A scale-invariant model of statistical mechanics is applied to describe modified forms of zeroth, first, second, and third laws of classical thermodynamics. Following Helmholtz, the total thermal energy of the thermodynamic system is decomposed into free heat U and latent heat pV suggesting the modified form of the first law of thermodynamics Q = H = U + pV. Following Boltzmann, entropy of ideal gas is expressed in terms of the number of Heisenberg–Kramers virtual oscillators as S = 4 Nk. Through introduction of stochastic definition of Planck and Boltzmann constants, Kelvin absolute temperature scale T (degree K) is identified as a length scale T (m) that is related to de Broglie wavelength of particle thermal oscillations. It is argued that rather than relating to the surface area of its horizon suggested by Bekenstein (1973, “Black Holes and Entropy,” Phys. Rev. D, 7(8), pp. 2333–2346), entropy of black hole should be related to its total thermal energy, namely, its enthalpy leading to S = 4Nk in exact agreement with the prediction of Major and Setter (2001, “Gravitational Statistical Mechanics: A Model,” Classical Quantum Gravity, 18, pp. 5125–5142).
Issue Section:
Energy Systems Analysis
References
1.
de Broglie
, L.
, 1926
, “Interference and Corpuscular Light
,” Nature
, 118
(2969
), pp. 441
–442
.2.
de Broglie
, L.
, 1927
, “Sur la possibilité de relier les phénomènes d'interférence et de diffraction à la théorie des quanta de lumière
,” C. R. Acad. Sci. Paris
, 183
, pp. 447
–448
.3.
de Broglie
, L.
, 1927
, “La structure atomique de la matière et du rayonnement et la mécanique Ondulatoire
,” C. R. Acad. Sci. Paris, 184
, pp. 273
–274
.4.
de Broglie
, L.
, 1927
, “Sur le rôle des ondes continues en mécanique ondulatoire
,” C. R. Acad. Sci. Paris
, 185
, pp. 380
–382
.5.
de Broglie
, L.
, 1960
, Non-Linear Wave Mechanics: A Causal Interpretation
, Elsevier
, New York
.6.
de Broglie
, L.
, 1970
, “The Reinterpretation of Wave Mechanics
,” Found. Phys.
, 1
(5
), pp. 5
–15
.7.
Madelung
, E.
, 1926
, “Quantentheorie in Hydrodynamischer Form
,” Z. Phys.
, 40
(3–4), pp. 322
–326
.8.
Schrödinger
, E.
, 1931
, “Über die Umkehrung der Naturgesetze
,” Sitzungsber. Preuss. Akad. Wiss. Phys. Math. Kl
, 144
, pp. 144
–153
.9.
Fürth
, R.
, 1933
, “Über Einige Beziehungen zwischen klassischer Staristik und Quantenmechanik
,” Z. Phys.
, 81
(3
), pp. 143
–162
.10.
Bohm
, D.
, 1952
, “A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables. I
,” Phys. Rev.
, 85
(2
), pp. 166
–179
.11.
Takabayasi
, T.
, 1952
, “On the Foundation of Quantum Mechanics Associated With Classical Pictures
,” Prog. Theor. Phys.
, 8
(2
), pp. 143
–182
.12.
Bohm
, D.
, and Vigier
, J. P.
, 1954
, “Model of the Causal Interpretation of Quantum Theory in Terms of a Fluid With Irregular Fluctuations
,” Phys. Rev.
, 96
(1
), pp. 208
–217
.13.
Nelson
, E.
, 1966
, “Derivation of the Schrödinger Equation From Newtonian Mechanics
,” Phys. Rev.
, 150
(4
), pp. 1079
–1085
.14.
Nelson
, E.
, 1985
, Quantum Fluctuations
, Princeton University Press
, Princeton, NJ
.15.
de la Peña
, L.
, 1969
, “New Foundation of Stochastic Theory of Quantum Mechanics
,” J. Math. Phys.
, 10
(9
), pp. 1620
–1630
.16.
de la Peña
, L.
, and Cetto
, A. M.
