Quantum mechanics, superfluids, and capillary fluids are closely related: It is thermodynamics that links them. In this paper, the Liu procedure is used to analyze the thermodynamic requirements. A comparison with the traditional method of divergence separation highlights the role of spacetime. It is shown that perfect Korteweg fluids are holographic. The conditions under which a complex field can represent the density and velocity fields of the fluid, and where the complex scalar field becomes a wave function of quantum mechanics, are explored. The bridge between the field and particle representations of a physical system is holography, and the key to holography is the Second Law of Thermodynamics.
REFERENCES
1.
2.
J. D.
Bekenstein
, “
Information in the holographic universe
,” Sci. Am.
289
(2
), 58
–65
(2003
).3.
R.
Bousso
, “
The holographic principle
,” Rev. Mod. Phys.
74
(3
), 825
(2002
).4.
E. P.
Verlinde
, “
On the origin of gravity and the laws of Newton
,” J. High Energy Phys.
2011
(04
), 29
.5.
S.
Hossenfelder
, “
Comments on and comments on comments on Verlinde's paper ‘On the origin of gravity and the laws of Newton
,’” arXiv:1003.1015 (2010
).6.
P.
Ván
and
S.
Abe
, “
Emergence of modified Newtonian gravity from thermodynamics
,” Physica A
588
, 126505
(2022
).7.
S.
Abe
and
P.
Ván
, “
Crossover in extended Newtonian gravity emerging from thermodynamics
,” Symmetry
14
(7
), 1048
(2022
).8.
M.
Szücs
and
P.
Ván
, “
Holographic property of fluid systems with gradient internal variable
” (unpublished).9.
A. A.
Friedmann
, “
Oput gidromechaniki zsimaemoj zsidkosty
,” Ph.D. thesis
(University of Petrograd
, 1922
)A. A.
Friedmann
, [“An essay on hydrodynamics of compressible fluid” (in Russian)].10.
E.
Madelung
, “
Quantentheorie in hydrodynamischer form
,” Z. Phys.
40
, 322
–326
(1926
) (in German).11.
P. R.
Holland
, The Quantum Theory of Motion
(
Cambridge University Press
,
Cambridge
, 1993
).12.
13.
B.
Bistrovic
,
R.
Jackiw
,
H.
Li
,
V. P.
Nair
, and
S.-Y.
Pi
, “
Non-Abelian fluid dynamics in Lagrangian formulation
,” Phys. Rev. D
67
(2
), 025013
–025011
(2003
).14.
R.
Jackiw
,
V. P.
Nair
,
S.-Y.
Pi
, and
A. P.
Polychronakos
, “
Perfect fluid theory and its extensions
,” J. Phys. A
37
, R327
–R432
(2004
).15.
J. D.
van der Waals
, “
Thermodynamische theorie der kapillarität unter voraussetzung stetiger dichteänderung
,” Z. Phys. Chem.
13U
, 657
–725
(1894
).16.
D. J.
Korteweg
, “
Sur la forme que prennant les équations du mouvement des fluides si l'on tient compte des froces capillaires causées par des variations de densité
,” Arch. Néerl. Sci. Exactes Nat., Ser. II
6
, 1
–16
(1901
).17.
L.
de Sobrino
, “
Some thermodynamic and stability properties of a fluid with gradient dependent free energy
,” Can. J. Phys.
54
(2
), 105
–117
(1976
).18.
J. E.
Dunn
and
J.
Serrin
, “
On the thermomechanics of interstitial working
,” Arch. Ration. Mech. Anal.
88
, 95
–133
(1985
).19.
L. K.
Antanovskii
, “
Microscale theory of surface tension
,” Phys. Rev. E
54
(6
), 6285
(1996
).20.
P.
Ván
and
T.
Fülöp
, “
Weakly nonlocal fluid mechanics: The Schrödinger equation
,” Proc. R. Soc. London, Ser. A
462
(2066
), 541
–557
(2006
).21.
D. M.
Anderson
,
G. B.
McFadden
, and
A. A.
Wheeler
, “
Diffuse-interface methods in fluid mechanics
,” Annu. Rev. Fluid Mech.
30
, 139
–165
(1998
).22.
23.
I. M.
Khalatnikov
, An Introduction to the Theory of Superfluidity
(
CRC Press
, 2018
).24.
V. L.
Ginzburg
, “
Superconductivity and superfluidity (what was done and what was not)
,” Phys.-Usp.
