This paper contains a fully geometric formulation of the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC). Although GENERIC, which is the sum of Hamiltonian mechanics and gradient dynamics, is a framework unifying a vast range of models in non-equilibrium thermodynamics, it has unclear geometric structure due to the diverse geometric origins of Hamiltonian mechanics and gradient dynamics. The difference can be overcome by cotangent lifts of the dynamics, which leads, for instance, to a Hamiltonian form of gradient dynamics. Moreover, the lifted vector fields can be split into their holonomic and vertical representatives, which provides a geometric method of dynamic reduction. The lifted dynamics can be also given physical meaning, here called the rate-GENERIC. Finally, the lifts can be formulated within contact geometry, where the second law of thermodynamics is explicitly contained within the evolution equations.
REFERENCES
1.
L.
Euler
, “Principes généraux du mouvement des fluides
,” Académie Royale des Sciences et des Belles-Lettres de Berlin, Mémoires
(1755
), Vol. 11, English translation is available in Physica D 237, 1825–1839 (2008).2.
H.
Callen
, Thermodynamics: An Introduction to the Physical Theories of Equilibrium Thermostatics and Irreversible Thermodynamics
(Wiley
, 1960
).3.
L.
Wolf
, “Face recognition, geometric vs. appearance-based
,” in Encyclopedia of Biometrics
, edited by S. Z.
Li
and A.
Jain
(Springer US
, Boston, Massachusetts
, 2009
), pp. 347
–352
.4.
J.
Liouville
, “Sur la theorie de la variation des constantes arbitraires
,” J. Math. Pures Appl.
3
, 342
–349
(1838
); available at http://eudml.org/doc/234417.5.
T.
Carleman
, “Application de la théorie des équations intégrales linéaires aux syst‘emes d’équations différentielles non linéaires
,” Acta Math.
59
, 63
(1932
).6.
B.
Koopman
, “Hamiltonian systems and transformations in Hilbert space
,” Proc. Natl. Acad. Sci. U. S. A.
17
, 315
(1931
).7.
8.
9.
R.
Mrugala
, “On contact and metric structures on thermo-dynamic spaces
,” RIMS Kokyuroku
(Kyoto
) 1142
, 167
(2000
).10.
M.
Grmela
, “Multiscale equilibrium and nonequilibrium thermodynamics in chemical engineering
,” Adv. Chem. Eng.
39
, 75
–129
(2010
).11.
M.
Grmela
and H. C.
Öttinger
, “Dynamics and thermodynamics of complex fluids. I. Development of a general formalism
,” Phys. Rev. E
56
, 6620
–6632
(1997
).12.
H. C.
Öttinger
and M.
Grmela
, “Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism
,” Phys. Rev. E
56
, 6633
–6655
(1997
).13.
H.
Öttinger
, Beyond Equilibrium Thermodynamics
(Wiley
, New York
, 2005
).14.
M.
Pavelka
, V.
Klika
, and M.
Grmela
, Multiscale Thermo-Dynamics
(de Gruyter
, Berlin
, 2018
).15.
M.
Fecko
, Differential Geometry and Lie Groups for Physicists
(Cambridge University Press
, 2006
).16.
O.
Esen
, M.
Grmela
, and M.
Pavelka
, “On the role of geometry in statistical mechanics and thermodynamics II: Thermodynamic perspective
,” J. Math. Phys.
63
, 123305
(2022
).17.
P. J.
Morrison
, “Bracket formulation for irreversible classical fields
,” Phys. Lett. A
100
, 423
–427
(1984
).18.
R.
Abraham
and J. E.
Marsden
, Foundations of Mechanics
(Benjamin/Cummings Publishing Co., Inc., Advanced Book Program
, Reading, Mass
, 1978
), pp. xxii+m
–xvi+806
, second edition, revised and enlarged, With the assistance of Tudor Raţiu and Richard Cushman.19.
M.
de León
and P. R.
Rodrigues
, Methods of differential geometry in analytical mechanics
, North-Holland Mathematics Studies (North-Holland Publishing
, Amsterdam
, 1989
), Vol. 158, pp. x+483
.20.
D. D.
Holm
, Geometric Mechanics. Part I: Dynamics and Symmetry
, 2nd ed. (Imperial College Press
, London
, 2011
), pp. xxiv+441
.21.
P.
Libermann
and C.-M.
