories assume. We then review how such noise can become multiplicative via off-diagonal couplings to additional fields that thermodynamically encode the underlying chemical processes responsible for activity. In this case, the spurious drift terms need careful accounting, not just to evaluate correctly the EPR but also to numerically implement the Langevin dynamics itself.

Original languageEnglish
Article number254
Issue number2
StatePublished - Feb 2022
Externally publishedYes


FundersFunder number
National Science Foundation Center for Theoretical Biological PhysicsPHY-2019745
Horizon 2020 Framework Programme885146, 740269
Royal Society
European Research Council
Fonds National de la Recherche Luxembourg


    • Active field theories
    • Active matter
    • Entropy production
    • Stochastic thermodynamics


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