40
(4
), 407
–432
(1997
).25.
E. P.
Gross
, “
Structure of a quantized vortex in boson systems
,” Nuovo Cimento
20
(3
), 454
–477
(1961
).26.
L. P.
Pitaevskii
, “
Vortex lines in an imperfect Bose gas
,” Sov. Phys. JETP
13
(2
), 451
–454
(1961
).27.
V. L.
Ginzburg
and
A. A.
Sobaynin
, “
Superfluidity of helium II near the λ point
,” Sov. Phys. Usp.
31
(4
), 289
(1988
).28.
I.
Bialynicki-Birula
and
J.
Mycielski
, “
Nonlinear wave mechanics
,” Ann. Phys.
100
, 62
–93
(1976
).29.
T. C.
Scott
and
K. G.
Zloshchastiev
, “
Resolving the puzzle of sound propagation in liquid helium at low temperatures
,” Low Temp. Phys.
45
(12
), 1231
–1236
(2019
).30.
P.
Ván
and
R.
Kovács
, “
Variational principles and nonequilibrium thermodynamics
,” Philos. Trans. R. Soc., A
378
(2170
), 20190178
(2020
).31.
T.
Matolcsi
, Spacetime without Reference Frames
, 2nd ed.
(
Minkowski Institute Press
, 2020
).32.
T.
Ruggeri
, “
Galilean invariance and entropy principle for systems of balance laws
,” Continuum Mech. Thermodyn.
1
(1
), 3
–20
(1989
).33.
W.
Muschik
and
L.
Restuccia
, “
Changing the observer and moving materials in continum physics: Objectivity and frame-indifference
,” Tech. Mech.
22
(3
), 152
–160
(2002
).34.
T.
Matolcsi
and
P.
Ván
, “
Can material time derivative be objective?
,” Phys. Lett. A
353
(2–3
), 109
–112
(2006
).35.
W.
Noll
, “
Space-time structures in classical mechanics
,” in The Foundations of Mechanics and Thermodynamics: Selected Papers
(
Springer Verlag
,
Berlin-Heidelberg-New York
, 1974
), pp. 204
–210
; originally, Delaware Seminar in the Foundations of Physics (Springer, Berlin-Heidelberg-New York, 1967), pp. 28–34.36.
M.
Friedman
, Foundations of Space-Time Theories: Relativistic Physics and Philosophy of Science
(
Princeton University Press
,
Princeton, New Jersey
, 1983
).37.
P.
Ván
,
M.
Pavelka
, and
M.
Grmela
, “
Extra mass flux in fluid mechanics
,” J. Non-Equilib. Thermodyn.
42
(2
), 133
–151
(2017
).38.
P.
Ván
, “
Galilean relativistic fluid mechanics
,” Continuum Mech. Thermodyn.
29
(2
), 585
–610
(2017
) (v1- Hungarian; v2- English).39.
The following is a simple explanation with relative quantities. The covector maps a vector to an invariant scalar. Let us assume that the transformation rule of a four-vector A is Galilean; therefore, in a reference frame K, it has the time and spacelike components
, and in the other reference frame
, moving with a relative velocity
related to K, the time and spacelike components are
. The timelike part does not change, but the spacelike part is transformed via Galilean transformation. If the time and spacelike components of a four-covector B in the reference frames K and
are represented by
and
, respectively, the transformation rules for the covector components follow from the invariance of
:
Therefore,
, the spacelike part of the four-covector does not transform, and the transformation rule of the timelike part is
.
40.
41.
W.
Noll
, see www.math.cmu.edu/ ∼wn0g/noll/FC.pdf for “
Five contributions to natural philosophy
,” 2004
.42.
W.
Noll
, “
A frame free formulation of elasticity
,” J. Elasticity
83
, 291
–307
(2006
).43.
W.
Noll
and
B.
Seguin
, “
Basic concepts of thermomechanics
,” J. Elasticity
101
, 121
–151
(2010
).44.
45.
Remarkably, Noll himself joined the critics of Noll objectivity to complain about the insufficient formulation and argued that the concept must be further developed; the Descartes product of space and time is not a correct representation of spacetime in nonrelativistic physics.41–43
46.
A.
Einstein
, “
Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen
,” Jahrb. Radioakt. Elektron.
4
, 411
–462
(1907
).47.
M.
Planck
, “
Zur dynamik bewegter systeme
,” Ann. Phys.
331
(6
), 1
–34
(1908
).48.
H.
