Stochastic Hydrodynamics of Complex Fluids: Discretisation and Entropy Production
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 nu...
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MDPI AG
2022-02-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/24/2/254 |
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| author | Michael E. Cates Étienne Fodor Tomer Markovich Cesare Nardini Elsen Tjhung |
| author_facet | Michael E. Cates Étienne Fodor Tomer Markovich Cesare Nardini Elsen Tjhung |
| author_sort | Michael E. Cates |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-09T22:01:44Z |
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| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-09T22:01:44Z |
| publishDate | 2022-02-01 |
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| series | Entropy |
| spelling | doaj.art-9481f1e38c3741c285dc52d6ffce253e2023-11-23T19:48:26ZengMDPI AGEntropy1099-43002022-02-0124225410.3390/e24020254Stochastic Hydrodynamics of Complex Fluids: Discretisation and Entropy ProductionMichael E. Cates0Étienne Fodor1Tomer Markovich2Cesare Nardini3Elsen Tjhung4Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UKDepartment of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, LuxembourgCenter for Theoretical Biological Physics, Rice University, Houston, TX 77005, USAService de Physique de l’Etat Condensé, CEA, CNRS Université Paris-Saclay, CEA-Saclay, 91191 Gif-sur-Yvette, FranceDepartment of Physics, University of Durham, Science Laboratories, South Road, Durham DH1 3LE, UKMany 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.https://www.mdpi.com/1099-4300/24/2/254active matterstochastic thermodynamicsentropy productionactive field theories |
| spellingShingle | Michael E. Cates Étienne Fodor Tomer Markovich Cesare Nardini Elsen Tjhung Stochastic Hydrodynamics of Complex Fluids: Discretisation and Entropy Production Entropy active matter stochastic thermodynamics entropy production active field theories |
| title | Stochastic Hydrodynamics of Complex Fluids: Discretisation and Entropy Production |
| title_full | Stochastic Hydrodynamics of Complex Fluids: Discretisation and Entropy Production |
| title_fullStr | Stochastic Hydrodynamics of Complex Fluids: Discretisation and Entropy Production |
| title_full_unstemmed | Stochastic Hydrodynamics of Complex Fluids: Discretisation and Entropy Production |
| title_short | Stochastic Hydrodynamics of Complex Fluids: Discretisation and Entropy Production |
| title_sort | stochastic hydrodynamics of complex fluids discretisation and entropy production |
| topic | active matter stochastic thermodynamics entropy production active field theories |
| url | https://www.mdpi.com/1099-4300/24/2/254 |
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