Marle
, Symplectic Geometry and Analytical Mechanics, Mathematics and its Applications
(D. Reidel Publishing
, Dordrecht
, 1987
), Vol. 35, pp. xvi+526
, translated from the French by Bertram Eugene Schwarzbach.22.
C.
Laurent-Gengoux
, A.
Pichereau
, and P.
Vanhaecke
, Poisson structures, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences)
(Springer
, Heidelberg
, 2013
), Vol. 347, pp. xxiv+461
.23.
I.
Vaisman
, Lectures on the geometry of Poisson manifolds
, Progress in Mathematics (Birkhäuser Verlag
, Basel
, 1994
), Vol. 118, pp. viii+205
.24.
A.
Weinstein
, “The local structure of Poisson manifolds
,” J. Differ. Geom.
18
, 523
–557
(1983
).25.
K.
Yano
and E. M.
Patterson
, “Vertical and complete lifts from a manifold to its cotangent bundle
,” J. Math. Soc. Jpn.
19
, 91
–113
(1967
).26.
O.
Esen
and H.
Gümral
, “Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields
,” J. Geom. Mech.
4
, 239
–269
(2012
).27.
J.
Marsden
and T.
Ratiu
, Introduction to Mechanics and symmetry
, Texts in Applied Mathematics, 2nd ed. (Springer-Verlag
, New York
, 1999
), Vol. 17 a basic exposition of classical mechanical systems.28.
V. I.
Arnold
, Mathematical Methods of Classical mechanics
, Graduate Texts in Mathematics, 2nd ed. (Springer-Verlag
, New York
, 1989
), Vol. 60, pp. xvi+508
, translated from the Russian by K. Vogtmann and A. Weinstein.29.
H.
Goldstein
, Classical Mechanics
, Addison-Wesley Series in Physics, 2nd ed., (Addison-Wesley Publishing Co.
, Reading, Mass
, 1980
), pp. xiv+672
.30.
J. F.
Cariñena
, X.
Gràcia
, G.
Marmo
, E.
Martínez
, M. C.
Muñoz-lecanda
, and N.
Román-Roy
, “Geometric Hamilton–Jacobi theory
,” Int. J. Geom. Methods Mod. Phys.
03
, 1417
–1458
(2006
).31.
L.
Colombo
, M.
de León
, P. D.
Prieto-Martínez
, and N.
Román-Roy
, “Geometric Hamilton-Jacobi theory for higher-order autonomous systems
,” J. Phys. A: Math. Theor.
47
, 235203
(2014
).32.
M.
de León
, J. C.
Marrero
, and D. M.
de Diego
, “A geometric Hamilton-Jacobi theory for classical field theories
,” Variations, Geometry and Physics
(Nova Science Publishers
, New York
, 2009
), pp. 129
–140
.33.
M.
De León
and S.
Vilariño
, “Hamilton-Jacobi theory in k-cosymplectic field theories
,” Int. J. Geom. Methods Mod. Phys.
11
, 1450007
(2014
).34.
M.
De León
, D. M.
De Diego
, J. C.
Marrero
, M.
Salgado
, and S.
Vilariño
, “Hamilton-Jacobi theory in k-symplectic field theories
,” Int. J. Geom. Methods Mod. Phys.
07
, 1491
–1507
(2010
).35.
O.
Esen
, M.
de León
, and C.
Sardón
, “A Hamilton-Jacobi theory for implicit differential systems
,” J. Math. Phys.
59
, 022902
(2018
).36.
O.
Esen
, M.
de León
, and C.
Sardón
, “A Hamilton-Jacobi formalism for higher order implicit Lagrangians
,” J. Phys. A: Math. Theor.
53
, 075204
(2020
).37.
J. F.
Cariñena
, X.
Gràcia
, G.
Marmo
, E.
Martínez
, M. C.
Muñoz-lecanda
, and N.
ROMÁN-ROY
, “Geometric Hamilton-Jacobi theory for nonholonomic dynamical systems
,” Int. J. Geom. Methods Mod. Phys.
07
, 431
–454
(2010
).38.
O.
Esen
, V. M.
Jiménez
, M.
de León
, and C.
Sardón
, “Reduction of a Hamilton–Jacobi equation for nonholonomic systems
,” Regul. Chaotic Dyn.
24
, 525
–559
(2019
).39.
O.
