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Generalization of the dynamical lack-of-fit reduction from GENERIC to GENERIC. (English) Zbl 1453.82040

Summary: The lack-of-fit statistical reduction, developed and formulated first by B. Turkington [J. Stat. Phys. 152, No. 3, 569–597 (2013; Zbl 1274.82035)], is a general method taking Liouville equation for probability density (detailed level) and transforming it to reduced dynamics of projected quantities (less detailed level). In this paper the method is generalized. The Hamiltonian Liouville equation is replaced by an arbitrary Hamiltonian evolution combined with gradient dynamics (GENERIC), the Boltzmann entropy is replaced by an arbitrary entropy, and the kinetic energy by an arbitrary energy. The gradient part is a generalized gradient dynamics generated by a dissipation potential. The reduced evolution of the projected state variables is shown to preserve the GENERIC structure of the original (detailed level) evolution. The dissipation potential is obtained by solving a Hamilton-Jacobi equation. In summary, the lack-of-fit reduction can start with GENERIC and obtain GENERIC for the reduced state variables.

MSC:

82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
82C35 Irreversible thermodynamics, including Onsager-Machlup theory
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
70H20 Hamilton-Jacobi equations in mechanics
35F21 Hamilton-Jacobi equations

Citations:

Zbl 1274.82035

Software:

GENERIC
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Full Text: DOI arXiv

References:

