Stochastic Hydrodynamics of Complex Fluids: Discretisation and Entropy Production

Abstract

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 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 encode thermodynamically 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.

27 pages, 4 figures

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• Entropy production and its large deviations in an active lattice gas
• Interface roughening in nonequilibrium phase-separated systems
• Hyperuniformity in phase ordering: the roles of activity, noise, and non-constant mobility
• Effective estimation of entropy production with lacking data
• Most probable path of an active Brownian particle
• Second Law of Thermodynamics without Einstein Relation
• Entropy production rate in thermodynamically consistent flocks
• Heat production in a stochastic system with nonlinear time-delayed feedback
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• Effective Energy, Interactions And Out Of Equilibrium Nature Of Scalar Active Matter
• Intracellular phagosome shell is rigid enough to transfer outside torque to the inner spherical particle
• Consistent expansion of the Langevin propagator with application to entropy production