Metriplectic four-bracket algorithm for constructing thermodynamically consistent dynamical systems

Zaidni, Azeddine; Morrison, Philip J.

doi:10.1103/r2lb-xkq6

Metriplectic four-bracket algorithm for constructing thermodynamically consistent dynamical systems

Journal Article · Fri Aug 01 00:00:00 EDT 2025 · Physical Review E

DOI:https://doi.org/10.1103/r2lb-xkq6· OSTI ID:2998046

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Mohammed VI Polytechnic University, Ben Guerir (Morocco)
The University of Texas at Austin, TX (United States)

A unified thermodynamic algorithm is presented for constructing thermodynamically consistent dynamical systems, i.e., systems that have Hamiltonian and dissipative parts that conserve energy while producing entropy. The algorithm is based on the metriplectic 4-bracket given in Morrison and Updike [Phys. Rev. E 109, 045202 (2024)]. A feature of the unified thermodynamic algorithm is the force-flux relation 𝐉^𝛼=−𝐿^𝛼⁢𝛽𝛁(𝛿⁢𝐻⁡/𝛿⁢𝜉^𝛽) for phenomenological coefficients 𝐿^𝛼⁢𝛽, Hamiltonian 𝐻, and dynamical variables 𝜉^𝛽. The algorithm is applied to the Navier-Stokes-Fourier, the Cahn-Hilliard-Navier-Stokes, and Brenner-Navier-Stokes-Fourier systems, and significant generalizations of these systems are obtained.

View Accepted Manuscript (DOE)

Research Organization:: The University of Texas at Austin, TX (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Fusion Energy Sciences (FES)

Grant/Contract Number:: FG02-04ER54742

OSTI ID:: 2998046

Journal Information:: Physical Review E, Journal Name: Physical Review E Journal Issue: 2 Vol. 112; ISSN 2470-0053; ISSN 2470-0045

Publisher:: American Physical Society (APS)Copyright Statement

Country of Publication:: United States

Language:: English

References (39)

