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Vanishing vacuum energy
Vanishing vacuum energy
Journal Pre-proof
Vanishing vacuum energy Gregory Ryskin PII: DOI: Reference:
S0927-6505(19)30190-2 https://doi.org/10.1016/j.astropartphys.2019.102387 ASTPHY 102387
To appear in:
Astroparticle Physics
Received date: Accepted date:
5 August 2019 1 September 2019
Please cite this article as: Gregory Ryskin, Vanishing vacuum energy, Astroparticle Physics (2019), doi: https://doi.org/10.1016/j.astropartphys.2019.102387
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
Vanishing vacuum energy
time curvature, vacuum energy must enter the Einstein field equation as a source term, its stress-energy tensor being added to the stress-energy tensor of matter and non-gravitational fields.
Until 1998, most physicists believed that vacuum energy is zero. To quote J. Schwinger: “the vacuum is not Robert R. McCormick School of Engineering and only the state of minimum energy, it is the state of zero Applied Science, Northwestern University, Evanston, energy, zero momentum, zero angular momentum, zero Illinois 60208, United States; [email protected] charge, zero whatever” [1]. R. P. Feynman’s approach to quantum electrodynamics was born in an attempt to substantiate this belief: Feynman had hoped that a theKeywords: vacuum fluctuations; vacuum energy; cosmo- ory formulated directly in terms of particles and their logical constant; dark energy interactions would imply that vacuum energy is zero [2]. Schwinger developed his source theory for similar reasons [1]. Other approaches were tried as well, but no definitive results have emerged. (We do not review the Abstract literature here because our approach is fundamentally different.) The vacuum energy problem arises because quantum fields fluctuate even at the absolute zero of temperature. In 1998, dark energy was discovered by astronomers. These “zero-point” (T = 0) vacuum fluctuations have But dark energy may have a different origin, unrelated non-zero energy. Since in general relativity all types of to vacuum energy. energy generate the space-time curvature, vacuum en- In the present work, Feynman’s original idea – to deergy must enter the Einstein field equation as a source scribe the quantum vacuum in terms of virtual particles term. Here we find that vacuum energy is zero if and and their interactions – is combined with thermodynamonly if the vacuum equation of state embodies relativis- ics. (That this has not been done hitherto is perplexing.) tic invariance. Consequently, we find that vacuum energy is zero if and Gregory Ryskin
only if the vacuum equation of state embodies relativistic invariance.3 The vanishing of vacuum energy is the joint effect of relativistic invariance and the laws of therDue to the fluctuations of quantum fields, the vacuum modynamics. is filled with the momentarily-existing virtual particleantiparticle pairs of different kinds, interacting in all Relativistic invariance dictates the form of the stresspossible ways. Calculations based on this picture are energy tensor of the vacuum, viz., −ρvac gµν , where ρvac is in excellent agreement with the experimental results – the vacuum energy density, and gµν is the metric tensor the Lamb shift, the magnetic moment of the electron, [3]. In local Lorentz frame, the stress-energy tensor of etc. the vacuum is seen to be that of a perfect fluid, with energy density ρvac , pressure pvac , and the equation of The vacuum energy problem1 is inherent in this picture state [3] – it arises because quantum fields fluctuate at the absolute zero of temperature. These “zero-point” (T = 0) pvac = −ρvac , (1) vacuum fluctuations2 have non-zero energy. Since in general relativity all types of energy generate the spacewhich is a direct consequence of relativistic invariance. 1
Also known as the cosmological constant problem. Other conceivable contributions to cosmological constant – symmetry-breaking condensates, scalar fields, bare cosmological constant – are not considered here. 2 “Zero-point” means “at absolute zero”. Some authors use the word “vacuum” (with or without quotation marks) for thermal states, whose temperature is non-zero due to the Hawking-Unruh effect. Such states contain thermal (blackbody) radiation; they are not actually vacuum states.