Esen
, M.
de León
, M.
Lainz
, C.
Sardón
, and M.
Zając
, “Reviewing the geometric Hamilton-Jacobi theory concerning Jacobi and Leibniz identities
,” J. Phys. A: Math. Theor.
55
, 403001
(2022
).40.
S.
Benenti
, Hamiltonian Structures and Generating Families
, Universitext (Springer
, New York
, 2011
), pp. xiv+258
.41.
W. M.
Tulczyjew
and P.
Urbański
, “A slow and careful Legendre transformation for singular Lagrangians
,” Acta Phys. Pol. B
30
, 2909
(1999
).42.
D. J.
Saunders
, The Geometry of Jet Bundles
, London Mathematical Society Lecture Note Series (Cambridge University Press
, Cambridge
, 1989
), Vol. 142, pp. viii+293
.43.
O.
Esen
, M.
Grmela
, H.
Gümral
, and M.
Pavelka
, “Lifts of symmetric tensors: Fluids, plasma, and grad hierarchy
,” Entropy
21
, 907
(2019
).44.
O.
Esen
and H.
Gümral
, “Lifts, jets and reduced dynamics
,” Int. J. Geom. Methods Mod. Phys.
08
, 331
–344
(2011
).45.
P. J.
Olver
, Applications of Lie Groups to Differential Equations
, Graduate Texts in Mathematics (Springer-Verlag
, New York
, 1986
), Vol. 107, pp. xxvi+497
.46.
M.
Pavelka
, V.
Klika
, and M.
Grmela
, “Time reversal in nonequilibrium thermodynamics
,” Phys. Rev. E
90
, 062131
(2014
).47.
I.
Gyarmati
, Non-equilibrium Thermodynamics: Field Theory and Variational Principles
, Engineering Science Library (Springer
, Berlin, Heidelberg
, 1970
).48.
F.
Otto
, “The geometry of dissipative evolution equations: The porous medium equation
,” Commun. Partial Differ. Equ.
26
, 101
–174
(2001
).49.
B.
Leimkuhler
, S.
Reich
, and C. U.
Press
, Simulating Hamiltonian dynamics
, Cambridge Monographs on Applied and Computational Mathematics (Cambridge University Press
, 2004
).50.
L.
Onsager
, “Reciprocal relations in irreversible processes. I
,” Phys. Rev.
37
, 405
(1931
).51.
M.
Grmela
, “Fluctuations in extended mass-action-law dynamics
,” Physica D
241
, 976
–986
(2012
).52.
A.
Mielke
, M. A.
Peletier
, and D. R. M.
Renger
, “On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion
,” Potential Anal.
41
, 1293
–1327
(2014
).53.
C. M.
Guldberg
and P.
Waage
, Etudes sur les affinités chimiques
(Imprimerie de Brøgger & Christie
, 1867
).54.
L.
Onsager
and S.
Machlup
, “Fluctuations and irreversible processes
,” Phys. Rev.
91
, 1505
–1512
(1953
).55.
D. R. M.
Renger
, “Gradient and generic systems in the space of fluxes, applied to reacting particle systems
,” Entropy
20
, 596
(2018
).56.
R.
McLachlan
and M.
Perlmutter
, “Conformal Hamiltonian systems
,” J. Geom. Phys.
39
, 276
–300
(2001
).57.
D.
Jou
, J.
Casas-Vázquez
, and G.
Lebon
, Extended Irreversible Thermodynamics
, 4th ed. (Springer-Verlag
, New York
, 2010
).58.
S. K.
Godunov
, “An interesting class of quasi-linear systems
,” (Russian) Dokl. Akad. Nauk SSSR
139
, 521
–523
(1961
).59.
S.
Godunov
, “Symmetric form of the magnetohydrodynamic equation
,” Chislennye Metody Mekhaniki Sploshnoi Sredy
(1972
), Vol. 3, pp. 26
–34
; available at https://www.osti.gov/biblio/4275163.60.
S.
Godunov
and E.
Romensky
, “Thermodynamics, conservation laws and symmetric forms of differential equations in mechanics of continuous media
,” in (Wiley
: New York
, 1995
), Vol. 95, pp. 19
–31
.61.
I.
Peshkov
, M.
Pavelka
, E.
Romenski
, and M.
Grmela
, “Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations
,” Continuum Mech. Thermodyn.