, 1982
, “Does Quantum Mechanics Accept a Stochastic Support?
,” Found. Phys.
, 12
(10
), pp. 1017
–1037
.17.
Barut
, A. O.
, 1988
, “Schrödinger's Interpretation of Ψ as a Continuous Charge Distribution
,” Ann. Phys.
, 7
(4–5
), pp. 31
–36
.18.
Barut
, A. O.
, and Bracken
, A. J.
, 1981
, “Zitterbewegung and the Internal Geometry of the Electron
,” Phys. Rev. D
, 23
(10
), pp. 2454
–2463
.19.
Vigier
, J. P.
, 1980
, “De Broglie Waves on Dirac Aether: A Testable Experimental Assumption
,” Lett. Nuvo Cim.
, 29
(14
), pp. 467
–475
.20.
Gueret
, Ph.
, and Vigier
, J. P.
, 1982
, “De Broglie's Wave Particle Duality in the Stochastic Interpretation of Quantum Mechanics: A Testable Physical Assumption
,” Found. Phys.
, 12
(11
), pp. 1057
–1083
.21.
Cufaro Petroni
, C.
, and Vigier
, J. P.
, 1983
, “Dirac's Aether in Relativistic Quantum Mechanics
,” Found. Phys.
, 13
(2
), pp. 253
–286
.22.
Vigier
, J. P.
, 1995
, “Derivation of Inertia Forces From the Einstein-de Broglie-Bohm (E.d.B.B) Causal Stochastic Interpretation of Quantum Mechanics
,” Found. Phys.
, 25
(10
), pp. 1461
–1494
.23.
Arecchi
, F. T.
, and Harrison
, R. G.
, 1987
, Instabilities and Chaos in Quantum Optics
, Springer-Verlag
, Berlin
.24.
Reynolds
, O.
, 1895
, “On the Dynamical Theory of Incompressible Viscous Fluid and the Determination of the Criterion
,” Philos. Trans. R. Soc., A
, 186
, pp. 23
–164
.25.
Enskog
, D.
, 1917
, Kinetische Theorie der Vorgange in Massig Verdunnten Gasen
, Almqvist and Wiksells Boktryckeri-AB
, Uppsala, Sweden
(English Translation: Brush, G. S., 1965
, Kinetic Theory
, Vol. 1–3
, Pergamon Press
, New York)
.26.
Taylor
, G. I.
, 1935
, “Statistical Theory of Turbulence-Parts I–IV
,” Proc. R. Soc., A
, 151
(873
), pp. 421
–478
.27.
von Kármán
, T.
, and Howarth
, L.
, 1938
, “On the Statistical Theory of Isotropic Turbulence
,” Proc. R. Soc. A
, 164
(917
), pp. 192
–215
.28.
Robertson
, H. P.
, 1940
, “The Invariant Theory of Isotropic Turbulence
,” Proc. Cambridge Philos. Soc.
, 36
(02
), pp. 209
–223
.29.
Kolmogoroff
, A. N.
, 1941
, “Local Structure on Turbulence in Incompressible Fluid
,” C. R. Acad. Sci. U.R.S.S.
, 30
, pp. 301
–305
.30.
Kolmogoroff
, A. N.
, 1942
, “Dissipation of Energy in Locally Isotropic Turbulence
,” C. R. Acad. Sci. U.R.S.S.
, 32
, pp. 19
–21
.31.
Kolmogoroff
, A. N.
, 1962
, “A Refinement of Previous Hypothesis Concerning the Local Structure of Turbulence in a Viscous Incompressible Fluid at High Reynolds Number
,” J. Fluid Mech.
, 13
(01
), pp. 82
–85
.32.
Obukhov
, A. M.
, 1941
, “On the Distribution of Energy in the Spectrum of Turbulent Flow
,” C. R. Acad. Sci. U.R.S.S.
, 32
, pp. 19
–22
.33.
Obukhov
, A. M.
, 1962
, “Some Specific Features of Atmospheric Turbulence
,” J. Fluid Mech.