[1] de Groot, SR; Mazur, P., Non-equilibrium Thermodynamics (1984), New York: Dover Publications, New York · Zbl 1375.82004
[2] Jou, D.; Casas-Vázquez, J.; Lebon, G., Extended Irreversible Thermodynamics (2010), New York: Springer, New York · Zbl 1185.74002
[3] Berezovski, A.; Ván, P., Internal Variables in Thermoelasticity. Solid Mechanics and Its Applications (2017), New York: Springer, New York · Zbl 1371.74001
[4] Grmela, M.; Öttinger, HC, Dynamics and thermodynamics of complex fluids. I. Development of a general formalism, Phys. Rev. E, 56, 6620-6632 (1997)
[5] Öttinger, HC; Grmela, M., Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism, Phys. Rev. E, 56, 6633-6655 (1997)
[6] Öttinger, HC, Beyond Equilibrium Thermodynamics (2005), New York: Wiley, New York
[7] Pavelka, M.; Klika, V.; Grmela, M., Multiscale Thermo-Dynamics (2018), Berlin: de Gruyter, Berlin · Zbl 1419.80001
[8] Chinesta, F., Cueto, E., Grmela, M., Moya, B., Pavelka, M.: Learning physics from data: a thermodynamic interpretation (2019)
[9] Jaynes, ET; Bunge, M., Foundations of probability theory and statistical mechanics, Delaware Seminar in the Foundation of Physics (1967), New York: Springer, New York · Zbl 0156.23203
[10] Grmela, M.; Klika, V.; Pavelka, M., Reductions and extensions in mesoscopic dynamics, Phys. Rev. E, 92, 032111 (2015)
[11] Maes, C.; Netočný, K., Time-reversal and entropy, J. Stat. Phys., 110, 1, 269-310 (2003) · Zbl 1035.82035
[12] Turkington, B., An optimization principle for deriving nonequilibrium statistical models of hamiltonian dynamics, J. Stat. Phys., 152, 3, 569 (2012) · Zbl 1274.82035
[13] Gorban, AN; Karlin, IV, Invariant Manifolds for Physical and Chemical Kinetics. Lecture Notes in Physics (2005), New York: Springer, New York · Zbl 1086.82009
[14] Klika, V.; Pavelka, M.; Vágner, P.; Grmela, M., Dynamic maximum entropy reduction, Entropy, 21, 715 (2019)
[15] Chapman, S.; Cowling, TG; Burnett, D.; Cercignani, C., The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases (1990), Cambridge: Cambridge Mathematical Library. Cambridge University Press, Cambridge
[16] Dumbser, M.; Peshkov, I.; Romenski, E.; Zanotti, O., High order ader schemes for a unified first order hyperbolic formulation of continuum mechanics: viscous heat-conducting fluids and elastic solids, J. Comput. Phys., 314, 824-862 (2016) · Zbl 1349.76324
[17] Callen, HB, Thermodynamics: An Introduction to the Physical Theories of Equilibrium Thermostatics and Irreversible Thermodynamics (1960), New York: Wiley, New York · Zbl 0095.23301
[18] Ehrenfest, P.; Ehrenfest, T., The Conceptual Foundations of the Statistical Approach in Mechanics. Dover Books on Physics (1990), New York: Dover Publications, New York · JFM 43.0763.01
[19] Gorban, AN; Karlin, IV; Öttinger, HC; Tatarinova, LL, Ehrenfest’s argument extended to a formalism of nonequilibrium thermodynamics, Phys. Rev. E, 63, 066124 (2001)
[20] Karlin, IV; Tatarinova, LL; Gorban, AN; Öttinger, HC, Irreversibility in the short memory approximation, Physica A, 327, 3-4, 399-424 (2003) · Zbl 1031.82020
[21] Pavelka, M.; Klika, V.; Grmela, M., Thermodynamic explanation of landau damping by reduction to hydrodynamics, Entropy, 20, 457 (2018)
[22] Pavelka, M.; Klika, V.; Grmela, M., Ehrenfest regularization of hamiltonian systems, Physica D, 399, 193-210 (2019) · Zbl 1453.70008
[23] Grmela, M., Role of thermodynamics in multiscale physics, Comput. Math. Appl., 65, 10, 1457-1470 (2013) · Zbl 1342.82004
[24] Grmela, M., Contact geometry of mesoscopic thermodynamics and dynamics, Entropy, 16, 1652-1686 (2014)
[25] Grmela, M.; Klika, V.; Pavelka, M., Gradient and GENERIC evolution towards reduced dynamics, Philos. Trans. R. Soc. A, 378, 20190472 (2020) · Zbl 1462.80002
[26] Zwanzig, R., Nonequilibrium Statistical Mechanics (2001), Oxford: Oxford University Press, Oxford · Zbl 1267.82001
[27] Zubarev, D.N., Morozov, V.G., Röpke, G.: Statistical Mechanics of Nonequilibrium Processes: Relaxation and Hydrodynamic Processes. Statistical Mechanics of Nonequilibrium Processes. Akademie Verlag (1996) · Zbl 0890.00009
[28] Español, P.; de la Torre, JA; Duque-Zumajo, D., Solution to the plateau problem in the green-kubo formula, Phys. Rev. E, 99, 022126 (2019)