30
, 1343
–1378
(2018
).62.
R. J.
DiPerna
and P. L.
Lions
, “On the Cauchy problem for Boltzmann equations: Global existence and weak stability
,” Ann. Math.
130
(2), 321
–366
(1989
).63.
This works in finite-dimensional spaces, while in the infinite-dimensional case, it becomes more involved.83
64.
A.
Bravetti
, “Contact Hamiltonian dynamics: The concept and its use
,” Entropy
19
, 535
(2017
).65.
A.
Bravetti
, H.
Cruz
, and D.
Tapias
, “Contact Hamiltonian mechanics
,” Ann. Phys.
376
, 17
–39
(2017
).66.
M.
de León
and M.
Lainz Valcázar
, “Contact Hamiltonian systems
,” J. Math. Phys.
60
, 102902
(2019
).67.
O.
Esen
, M.
Lainz Valcázar
, M.
de León
, and J. C.
Marrero
, “Contact dynamics: Legendrian and Lagrangian submanifolds
,” Mathematics
9
, 2704
(2021
).68.
A. A.
Simoes
, D. M.
de Diego
, M. L.
Valcázar
, and M.
de León
, “The geometry of some thermodynamic systems
,” in Geometric Structures of Statistical Physics, Information Geometry, and Learning
, edited by F.
Barbaresco
and F.
Nielsen
(Springer International Publishing
, Cham
, 2021
), pp. 247
–275
.69.
A. A.
Simoes
, M.
de León
, M. L.
Valcázar
, and D. M.
de Diego
, “Contact geometry for simple thermodynamical systems with friction
,” Proc. R. Soc. A
476
, 20200244
(2020
).70.
P.
Cannarsa
, W.
Cheng
, K.
Wang
, and J.
Yan
, “Herglotz’generalized variational principle and contact type Hamilton-Jacobi equations
,” Trends Control Theory Partial Differ. Equ.
, 39
–67
(2019
).71.
M.
de León
, M.
Lainz
, and Á.
Muñiz-Brea
, “The Hamilton–Jacobi theory for contact Hamiltonian systems
,” Mathematics
9
, 1993
(2021
).72.
M.
de León
and C.
Sardón
, “Cosymplectic and contact structures for time-dependent and dissipative Hamiltonian systems
,” J. Phys. A: Math. Theor.
50
, 255205
(2017
).73.
O.
Esen
, M. L.
Valcázar
, M.
de León
, and C.
Sardón
, “Implicit contact dynamics and Hamilton-Jacobi theory
,” arXiv:2109.14921 (2021
).74.
S.
Grillo
and E.
Padrón
, “Extended Hamilton-Jacobi theory, contact manifolds, and integrability by quadratures
,” J. Math. Phys.
61
, 012901
(2020
).75.
I.
Prigogine
, Introduction to Thermodynamics of Irreversible Processes
(Thomas
, New York
, 1955
).76.
G.
Herglotz
, Berührungstransformationen
, Lectures at the University of Göttingen (University of Göttingen
, Göttingen
, 1930
).77.
M.
Pavelka
, V.
Klika
, and M.
Grmela
, “Generalization of the dynamical lack-of-fit reduction
,” J. Stat. Phys.
181
, 19
–52
(2020
).78.
A.
Bravetti
, “Contact geometry and thermodynamics
,” Int. J. Geom. Methods Mod. Phys.
16
, 1940003
(2019
).79.
M.
Grmela
, “Contact geometry of mesoscopic thermodynamics and dynamics
,” Entropy
16
, 1652
–1686
(2014
).80.
S.-i.
Goto
, “Legendre submanifolds in contact manifolds as attractors and geometric nonequilibrium thermodynamics
,” J. Math. Phys.
56
, 073301
(2015
).81.
H.
Matsuzoe
and M.
Henmi
, Geometric science of information
, Lecture Notes in Computer Science (Springer
, 2013
), Vol. 8085, pp. 275
–282
.82.
H.
Shima
and K.
Yagi
, “Geometry of Hessian manifolds
,” Differ. Geom. Appl.
7
, 277
–290
(1997
).83.
T.
Roubíček,
, Nonlinear Partial Differential Equations with Applications
, International Series of Numerical Mathematics (Birkhäuser
, Basel
, 2005
).© 2022 Author(s). Published under an exclusive license by AIP Publishing.
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