, 13
(01
), pp. 77
–81
.34.
Chandrasekhar
, S.
, 1943
, “Stochastic Problems in Physics and Astronomy
,” Rev. Mod. Phys.
, 151
(1
), pp. 1
–89
.35.
Chandrasekhar
, S.
, 1989
, Stochastic, Statistical, and Hydrodynamic Problems in Physics and Astronomy
, Selected Papers, Vol. 3
, University of Chicago Press
, Chicago, IL
, pp. 199
–206
.36.
Heisenberg
, W.
, 1948
, “On the Theory of Statistical and Isotropic Turbulence
,” Proc. R. Soc., A
, 195
(1042
), pp. 402
–406
.37.
Heisenberg
, W.
, 1948
, “Zur Statistischen Theorie der Turbulenz
,” Z. Phys.
, 124
(7–12
), pp. 628
–657
.38.
Batchelor
, G. K.
, 1953
, The Theory of Homogeneous Turbulence
, Cambridge University Press
, Cambridge
.39.
Landau
, L. D.
, and Lifshitz
, E. M.
, 1959
, Fluid Dynamics
, Pergamon Press
, New York
.40.
Tennekes
, H.
, and Lumley
, J. L.
, 1972
, A First Course in Turbulence
, MIT Press
, Boston, MA
.41.
Sohrab
, S. H.
, 1996
, “Transport Phenomena and Conservation Equations in Multicomponent Chemically-Reactive Ideal Gas Mixtures
,” 31st ASME National Heat Transfer Conference
, HTD-Vol. 328
, pp. 37
–60
.42.
Sohrab
, S. H.
, 1999
, “A Scale Invariant Model of Statistical Mechanics and Modified Forms of the First and the Second Laws of Thermodynamics
,” Rev. Gen. Therm.
, 38
(10
), pp. 845
–853
.43.
Sohrab
, S. H.
, 2008
, “The Nature of Mass, Dark Matter, and Dark Energy in Cosmology and the Foundation of Relativistic Thermodynamics
,” New Aspects of Heat Transfer, Thermal Engineering, and Environment
, S. H.
Sohrab
, H. J.
Catrakis
, and N.
Kobasko
, eds., WSEAS Press
, Rhodes, Greece
, pp. 434
–442
.44.
Sohrab
, S. H.
, 2014
, “Boltzmann Entropy of Thermodynamics Versus Shannon Entropy of Information Theory
,” Int. J. Mech.
, 8
, pp. 73
–84
.45.
Sohrab
, S. H.
, 2007
, “Invariant Planck Energy Distribution Law and Its Connection to the Maxwell-Boltzmann Distribution Function
,” WSEAS Trans. Math.
, 6
(2
), pp. 254
–262
.46.
Sohrab
, S. H.
, 2014
, “Invariant Forms of Conservation Equations and Some Examples of Their Exact Solutions
,” ASME J. Energy Resour. Technol.
, 136
(3
), p. 032002
.47.
Sohrab
, S. H.
, 2009
, “Universality of a Scale Invariant Model of Turbulence and Its Quantum Mechanical Foundation
,” Recent Advances in Fluid Mechanics & Aerodynamics
, S.
Sohrab
, H.
Catrakis
, and N.
Kobasko
, eds., WSEAS Press
, Moscow
, pp. 134
–140
.48.
Sohrab
, S. H.
, 2014
, “On a Scale Invariant Model of Statistical Mechanics and Derivation of Invariant Forms of Conservation Equations
,” WSEAS Trans. Heat Mass Transfer
, 9
, pp. 169
–194
.49.
Sohrab
, S. H.
, 2010
, “Quantum Theory of Fields From Planck to Cosmic Scales
,” WSEAS Trans. Math.
, 9
(8
), pp. 734
–756
.50.
Sohrab
, S. H.
, 2015
, “On a Scale Invariant Model of Statistical Mechanics, Kinetic Theory of Ideal Gas, and Riemann Hypothesis
,” Int. J. Mod. Commun. Res. Technol.