Beyond Equilibrium Thermodynamics Öttinger, Hans Christian John Wiley & Sons https://doi.org/10.1002/0471727903	book	January 2005
On the Bianchi identities Kulkarni, Ravindra S. Mathematische Annalen, Vol. 199, Issue 4 https://doi.org/10.1007/BF01429873	journal	December 1972
A paradigm for joined Hamiltonian and dissipative systems Morrison, Philip J. Physica D: Nonlinear Phenomena, Vol. 18, Issue 1-3 https://doi.org/10.1016/0167-2789(86)90209-5	journal	January 1986
Algebraic structure of the plasma quasilinear equations Kaufman, Allan N.; Morrison, Philip J. Physics Letters A, Vol. 88, Issue 8 https://doi.org/10.1016/0375-9601(82)90664-8	journal	April 1982
Dissipative hamiltonian systems: A unifying principle Kaufman, Allan N. Physics Letters A, Vol. 100, Issue 8 https://doi.org/10.1016/0375-9601(84)90634-0	journal	February 1984
Bracket formulation for irreversible classical fields Morrison, Philip J. Physics Letters A, Vol. 100, Issue 8 https://doi.org/10.1016/0375-9601(84)90635-2	journal	February 1984
II—mean curvature and weighted mean curvature Taylor, J. E. Acta Metallurgica et Materialia, Vol. 40, Issue 7 https://doi.org/10.1016/0956-7151(92)90091-R	journal	July 1992
A phase-field model of solidification with convection Anderson, D. M.; McFadden, G. B.; Wheeler, A. A. Physica D: Nonlinear Phenomena, Vol. 135, Issue 1-2 https://doi.org/10.1016/S0167-2789(99)00109-8	journal	January 2000
Classification and Casimir invariants of Lie–Poisson brackets Thiffeault, Jean-Luc; Morrison, P. J. Physica D: Nonlinear Phenomena, Vol. 136, Issue 3-4 https://doi.org/10.1016/S0167-2789(99)00155-4	journal	February 2000
Energetically consistent model reduction for metriplectic systems Gruber, Anthony; Gunzburger, Max; Ju, Lili Computer Methods in Applied Mechanics and Engineering, Vol. 404 https://doi.org/10.1016/j.cma.2022.115709	journal	February 2023
A thermodynamically consistent discretization of 1D thermal-fluid models using their metriplectic 4-bracket structure Barham, William; Morrison, Philip J.; Zaidni, Azeddine Communications in Nonlinear Science and Numerical Simulation, Vol. 145 https://doi.org/10.1016/j.cnsns.2025.108683	journal	June 2025
A collision operator for describing dissipation in noncanonical phase space Sato, Naoki; Morrison, Philip J. Fundamental Plasma Physics, Vol. 10 https://doi.org/10.1016/j.fpp.2024.100054	journal	June 2024
Beyond Navier–Stokes Brenner, Howard International Journal of Engineering Science, Vol. 54 https://doi.org/10.1016/j.ijengsci.2012.01.006	journal	May 2012
Kinematics of volume transport Brenner, Howard Physica A: Statistical Mechanics and its Applications, Vol. 349, Issue 1-2 https://doi.org/10.1016/j.physa.2004.10.033	journal	April 2005
Navier–Stokes revisited Brenner, Howard Physica A: Statistical Mechanics and its Applications, Vol. 349, Issue 1-2 https://doi.org/10.1016/j.physa.2004.10.034	journal	April 2005
Fluid mechanics revisited Brenner, Howard Physica A: Statistical Mechanics and its Applications, Vol. 370, Issue 2 https://doi.org/10.1016/j.physa.2006.03.066	journal	October 2006
On a possible difference between the barycentric velocity and the velocity that gives translational momentum in fluids Bedeaux, Dick; Kjelstrup, Signe; Christian Öttinger, Hans Physica A: Statistical Mechanics and its Applications, Vol. 371, Issue 2 https://doi.org/10.1016/j.physa.2006.05.023	journal	November 2006
Comment on “Navier–Stokes Revisited,” H. Brenner, Physica A 349 (2005) 60 Mills, A. F. Physica A: Statistical Mechanics and its Applications, Vol. 371, Issue 2 https://doi.org/10.1016/j.physa.2006.07.007	journal	November 2006
Thermodynamically consistent Cahn–Hilliard–Navier–Stokes equations using the metriplectic dynamics formalism Zaidni, Azeddine; Morrison, Philip J.; Benjelloun, Saad Physica D: Nonlinear Phenomena, Vol. 468 https://doi.org/10.1016/j.physd.2024.134303	journal	November 2024
A general metriplectic framework with application to dissipative extended magnetohydrodynamics Coquinot, Baptiste; Morrison, Philip J. Journal of Plasma Physics, Vol. 86, Issue 3 https://doi.org/10.1017/S0022377820000392	journal	May 2020
A thermodynamically consistent phase-field model for two-phase flows with thermocapillary effects Guo, Z.; Lin, P. Journal of Fluid Mechanics, Vol. 766 https://doi.org/10.1017/jfm.2014.696	journal	February 2015
Thermodynamics of Irreversible Processes. The Experimental Verification of the Onsager Reciprocal Relations. Miller, Donald G. Chemical Reviews, Vol. 60, Issue 1 https://doi.org/10.1021/cr60203a003	journal	February 1960
Free Energy of a Nonuniform System. I. Interfacial Free Energy Cahn, John W.; Hilliard, John E. The Journal of Chemical Physics, Vol. 28, Issue 2 https://doi.org/10.1063/1.1744102	journal	February 1958
Poisson brackets for fluids and plasmas Morrison, Philip J. AIP Conference Proceedings Volume 88 https://doi.org/10.1063/1.33633	conference	January 1982
Derivation of the Hall and extended magnetohydrodynamics brackets D'Avignon, Eric C.; Morrison, Philip J.; Lingam, Manasvi Physics of Plasmas, Vol. 23, Issue 6 https://doi.org/10.1063/1.4952641	journal	June 2016
Metriplectic integrators for the Landau collision operator Kraus, Michael; Hirvijoki, Eero Physics of Plasmas, Vol. 24, Issue 10 https://doi.org/10.1063/1.4998610	journal	October 2017
Hamiltonian formulation of reduced magnetohydrodynamics Morrison, P. J.; Hazeltine, R. D. Physics of Fluids, Vol. 27, Issue 4 https://doi.org/10.1063/1.864718	journal	January 1984
Relaxation to magnetohydrodynamics equilibria via collision brackets Bressan, C.; Kraus, M.; Morrison, P. J. Journal of Physics: Conference Series, Vol. 1125 https://doi.org/10.1088/1742-6596/1125/1/012002	journal	November 2018
Hamiltonian formalism of extended magnetohydrodynamics Abdelhamid, H. M.; Kawazura, Y.; Yoshida, Z. Journal of Physics A: Mathematical and Theoretical, Vol. 48, Issue 23 https://doi.org/10.1088/1751-8113/48/23/235502	journal	May 2015
Recasting Navier–Stokes equations Reddy, M. H. Lakshminarayana; Dadzie, S. Kokou; Ocone, Raffaella Journal of Physics Communications, Vol. 3, Issue 10 https://doi.org/10.1088/2399-6528/ab4b86	journal	October 2019
Particle and bracket formulations of kinetic equations Grmela, Miroslav Contemporary Mathematics, Vol. 28 https://doi.org/10.1090/conm/028/751978	book	January 1984
Reciprocal Relations in Irreversible Processes. I. Onsager, Lars Physical Review, Vol. 37, Issue 4 https://doi.org/10.1103/PhysRev.37.405	journal	February 1931
Inclusive curvaturelike framework for describing dissipation: Metriplectic 4-bracket dynamics Morrison, Philip J.; Updike, Michael H. Physical Review E, Vol. 109, Issue 4 https://doi.org/10.1103/PhysRevE.109.045202	journal	April 2024
Dynamics and thermodynamics of complex fluids. I. Development of a general formalism Grmela, Miroslav; Öttinger, Hans Christian Physical Review E, Vol. 56, Issue 6 https://doi.org/10.1103/PhysRevE.56.6620	journal	December 1997
Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism Öttinger, Hans Christian; Grmela, Miroslav Physical Review E, Vol. 56, Issue 6 https://doi.org/10.1103/PhysRevE.56.6633	journal	December 1997
Noncanonical Hamiltonian Density Formulation of Hydrodynamics and Ideal Magnetohydrodynamics Morrison, Philip J.; Greene, John M. Physical Review Letters, Vol. 45, Issue 10 https://doi.org/10.1103/PhysRevLett.45.790	journal	September 1980
On Onsager's Principle of Microscopic Reversibility Casimir, H. B. G. Reviews of Modern Physics, Vol. 17, Issue 2-3 https://doi.org/10.1103/RevModPhys.17.343	journal	April 1945
Hamiltonian description of the ideal fluid Morrison, P. J. Reviews of Modern Physics, Vol. 70, Issue 2 https://doi.org/10.1103/RevModPhys.70.467	journal	April 1998
Diffuse-Interface Methods in Fluid Mechanics Anderson, D. M.; McFadden, G. B.; Wheeler, A. A. Annual Review of Fluid Mechanics, Vol. 30, Issue 1 https://doi.org/10.1146/annurev.fluid.30.1.139	journal	January 1998

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