The vacuum energy contribution to the Einstein field equation is characterized by thermodynamic parameters – energy density, temperature, and pressure. “A true vacuum is defined as a state of thermal equilibrium at a temperature of absolute zero” [4]. The fundamental 3
1
Thermodynamics does not provide the equation of state.
thermodynamic relation is X dE = T dS − pdV + µi dNi ,
with T = 0 and the equation of state (1), Eq. (3) is satisfied identically. Thus for vacuum
(2)
X
i
µi dNi = 0. (4) where E is energy, T is temperature, S is entropy, p is i pressure, V is volume, µi is the chemical potential of a particle, or a similar constituent entity, of type i, and Since dNi are mutually independent, Eq. (4) implies that Ni is the number of such entities in the system. Thermodynamic equations that follow from the fundaµi = 0 (5) mental relation (2), in conjunction with the equation of state (1), will be sufficient to calculate vacuum energy. for all pair types i and all possible values of N . The i These equations are exact, and the result will be exact chemical potentials of virtual particle-antiparticle pairs also. in the quantum vacuum are all equal to zero.
The calculation proceeds as follows. µi is defined as the chemical potential of a virtual particle-antiparticle pair4 of type i, viewed as a single thermodynamic component. This definition is the most convenient one when conservation laws force particles to appear and disappear in pairs. With any other definition, the conservation laws would impose compatibility conditions on Ni corresponding to different i; this would greatly complicate the analysis. With the present definition, the numbers of virtual pairs Ni are mutually independent and freely variable.
Note the crucial role played by relativistic invariance in the above calculation. With an equation of state different from Eq. (1), the result in Eq. (5) would not follow.5 It is important to emphasize that µi are absolute chemical potentials, i.e., their values are counted not from some fiducial reference point, but from the true zero. In most applications of thermodynamics, a particular value of energy, usually that of the lowest quantum state, is chosen as the zero of the single-particle energy scale. Since chemical potentials are defined by Eq. (2), this provides the reference point for the chemical potential as well [5]. Then the energy E in Eq. (2) is not the true energy of the system; to obtain the latter, one must add the energy of the lowest quantum state ε0 times the number of particles. (To obtain the absolute chemical potential, one must add ε0 .) If E were the true energy of the system, counted from the true zero, chemical potentials defined by Eq. (2) would be absolute chemical potentials. But most phenomena studied in physics do not provide means of establishing the absolute energy scale.
On physical grounds, µi are expected to be zero; this can be seen as follows. The chemical potential difference between two systems is equal to the amount of work that must be performed in order to transfer a particle, or similar entity, from system A to system B. That is, µB − µA = WA→B , and so µB = µA + WA→B . Consider now the appearance in vacuum of a virtual particleantiparticle pair. The pair appears spontaneously, via a quantum fluctuation due to the uncertainty principle. It does not carry chemical potential from another system (µA = 0), and no work of transfer is performed (WA→B = 0). Thus the chemical potential of a virtual The present case is different: Evac is that property of pair is expected to be zero. the vacuum that enters the Einstein field equation as To prove that µi = 0, one can proceed as follows. Con- a source term and generates the space-time curvature. sider the thermodynamic equation [5, Eq. 16.5] That is, Evac is the true energy of the quantum vac uum by definition: Unlike the rest of physics, gravity ∂E ∂p =T − p. (3) responds to – and thereby defines – the true value of en∂V T ∂T V ergy. Thus µi defined by Eq. (2) are absolute chemical P This equation is valid if and only if the term µi dNi potentials; the result µ = 0 must be taken literally. i i in Eq. (2) is equal to zero [5, Section 16]. For vacuum, 5 Equation of state is normally obtained from experiment, or from a microscopic model by methods of statistical mechanics, and is necessarily approximate. The present case is unique in that neither experimental data nor a microscopic model are needed; the equation of state follows from relativistic invariance, and is exact.
4
Virtual, off-shell particles carry chemical potential just as the on-shell particles do; in high-energy quantum field theory [6], chemical potential appears naturally in the propagators of virtual particles. By contrast, the concepts of mass and rest energy have no meaning for off-shell particles.
2
In the case of a degenerate Fermi gas, the exchange interaction between fermions (the Pauli repulsion) causes the chemical potential at T = 0 to increase with N . If other interactions are negligible, the result is µ(N ) ∝ (N/V )2/3 [5, Section 57].