, 6
(6
), pp. 734
–756
.51.
Sohrab
, S. H.
, 1998
, “Physical Foundation of a Unified Statistical Theory of Fields and the Scale Invariant Schrödinger Equation
,” Bull. Am. Phys. Soc.
, 43
(1
), p. 781
.52.
Sohrab
, S. H.
, 2014
, “Some Implications of a Scale Invariant Model of Statistical Mechanics to Classical and Relativistic Thermodynamics
,” Int. J. Therm.
, 17
(4
), pp. 233
–248
.53.
Poincaré
, H.
, 2000
, “The Present and the Future of Mathematical Physics
,” Address Delivered Before the Section of Applied Mathematics of the International Congress of Arts and Science, St. Louis; Bull. Am. Math. Soc.
, 37
(1
), pp. 25
–38
.54.
Gouy
, M.
, 1888
, “Note Sur le Mouvement Brownien
,” J. Phys.
, 7
(1
), pp. 561
–564
.55.
Gouy
, M.
, 1889
, “Sur le Mouvement Brownien
,” C. R. Acad. Sci., Paris
, 109
, pp. 102
–105
.56.
Gouy
, M.
, 1895
, “Le Mouvement Brownien et les Mouvement Moléculaires
,” Rev. Gén. Sci.
, 1
, pp. 1
–7
.57.
Bachelier
, L.
, 1900
, “Théorie Mathématique du Jeux
,” Ann. Sci. Ec. Norm. Supér.
, 27
(3
), pp. 21
–86
.58.
Cercignani
, C.
, 1998
, Ludwig Boltzmann, The Man Who Trusted Atoms
, Oxford University Press
, Oxford
.59.
Sutherland
, A. W.
, 1897
, “Causes of Osmotic Pressure and of the Simplicity of the Laws of Dilute Solutions
,” Philos. Mag. Ser. 5
, 44
(5
), pp. 493
–498
.60.
Sutherland
, A. W.
, 1902
, “Ionization, Ionic Velocities, and Atomic Sizes
,” Philos. Mag.
, 4
(24
), pp. 625
–645
.61.
Sutherland
, A. W.
, 1905
, “A Dynamic Theory for Non-Electrolytes and the Molecular Mass of Albumin
,” Philos. Mag.
, 9
(54
), pp. 781
–785
.62.
Smoluchowski
, M.
, 1986
, Polish Men of Science
, R. S.
Ingarden
, ed., Polish Science Publishers
, Warszawa, Poland
.63.
Einstein
, A.
, 1956
, Investigations on the Theory of Brownian Movement
, R.
Fürth
, ed., Dover Publications
, New York
.64.
Perrin
, M. J.
, 1910
, Brownian Movement and Molecular Reality
, Taylor and Francis
, London
.65.
Uhlenbeck
, G. E.
, and Ornstein
, L. S.
, 1930
, “On the Theory of the Brownian Motion
,” Phys. Rev.
, 36
(5
), pp. 823
–841
.66.
Wang
, C. M.
, and Uhlenbeck
, G. E.
, 1945
, “On the Theory of the Brownian Motion II
,” Rev. Mod. Phys.
, 17
(2–3
), pp. 323
–342
.67.
Füchs
, N. A.
, 1964
, The Mechanics of Aerosols
, Dover
, New York
.68.
Nelson
, E.
, 1967
, Dynamical Theory of Brownian Motion
, Princeton University Press
, Princeton, NJ
.69.
Duplantier
, B.
, 2006
, “Brownian Motions, Diverse and Undulating
,” Einstein 1905–2005, Poincaré Seminar 2005
, T.
Damour
, O.
Darrigol
, B.
Duplantier
, and V.
Rivasseau
, eds., Birkhäuser
, Basel, Switzerland
, pp. 201
–293
.70.
de Groot
, R. S.
, and Mazur
, P.
, 1962
, Nonequilibrium Thermodynamics
, North-Holland, Amsterdam
.71.
Schlichting
, H.
, 1968
, Boundary-Layer Theory
, McGraw-Hill
, New York
.72.
