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 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.

Entropy production and its large deviations in an active lattice gas

“…This is proven in appendix B. Thus both the PS and TB regions are accessible by this quasi-thermodynamic approach to the minimization in (40), essentially because in each case the minimiser consists of two phases separated by a narrow domain wall whose contribution to I H is sub-leading at small . The quantitative results of this procedure are detailed in section 7 below.…”

“…The interfacial energy in such a phase-separated state scales as and is negligible for 1. Notice however, that for such a sharply phase separated solution, the gradient expansion (40) breaks down in the region of the sharp interface where higher order gradient terms contribute. Nonetheless, the interfacial contribution is sub-extensive so that the minimized rate function I consists, to leading order in , of the convex hull of I H .…”

“…This means that the local contribution to the action functional should match that for homogeneous profiles (38), yet now these profiles can vary in space. Meanwhile the derivation of the sub-leading term in (40) necessitates retaining the gradient terms in the functional (11) and is derived in appendix B.…”

“…To perform the minimization in (40), we again take inspiration from equilibrium statistical mechanics: I H (ρ, α) can be interpreted as an analogue of a bulk free energy density as a function of two order parameters α and ρ that are spatially conserved due to (41). This is analogous to globally minimizing the free energy of a ternary fluid mixture, which is a well-studied problem in chemical physics (see, e.g.…”

Entropons as collective excitations in active solids

“…Quantifying the nonequilibrium character of active systems via entropy production has represented a topic of central interest in recent years [42][43][44][45]. This topic has been investigated numerically, simulating both active field theories [46][47][48][49], active particle dynamics in interacting systems [50][51][52], as well as colloids in the presence of an active bath [53,54]. Particular attention has been devoted to phase-separated configurations where the main contribution to the spatial profile of the entropy production has been observed at the interface between dense and dilute phases [46,55,56].…”