Ott
, “
Lorentz-transformation der Wärme und der temperatur
,” Z. Phys.
175
, 70
–104
(1963
).49.
N. G.
van Kampen
, “
Relativistic thermodynamics of moving systems
,” Phys. Rev.
173
, 295
–301
(1968
).50.
T. S.
Bíró
and
P.
Ván
, “
About the temperature of moving bodies
,” Europhys. Lett.
89
, 30001
(2010
).51.
B. D.
Coleman
and
W.
Noll
, “
The thermodynamics of elastic materials with heat conduction and viscosity
,” Arch. Ration. Mech. Anal.
13
, 167
–178
(1963
).52.
I.-S.
Liu
, “
Method of Lagrange multipliers for exploitation of the entropy principle
,” Arch. Ration. Mech. Anal.
46
, 131
–148
(1972
).53.
F.
Gy
, “
A Fourier-féle mechanikai elv alkalmazásai
,” Math. Természettudományi Értesítő
12
, 457
–472
(1894
) (in Hungarian).54.
J.
Farkas
, “
Theorie der einfachen Ungleichungen
,” J. Angew. Math. (Crelles J.)
124
, 1
–27
(1902
).55.
R. A.
Hauser
and
N. P.
Kirchner
, “
A historical note on the entropy principle of Müller and Liu
,” Continuum Mech. Thermodyn.
14
, 223
–226
(2002
).56.
W.
Muschik
,
C.
Papenfuss
, and
H.
Ehrentraut
, “
A sketch of continuum thermodynamics
,” J. Non-Newtonian Fluid Mech.
96
, 255
–290
(2001
).57.
P.
Ván
, “
Exploiting the Second Law in weakly nonlocal continuum physics
,” Period. Polytech., Mech. Eng.
49
(1
), 79
–94
(2005
); available at https://pp.bme.hu/me/article/view/1344.58.
V. A.
Cimmelli
, “
An extension of Liu procedure in weakly nonlocal thermodynamics
,” J. Math. Phys.
48
, 113510
(2007
).59.
C.
Eckart
, “
The thermodynamics of irreversible processes. I. The simple fluid
,” Phys. Rev.
58
, 267
–269
(1940
).60.
S. R.
de Groot
and
P.
Mazur
, Non-Equilibrium Thermodynamics
(
North-Holland Publishing Company
,
Amsterdam
, 1962
).61.
A.
Berezovski
and
P.
Ván
, Internal Variables in Thermoelasticity
(
Springer
, 2017
).62.
L. K.
Antanovskii
,
C.
Rogers
, and
W. K.
Schief
, “
A note on a capillarity model and the nonlinear Schrödinger equation
,” J. Phys. A
30
, L555
–L557
(1997
).63.
S.
Kjelstrup
and
D.
Bedeaux
, Non-Equilibrium Thermodynamics of Heterogeneous Systems
(
World Scientific
,
Berlin, New Jersey
, 2008
).64.
S.
Kjelstrup
,
D.
Bedeaux
,
E.
Johannessen
, and
J.
Gross
, Non-Equilibrium Thermodynamics for Engineers
(
World Scientific
, 2010
).65.
M.
Szücs
and
P.
Ván
, “
About the heat flux in nonequilibrium thermodynamics” (unpublished).
66.
E. T.
Jaynes
, “
Information theory and statistical mechanics
,” Phys. Rev.
106
(4
), 620
–630
(1957
).67.
A.
Rényi
, “
On measures of entropy and information
,” in Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, 1960
, edited by
J.
Neyman
(
University of California Press
,
Berkeley
, 1961
), Vol.
I
, pp. 547
–561
.68.
T.
Takabayasi
, “
Relativistic hydrodynamics of the Dirac matter. Part I. General theory
,” Suppl. Prog. Theor. Phys.
4
, 1
–80
(1957
).69.
J. W. M.
Bush
, “
Pilot-wave hydrodynamics
,” Annu. Rev. Fluid Mech.
47
, 269
–292
(2015
).70.
V.
Frumkin
,
D.
Darrow
,
J. W. M.
Bush
, and
W.
Struyve
, “
Real surreal trajectories in pilot-wave hydrodynamics
,” Phys. Rev. A
106
(1
), L010203
(2022
).71.
K. O.
Friedrichs
and
P. D.
Lax
, “
Systems of conservation equations with a convex extension
,” Proc. Natl. Acad. Sci.
68
(8
), 1686
–1688
(1971
).72.
P.