An alternative derivation of this important result may be of interest. Consider volumes V and V + dV , where dV is a small variation. The total amount of vacuum energy in V is ρvac V ; in V + dV , it is ρvac (V + dV ). In Eq. (2) for this variation, replace dEvac on the left-hand side by ρvac dV . By virtue of the equation of state (1), Eq. (4) follows immediately. This again leads to Eq. (5).
But in the case of the quantum vacuum, the chemical potentials of virtual pairs are all equal to zero. Equation (9) then yields Evac = 0.
Consider now the general thermodynamic relation [7, Thus Evac = 0, and so Eq. 3.6] X ρvac = 0. E = T S − pV + µi Ni . (6)
(10)
i
The vacuum energy density is zero. The stress-energy In the present case, with T = 0 and µi = 0, Eq. (6) tensor of the vacuum, −ρvac gµν , is a zero tensor. becomes The above results have been obtained in a local Lorentz Evac = −pvac V. (7) frame, where Eqs. (1,2) are valid. A tensor that vanThis agrees precisely with the equation of state (1), ishes in one coordinate system, vanishes in all. Thus the stress-energy tensor of the vacuum is zero in all referdemonstrating the consistency of the present analysis. ence frames. The origin of dark energy must be sought To calculate Evac , proceed as follows. (The argument elsewhere. will follow closely the standard treatment of a degenerate quantum gas at absolute zero.) Consider a single- There is, of course, no conflict between the existence component system of interacting particles, or similar en- of vacuum fluctuations and their energies on the mitities, in volume V , kept at constant temperature T . croscale, and the vanishing of vacuum energy on the (Easily generalized to multiple species.) The system is macroscale. Vacuum energy is not simply the sum of built up, starting from zero and adding particles one by the bare energies of vacuum fluctuations, because interone. The free energy of the system is F = E − T S; for actions between fluctuations (virtual particles) cannot constant volume and temperature, dF = µdN . Thus be neglected. The above results indicate that the enerthe free energy of the system containing N particles is gies of the fluctuations and of their interactions sum up to zero. given by Z
The present treatment accounts for interactions autoµ(N 0 )dN 0 = µ(1)+µ(2)+µ(3)+...+µ(N ), (8) matically because it uses chemical potentials, which re0 flect interactions. The equation of state follows from relwhere µ(N ) is the chemical potential of the N th particle ativistic invariance; knowledge of the equation of state added to the system. (After this N th particle becomes allows direct calculation of chemical potentials. part of the system, all the particles in the system have the same chemical potential equal to µ(N ).) The inte- I thank Stephen L. Adler, John Joseph M. Carrasco, gral in Eq. (8) is an approximation, useful for large N ; Andr´e de Gouvˆea, Anthony J. Leggett and Rainer Weiss the correct expression is the sum, because particles are for comments on the manuscript. countable entities. F =
N
By definition of chemical potentials, their values reflect all interactions between particles (including the quantum exchange interactions). Generally, µ(N ) is a function of N . In exceptional cases, while still fully accounting for interactions, µ(N ) can be independent of N .
References [1] J. Schwinger, A report on quantum electrodynamics. In: J. Mehra (Ed.), The Physicist’s Conception of Nature (D. Reidel Publishing Company, Dordrecht, Holland, 1973), pp. 413-429 https://doi.org/10.1007/ 978-94-010-2602-4_20
At absolute zero, F = E, so the energy of the system is given by the same expression E = µ(1) + µ(2) + µ(3) + ... + µ(N ).