[29] Turkington, B.; Chen, Q-Y; Thalabard, S., Coarse-graining two-dimensional turbulence via dynamical optimization, Nonlinearity, 29, 10, 2961-2989 (2016) · Zbl 1349.76108
[30] Maack, J.; Turkington, B., Reduced models of point vortex systems, Entropy, 20, 12, 914 (2018)
[31] Thalabard, S.; Turkington, B., Optimal response to non-equilibrium disturbances under truncated burgers-hopf dynamics, J. Phys. A, 50, 17, 175502 (2017) · Zbl 1454.82039
[32] Kleeman, R., A path integral formalism for non-equilibrium hamiltonian statistical systems, J. Stat. Phys., 158, 6, 1271-1297 (2015) · Zbl 1319.82018
[33] Kleeman, R.: A non-equilibrium theoretical framework for statistical physics with application to turbulent systems and their predictability (2019)
[34] Pavelka, M.; Klika, V.; Grmela, M., Time reversal in nonequilibrium thermodynamics, Phys. Rev. E, 90, 062131 (2014)
[35] Málek, J.; Průša, V.; Giga, Y.; Novotný, A., Derivation of equations for continuum mechanics and thermodynamics of fluids, Handbook of Mathematical Analysis in Mechanics of Viscus Fluids (2016), New York: Springer, New York
[36] Janečka, A.; Pavelka, M., Non-convex dissipation potentials in multiscale non-equilibrium thermodynamics, Contin. Mech. Thermodyn., 30, 4, 917-941 (2018) · Zbl 1396.76019
[37] Janečka, A.; Pavelka, M., Gradient dynamics and entropy production maximization, J. Non-equilibrium Thermodyn., 43, 1, 1-19 (2018)
[38] Rajagopal, KR; Srinivasa, AR, On thermomechanical restrictions of continua, Proc. R. Soc. Lond. A, 460, 2042, 631-651 (2004) · Zbl 1041.74002
[39] Málek, J.; Rajagopal, KR; Tůma, K., On a variant of the maxwell and oldroyd-b models within the context of a thermodynamic basis, Int. J. Non-Linear Mech., 76, 42-47 (2015)
[40] Montefusco, A.; Consonni, F.; Beretta, GP, Essential equivalence of the general equation for the nonequilibrium reversible-irreversible coupling (generic) and steepest-entropy-ascent models of dissipation for nonequilibrium thermodynamics, Phys. Rev. E, 91, 042138 (2015)
[41] Mielke, A.; Peletier, MA; Renger, DRM, On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion, Potential Anal., 41, 4, 1293-1327 (2014) · Zbl 1304.35692
[42] Mielke, A.; Renger, DRM; Peletier, MA, A generalization of Onsager’s reciprocity relations to gradient flows with nonlinear mobility, J. Non-equilibrium Thermodyn., 41, 2, 141 (2016)
[43] Bulíček, M., Málek, J., Průša, V.: Thermodynamics and stability of non-equilibrium steady states in open systems. Entropy 21(7), 704 (2019)
[44] Arnold, VI, Mathematical Methods of Classical Mechanics (1989), New York: Springer, New York · Zbl 0692.70003
[45] Esen, O.; Gümral, H., Geometry of plasma dynamics ii: Lie algebra of hamiltonian vector fields, J. Geom. Mech., 4, 3, 239 (2012) · Zbl 1375.76222
[46] Hermann, R., Geometry, Physics and Systems (1984), New York: Marcel Dekker, New York
[47] Pavelka, M.; Klika, V.; Esen, O.; Grmela, M., A hierarchy of Poisson brackets in non-equilibrium thermodynamics, Physica D, 335, 54-69 (2016) · Zbl 1415.82010
[48] Gelfand, IM; Fomin, SV; Silverman, RA, Calculus of Variations. Dover Books on Mathematics (2000), Mineola, NY: Dover Publications, Mineola, NY · Zbl 0964.49001
[49] Kraaij, R.; Lazarescu, A.; Maes, C.; Peletier, M., Deriving generic from a generalized fluctuation symmetry, J. Stat. Phys., 170, 492-508 (2018) · Zbl 1390.82045
[50] Ellero, M.; Espanol, P., Everything you always wanted to know about sdpd *( but were afraid to ask)*, Appl. Math. Mech. -Engl. Ed., 39, 1, 103-124 (2018)
[51] Kučera, V., A review of the matrix riccati equation, Kybernetika, 9, 1, 42 (1973) · Zbl 0279.49015
[52] Grmela, M., Particle and bracket formulations of kinetic equations, Contemp. Math., 28, 125-132 (1984) · Zbl 0558.58012
[53] Roubicek, T., Nonlinear Partial Differential Equations with Applications. International Series of Numerical Mathematics (2005), Basel: Birkhäuser, Basel · Zbl 1087.35002
[54] Mielke, A., Formulation of thermoelastic dissipative material behavior using GENERIC, Contin. Mech. Thermodyn., 23, 3, 233-256 (2011) · Zbl 1272.74137
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