Glansdorff
and
I.
Prigogine
, Thermodynamic Theory of Structure, Stability and Fluctuations
(
Wiley-Interscience
, London
, 1971
).73.
T.
Matolcsi
, Ordinary Thermodynamics
, 2nd ed. (Society for the Unity of Science and Technology, 2017); available at: https://oszkdk.oszk.hu/storage/00/01/98/77/dd/1/Matolcsi_Ordinary_Thermodynamics_2017-04-26.pdf.74.
P.
Kovtun
,
D. T.
Son
, and
A. O.
Starinets
, “
Holography and hydrodynamics: Diffusion on stretched horizons
,” J. High Energy Phys.
2003
(10
), 064
.75.
T.
Nishioka
,
S.
Ryu
, and
T.
Takayanagi
, “
Holographic entanglement entropy: An overview
,” J. Phys. A: Math. Theor.
42
(50
), 504008
(2009
).76.
M.
Baggioli
and
B.
Goutéraux
, “
Colloquium: Hydrodynamics and holography of charge density wave phases
,” Rev. Mod. Phys.
95
(1
), 011001
(2023
).77.
J.
de Boer
,
J.
Hartong
,
N.
Obers
,
W.
Sybesma
, and
S.
Vandoren
, “
Perfect fluids
,” SciPost Phys.
5
(1
), 003
(2018
).78.
J.
Armas
and
A.
Jain
, “
Viscoelastic hydrodynamics and holography
,” J. High Energy Phys.
2020
(1
), 1
–61
.79.
J.
de Boer
,
J.
Hartong
,
N.
Obers
,
W.
Sybesma
, and
S.
Vandoren
, “
Hydrodynamic modes of homogeneous and isotropic fluids
,” SciPost Phys.
5
(2
), 014
(2018
).80.
81.
M.
Pavelka
,
V.
Klika
, and
M.
Grmela
, Multiscale Thermo-Dynamics: Introduction to GENERIC
(
Walter de Gruyter GmbH & Co. KG
, 2018
).82.
M. D.
Kostin
, “
On the Schrödinger–Langevin equation
,” J. Chem. Phys.
57
(9
), 3589
–3590
(1972
).83.
K.
Yasue
, “
Stochastic quantization: A review
,” Int. J. Theor. Phys.
18
(12
), 861
–913
(1979
).84.
A. L.
Sanin
and
A. A.
Smirnovsky
, “
Oscillatory motion in confined potential systems with dissipation in the context of the Schrödinger–Langevin–Kostin equation
,” Phys. Lett. A
372
(1
), 21
–27
(2007
).85.
H.
Losert
,
F.
Ullinger
,
M.
Zimmermann
,
M. A.
Efremov
,
E. M.
Rasel
, and
W. P.
Schleich
, “
The Kostin equation, the deceleration of a quantum particle and coherent control
,” J. Low Temp. Phys.
210
, 4–50
(2023
).86.
I.
Fényes
, “
Eine wahrscheinlichkeitstheoretische begründung und interpretation der quantenmechanik
,” Z. Phys.
132
, 81
–106
(1952
) (
in German)
.87.
E.
Nelson
, “
Derivation of the Schrödinger equation from Newtonian mechanics
,” Phys. Rev.
150
(4
), 1079
–1085
(1966
).88.
D.
Bohm
and
B. J.
Hiley
, The Undivided Universe: An Ontological Interpretation of Quantum Theory
(
Routledge
, 2006
).89.
C. M.
Bender
, “
Making sense of non-Hermitian Hamiltonians
,” Rep. Prog. Phys.
70
(6
), 947
(2007
).90.
91.
D.
Anchishkin
,
I.
Mishustin
, and
H.
Stoecker
, “
Phase transition in an interacting boson system at finite temperatures
,” J. Phys. G: Nucl. Part. Phys.
46
(3
), 035002
(2019
).92.
D.
Anchishkin
,
V.
Gnatovskyy
,
D.
Zhuravel
, and
V.
Karpenko
, “
Self-interacting particle-antiparticle system of bosons
,” Phys. Rev. C
105
(4
), 045205
(2022
).93.
H.
Brenner
, “
Kinematics of volume transport
,” Physica A
349
, 11
–59
(2005
).94.
H. C.
Öttinger
,
H.
Struchtrup
, and
M.
Liu
, “
Inconsistency of a dissipative contribution to the mass flux in hydrodynamics
,” Phys. Rev. E
80
(5
), 056303
(2009
).95.