(9)
3
[2] F. Wilczek, The Lightness of Being: Mass, Ether, and the Unification of Forces (Basic Books, New York, 2008), pp. 83-84 [3] S. Weinberg, Cosmology (Oxford University Press, Oxford, UK, 2008), p. 40 [4] B. S. DeWitt, Quantum gravity, Scientific American 249 (Issue 6), 112-129 (1983) https://doi.org/10.1038/ scientificamerican1283-112 [5] L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1 (Third Edition, ButterworthHeinemann, Oxford, UK, 1980) [6] M. Le Bellac, Thermal Field Theory (Cambridge University Press, Cambridge, UK, 2000) [7] H. B. Callen, Thermodynamics and an Introduction to Thermostatistics (Second Edition, Wiley, New York, 1985)
4
Journal Pre-proof
Vanishing vacuum energy Gregory Ryskin PII: DOI: Reference:
S0927-6505(19)30190-2 https://doi.org/10.1016/j.astropartphys.2019.102387 ASTPHY 102387
To appear in:
Astroparticle Physics
Received date: Accepted date:
5 August 2019 1 September 2019
Please cite this article as: Gregory Ryskin, Vanishing vacuum energy, Astroparticle Physics (2019), doi: https://doi.org/10.1016/j.astropartphys.2019.102387
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
Vanishing vacuum energy
time curvature, vacuum energy must enter the Einstein field equation as a source term, its stress-energy tensor being added to the stress-energy tensor of matter and non-gravitational fields.
Until 1998, most physicists believed that vacuum energy is zero. To quote J. Schwinger: “the vacuum is not Robert R. McCormick School of Engineering and only the state of minimum energy, it is the state of zero Applied Science, Northwestern University, Evanston, energy, zero momentum, zero angular momentum, zero Illinois 60208, United States; [email protected] charge, zero whatever” [1]. R. P. Feynman’s approach to quantum electrodynamics was born in an attempt to substantiate this belief: Feynman had hoped that a theKeywords: vacuum fluctuations; vacuum energy; cosmo- ory formulated directly in terms of particles and their logical constant; dark energy interactions would imply that vacuum energy is zero [2]. Schwinger developed his source theory for similar reasons [1]. Other approaches were tried as well, but no definitive results have emerged. (We do not review the Abstract literature here because our approach is fundamentally different.) The vacuum energy problem arises because quantum fields fluctuate even at the absolute zero of temperature. In 1998, dark energy was discovered by astronomers. These “zero-point” (T = 0) vacuum fluctuations have But dark energy may have a different origin, unrelated non-zero energy. Since in general relativity all types of to vacuum energy. energy generate the space-time curvature, vacuum en- In the present work, Feynman’s original idea – to deergy must enter the Einstein field equation as a source scribe the quantum vacuum in terms of virtual particles term. Here we find that vacuum energy is zero if and and their interactions – is combined with thermodynamonly if the vacuum equation of state embodies relativis- ics. (That this has not been done hitherto is perplexing.) tic invariance. Consequently, we find that vacuum energy is zero if and Gregory Ryskin
only if the vacuum equation of state embodies relativistic invariance.3 The vanishing of vacuum energy is the joint effect of relativistic invariance and the laws of therDue to the fluctuations of quantum fields, the vacuum modynamics. is filled with the momentarily-existing virtual particleantiparticle pairs of different kinds, interacting in all Relativistic invariance dictates the form of the stresspossible ways. Calculations based on this picture are energy tensor of the vacuum, viz., −ρvac gµν , where ρvac is in excellent agreement with the experimental results – the vacuum energy density, and gµν is the metric tensor the Lamb shift, the magnetic moment of the electron, [3]. In local Lorentz frame, the stress-energy tensor of etc. the vacuum is seen to be that of a perfect fluid, with energy density ρvac , pressure pvac , and the equation of The vacuum energy problem1 is inherent in this picture state [3] – it arises because quantum fields fluctuate at the absolute zero of temperature. These “zero-point” (T = 0) pvac = −ρvac , (1) vacuum fluctuations2 have non-zero energy. Since in general relativity all types of energy generate the spacewhich is a direct consequence of relativistic invariance. 1
Also known as the cosmological constant problem. Other conceivable contributions to cosmological constant – symmetry-breaking condensates, scalar fields, bare cosmological constant – are not considered here. 2 “Zero-point” means “at absolute zero”. Some authors use the word “vacuum” (with or without quotation marks) for thermal states, whose temperature is non-zero due to the Hawking-Unruh effect. Such states contain thermal (blackbody) radiation; they are not actually vacuum states.