Williams
, F. A.
, 1985
, Combustion Theory
, 2nd ed., Addison Wesley
, Menlo Park, CA
.73.
Hirschfelder
, J. O.
, Curtiss
, C. F.
, and Bird
, R. B.
, 1954
, Molecular Theory of Gases and Liquids
, Wiley
, New York
.74.
Chapman
, S.
, and Cowling
, T. G.
, 1953
, The Mathematical Theory of Non-Uniform Gases
, Cambridge University Press
, Cambridge
.75.
Planck
, M.
, 1901
, “On the Law of the Energy Distribution in the Normal Spectrum
,” Ann. Phys.
, 4
, pp. 553
–558
.76.
Planck
, M.
, 1959
, The Theory of Heat Radiation
, Dover
, New York
.77.
van der Waerden
, B. L.
, 1967
, “Towards Quantum Mechanics
,” Sources of Quantum Mechanics
, B. L.
van der Waerden
, ed., Dover
, New York
, pp. 1
–59
.78.
Planck
, M.
, 1981
, Where Is Science Going
, Ox Bow Press
, Woodbridge, CT
.79.
Casimir
, H. B. G.
, 1948
, “On the Attraction Between Two Perfectly Conducting Plates
,” Proc. K. Ned. Akad. Wet.
, 51
, pp. 793
–795
.80.
Darrigol
, O.
, 1988
, “Statistics and Combinatorics in Early Quantum Theory
,” Hist. Stud. Phys. Biol. Sci.
, 19
(1
), pp. 17
–80
.81.
Darrigol
, O.
, 1991
, “Statistics and Combinatorics in Early Quantum Theory, II: Early Symptoms of Indistinguishability and Holism
,” Hist. Stud. Phys. Biol. Sci.
, 21
(2
), pp. 237
–298
.82.
Lochak
, G.
, 1984
, “The Evolution of the Ideas of Louis de Broglie on the Interpretation of Wave Mechanics
,” Quantum, Space and Time-The Quest Continues
, Asim O.
Barut
, Alwyn
van der Merwe
, and Jean-Pierre
Vigier
, eds., Cambridge University Press
, Cambridge
, pp. 11
–33
.83.
Lorentz
, H. A.
, 1904
, “Electromagnetic Phenomena in a System Moving With Any Velocity Less Than That of Light
,” Proc. Acad. Sci. Amsterdam
, 6
, pp. 809
–831
.84.
Poincaré
, H.
, 1900
, “La Théorie de Lorentz et le Principe de Réaction
,” Arch. Neerland.
, 5
, pp. 252
–278
.85.
Heisenberg
, W.
, 1949
, The Physical Principles of Quantum Theory
, Dover
, New York
.86.
Huygens
, C.
, 1912
, Treatise on Light
, Dover
, New York
, p. 14
.87.
Kardar
, M.
, 2007
, Statistical Physics of Particles
, Cambridge University Press
, New York
.88.
Pauli
, W.
, 1973
, Pauli Lectures on Physics
, Vol. 3
, MIT Press
, New York
, p. 14
.89.
Long
, C. A.
, and Sohrab
, S. H.
, 2008
, “The Power of Two, Speed of Light, Force and Energy and the Universal Gas Constant
,” Recent Advances on Applied Mathematics
, C. A.
Long
, S. H.
Sohrab
, G.
Bognar
, and L.
Perlovsky
, eds., WSEAS Press
, Athens, Greece
, pp. 434
–442
.90.
De Pretto
, O.
, 1904
, “Ipotesi dell'Etere Nella Vita dell'Universo
,” R. Inst. Veneto Sci. Lett. Arti
, 63
(2
), pp. 439
–500
.91.
Hasenöhrl
, F.
, 1904
, “Zur Theorie der Strahlung in bewegten Körpern
,” Ann. Phys.
, 15
(4
), pp. 344
–370
.92.
Hasenöhrl
, F.
, 1905
, “Zur Theorie der Strahlung in bewegten Körpern
,” Ann. Phys.