W. A.
Hiscock
and
L.
Lindblom
, “
Generic instabilities in first-order dissipative relativistic fluid theories
,” Phys. Rev. D
31
(4
), 725
–733
(1985
).96.
P.
Ván
and
T. S.
Biró
, “
First order and generic stable relativistic dissipative hydrodynamics
,” Phys. Lett. B
709
(1–2
), 106
–110
(2012
).97.
P.
Ván
, “
Generic stability of dissipative non-relativistic and relativistic fluids
,” J. Stat. Mech.: Theory Exp.
2009
, P02054
.98.
M. A.
Goodman
and
S. C.
Cowin
, “
A continuum theory for granular materials
,” Arch. Ration. Mech. Anal.
44
, 249
–266
(1972
).99.
M. M.
Mehrabadi
,
S. C.
Cowin
, and
M.
Massoudi
, “
Conservation laws and constitutive relations for density-gradient-dependent viscous fluids
,” Continuum Mech. Thermodyn.
17
(2
), 183
–200
(2005
).100.
M.
Fabrizio
,
B.
Lazzari
, and
R.
Nibbi
, “
Thermodynamics of non-local materials: Extra fluxes and internal powers
,” Continuum Mech. Thermodyn.
23
, 509
–525
(2011
).101.
M.
Fabrizio
,
F.
Franchi
, and
R.
Nibbi
, “
Nonlocal continuum mechanics structures: The virtual powers method vs the extra fluxes topic
,” J. Therm. Stresses
46
, 75
–87
(2023
).102.
V. A.
Cimmelli
,
A.
Sellitto
, and
D.
Jou
, “
Nonlocal effects and second sound in a nonequilibrium steady state
,” Phys. Rev. B
79
, 014303
(2009
).103.
V. A.
Cimmelli
,
F.
Oliveri
, and
A. R.
Pace
, “
On the thermodynamics of Korteweg fluids with heat conduction and viscosity
,” J. Elasticity
104
(1
), 115
–131
(2011
).104.
V. A.
Cimmelli
,
F.
Oliveri
, and
A. R.
Pace
, “
Phase-field evolution in Cahn–Hilliard–Korteweg fluids
,” Acta Mech.
227
(8
), 2111
–2124
(2016
).105.
V. A.
Cimmelli
,
M.
Gorgone
,
F.
Oliveri
, and
A. R.
Pace
, “
Weakly nonlocal thermodynamics of binary mixtures of Korteweg fluids with two velocities and two temperatures
,” Eur. J. Mech., B: Fluids
83
, 58
–65
(2020
).106.
M.
Gorgone
,
F.
Oliveri
, and
P.
Rogolino
, “
Thermodynamical analysis and constitutive equations for a mixture of viscous Korteweg fluids
,” Phys. Fluids
33
(9
), 093102
(2021
).107.
K. R.
Rajagopal
, “
On implicit constitutive theories for fluids
,” J. Fluid Mech.
550
, 243
–249
(2006
).108.
M.
Heida
and
J.
Málek
, “
On compressible Korteweg fluid-like materials
,” Int. J. Eng. Sci.
48
(11
), 1313
–1324
(2010
).109.
P. C.
Hohenberg
and
B. I.
Halperin
, “
Theory of dynamic critical phenomena
,” Rev. Mod. Phys.
49
(3
), 435
–479
(1977
).110.
O.
Penrose
and
P. C.
Fife
, “
Thermodynamically consistent models of phase-field type for the kinetics of phase transitions
,” Physica D
43
, 44
–62
(1990
).111.
C.
Eringen
, Microcontinuum Field Theories I. Foundations and Solids
, 3rd ed.
(
Springer-Verlag
, Berlin
, 1999
).112.
R.
Kovács
and
P.
Ván
, “
Generalized heat conduction in heat pulse experiments
,” Int. J. Heat Mass Transfer
83
, 613
–620
(2015
).113.
Á.
Pozsár
,
M.
Szücs
,
R.
Kovács
, and
T.
Fülöp
, “
Four spacetime dimensional simulation of rheological waves in solids and the merits of thermodynamics
,” Entropy
22
(12
), 1376
(2020
).114.
D. M.
Takács
,
Á.
Pozsár
, and
T.
Fülöp
, “
Thermodynamically extended symplectic numerical simulation of viscoelastic, thermal expansion and heat conduction phenomena in solids
,” arXiv:2211.12120 (2022
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
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