The vacuum energy contribution to the Einstein field equation is characterized by thermodynamic parameters – energy density, temperature, and pressure. “A true vacuum is defined as a state of thermal equilibrium at a temperature of absolute zero” [4]. The fundamental 3
1
Thermodynamics does not provide the equation of state.
thermodynamic relation is X dE = T dS − pdV + µi dNi ,
with T = 0 and the equation of state (1), Eq. (3) is satisfied identically. Thus for vacuum
(2)
X
i
µi dNi = 0. (4) where E is energy, T is temperature, S is entropy, p is i pressure, V is volume, µi is the chemical potential of a particle, or a similar constituent entity, of type i, and Since dNi are mutually independent, Eq. (4) implies that Ni is the number of such entities in the system. Thermodynamic equations that follow from the fundaµi = 0 (5) mental relation (2), in conjunction with the equation of state (1), will be sufficient to calculate vacuum energy. for all pair types i and all possible values of N . The i These equations are exact, and the result will be exact chemical potentials of virtual particle-antiparticle pairs also. in the quantum vacuum are all equal to zero.
The calculation proceeds as follows. µi is defined as the chemical potential of a virtual particle-antiparticle pair4 of type i, viewed as a single thermodynamic component. This definition is the most convenient one when conservation laws force particles to appear and disappear in pairs. With any other definition, the conservation laws would impose compatibility conditions on Ni corresponding to different i; this would greatly complicate the analysis. With the present definition, the numbers of virtual pairs Ni are mutually independent and freely variable.
Note the crucial role played by relativistic invariance in the above calculation. With an equation of state different from Eq. (1), the result in Eq. (5) would not follow.5 It is important to emphasize that µi are absolute chemical potentials, i.e., their values are counted not from some fiducial reference point, but from the true zero. In most applications of thermodynamics, a particular value of energy, usually that of the lowest quantum state, is chosen as the zero of the single-particle energy scale. Since chemical potentials are defined by Eq. (2), this provides the reference point for the chemical potential as well [5]. Then the energy E in Eq. (2) is not the true energy of the system; to obtain the latter, one must add the energy of the lowest quantum state ε0 times the number of particles. (To obtain the absolute chemical potential, one must add ε0 .) If E were the true energy of the system, counted from the true zero, chemical potentials defined by Eq. (2) would be absolute chemical potentials. But most phenomena studied in physics do not provide means of establishing the absolute energy scale.
On physical grounds, µi are expected to be zero; this can be seen as follows. The chemical potential difference between two systems is equal to the amount of work that must be performed in order to transfer a particle, or similar entity, from system A to system B. That is, µB − µA = WA→B , and so µB = µA + WA→B . Consider now the appearance in vacuum of a virtual particleantiparticle pair. The pair appears spontaneously, via a quantum fluctuation due to the uncertainty principle. It does not carry chemical potential from another system (µA = 0), and no work of transfer is performed (WA→B = 0). Thus the chemical potential of a virtual The present case is different: Evac is that property of pair is expected to be zero. the vacuum that enters the Einstein field equation as To prove that µi = 0, one can proceed as follows. Con- a source term and generates the space-time curvature. sider the thermodynamic equation [5, Eq. 16.5] That is, Evac is the true energy of the quantum vac uum by definition: Unlike the rest of physics, gravity ∂E ∂p =T − p. (3) responds to – and thereby defines – the true value of en∂V T ∂T V ergy. Thus µi defined by Eq. (2) are absolute chemical P This equation is valid if and only if the term µi dNi potentials; the result µ = 0 must be taken literally. i i in Eq. (2) is equal to zero [5, Section 16]. For vacuum, 5 Equation of state is normally obtained from experiment, or from a microscopic model by methods of statistical mechanics, and is necessarily approximate. The present case is unique in that neither experimental data nor a microscopic model are needed; the equation of state follows from relativistic invariance, and is exact.
4
Virtual, off-shell particles carry chemical potential just as the on-shell particles do; in high-energy quantum field theory [6], chemical potential appears naturally in the propagators of virtual particles. By contrast, the concepts of mass and rest energy have no meaning for off-shell particles.
