Stochastic Hydrodynamics of Complex Fluids: Discretisation and Entropy Production
English
Cates, Michael E.[Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK]
Fodor, Etienne[University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Physics and Materials Science (DPHYMS) >]
Nardini, Cesare[Service de Physique de l’Etat Condensé, CEA, CNRS Université Paris-Saclay, CEA-Saclay, 91191 Gif-sur-Yvette, France; > > > ; Laboratoire de Physique Théorique de la Matière Condensée, Sorbonne Université, CNRS, 75005 Paris, France]
Tjhung, Elsen[Department of Physics, University of Durham, Science Laboratories, South Road, Durham DH1 3LE, UK > > > ; School of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK]
[en] 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.
[University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Physics and Materials Science (DPHYMS) >]