, 16
(4
), pp. 589
–592
.93.
Einstein
, A.
, 1905
, “Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?
,” Ann. Phys. (Leipzig)
, 18
(13
), pp. 639
–641
.94.
Helmholtz
, H.
, 1847
, “Über der Ehaltung der Kraft
,” Eine Physikalische Abhandlung
, G. Reiner
, Berlin
(English Translation: Brush, G. S., 1965, Kinetic Theory, Vol. 1–3, Pergamon Press, New York).95.
Clausius
, R.
, 1857
, “Über die Art der Bewegung, welche wir Wärme nennen
,” Ann. Phys.
, 176
(3
), pp. 353
–380
.96.
Clausius
, R.
, 1870
, “Ueber einen auf die Wärme anwendbaren mechanischen Satz
,” Sitzungsberichte der Niedderrheinischen Gesellschaft
, Bonn
, Germany
, pp. 114
–119
.97.
Sommerfeld
, A.
, 1956
, Thermodynamics and Statistical Mechanics
, Academic Press
, New York
.98.
Sonntag
, R. E.
, and van Wylen
, G. E.
, 1966
, Fundamentals of Statistical Thermodynamics
, Wiley
, New York
.99.
Yourgrau
, W.
, van der Merwe
, A.
, and Raw
, G.
, 1982
, Treatise on Irreversible and Statistical Thermodynamics
, Dover
, New York
.100.
Boltzmann
, L.
, 1872
, “Weitere Studien uber das Warmegleichgewicht unter Gasmoleculen
,” Sitzungsber. Akad.Wiss., Vienna, Part II
, 66
, pp. 275
–370
(English Translation: Brush, G. S., 1965, Kinetic Theory, Vol. 1–3, Pergamon Press, New York, pp. 88–175).101.
Boltzmann
, L.
, 1964
, Lectures on Gas Theory
, Dover
, New York
.102.
Onsager
, H. B. G.
, 1931
, “Reciprocity Relations in Irreversible Processes. I
,” Phys. Rev.
, 37
(4
), pp. 405
–425
.103.
Riek
, R.
, 2014
, “A Derivation of a Microscopic Entropy and Time Irreversibility From the Discreteness of Time
,” Entropy
, 16
(6
), pp. 3149
–3172
.104.
Bardeen
, J. M.
, Carter
, B.
, and Hawking
, S. W.
, 1973
, “The Four Laws of Black Hole Mechanics
,” Commun. Math. Phys.
, 31
(2
), pp. 161
–170
.105.
Bekenstein
, J. D.
, 1973
, “Black Holes and Entropy
,” Phys. Rev. D
, 7
(8
), pp. 2333
–2346
.106.
Hawking
, S. W.
, 1976
, “Black Holes and Thermodynamics
,” Phys. Rev. D
, 13
(2
), pp. 191
–197
.107.
Hawking
, S. W.
, 1974
, “Black Hole Explosions
,” Nature
, 248
(5443
), pp. 30
–31
.108.
‘
t Hooft
, G.
, 1985
, “On the Quantum Structure of a Black Hole
,” Nucl. Phys. B
, 256
, pp. 727
–745
.109.
Wald
, R. M.
, 1998
, “Black Holes and Thermodynamics
,” Black Holes and Relativistic Stars
, R. M.
Wald
, ed., University of Chicago Press
, Chicago, IL
, pp. 155
–176
.110.
Grumiller
, D.
, McNees
, R.
, and Salzer
, J.
, 2015
, “Black Hole Thermodynamics: The First Half Century
,” Quantum Aspects of Black Holes
, X.
Calmet
, ed., Fundamentals of Theoretical Physics 178, Springer
, Switzerland
.111.
Major
, S. A.
, and Setter
, K. L.
, 2001
, “Gravitational Statistical Mechanics: A Model
,” Classical Quantum Gravity
, 18
(23
), pp. 5125
–5142
.112.
Poincaré
, H.
, 1952
, Science and Hypothesis
, Dover
, New York
, p. 65
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