2
In the case of a degenerate Fermi gas, the exchange interaction between fermions (the Pauli repulsion) causes the chemical potential at T = 0 to increase with N . If other interactions are negligible, the result is µ(N ) ∝ (N/V )2/3 [5, Section 57].
An alternative derivation of this important result may be of interest. Consider volumes V and V + dV , where dV is a small variation. The total amount of vacuum energy in V is ρvac V ; in V + dV , it is ρvac (V + dV ). In Eq. (2) for this variation, replace dEvac on the left-hand side by ρvac dV . By virtue of the equation of state (1), Eq. (4) follows immediately. This again leads to Eq. (5).
But in the case of the quantum vacuum, the chemical potentials of virtual pairs are all equal to zero. Equation (9) then yields Evac = 0.
Consider now the general thermodynamic relation [7, Thus Evac = 0, and so Eq. 3.6] X ρvac = 0. E = T S − pV + µi Ni . (6)
(10)
i
The vacuum energy density is zero. The stress-energy In the present case, with T = 0 and µi = 0, Eq. (6) tensor of the vacuum, −ρvac gµν , is a zero tensor. becomes The above results have been obtained in a local Lorentz Evac = −pvac V. (7) frame, where Eqs. (1,2) are valid. A tensor that vanThis agrees precisely with the equation of state (1), ishes in one coordinate system, vanishes in all. Thus the stress-energy tensor of the vacuum is zero in all referdemonstrating the consistency of the present analysis. ence frames. The origin of dark energy must be sought To calculate Evac , proceed as follows. (The argument elsewhere. will follow closely the standard treatment of a degenerate quantum gas at absolute zero.) Consider a single- There is, of course, no conflict between the existence component system of interacting particles, or similar en- of vacuum fluctuations and their energies on the mitities, in volume V , kept at constant temperature T . croscale, and the vanishing of vacuum energy on the (Easily generalized to multiple species.) The system is macroscale. Vacuum energy is not simply the sum of built up, starting from zero and adding particles one by the bare energies of vacuum fluctuations, because interone. The free energy of the system is F = E − T S; for actions between fluctuations (virtual particles) cannot constant volume and temperature, dF = µdN . Thus be neglected. The above results indicate that the enerthe free energy of the system containing N particles is gies of the fluctuations and of their interactions sum up to zero. given by Z
The present treatment accounts for interactions autoµ(N 0 )dN 0 = µ(1)+µ(2)+µ(3)+...+µ(N ), (8) matically because it uses chemical potentials, which re0 flect interactions. The equation of state follows from relwhere µ(N ) is the chemical potential of the N th particle ativistic invariance; knowledge of the equation of state added to the system. (After this N th particle becomes allows direct calculation of chemical potentials. part of the system, all the particles in the system have the same chemical potential equal to µ(N ).) The inte- I thank Stephen L. Adler, John Joseph M. Carrasco, gral in Eq. (8) is an approximation, useful for large N ; Andr´e de Gouvˆea, Anthony J. Leggett and Rainer Weiss the correct expression is the sum, because particles are for comments on the manuscript. countable entities. F =
N
By definition of chemical potentials, their values reflect all interactions between particles (including the quantum exchange interactions). Generally, µ(N ) is a function of N . In exceptional cases, while still fully accounting for interactions, µ(N ) can be independent of N .
References [1] J. Schwinger, A report on quantum electrodynamics. In: J. Mehra (Ed.), The Physicist’s Conception of Nature (D. Reidel Publishing Company, Dordrecht, Holland, 1973), pp. 413-429 https://doi.org/10.1007/ 978-94-010-2602-4_20
At absolute zero, F = E, so the energy of the system is given by the same expression E = µ(1) + µ(2) + µ(3) + ... + µ(N ).
(9)
3
[2] F. Wilczek, The Lightness of Being: Mass, Ether, and the Unification of Forces (Basic Books, New York, 2008), pp. 83-84 [3] S. Weinberg, Cosmology (Oxford University Press, Oxford, UK, 2008), p. 40 [4] B. S. DeWitt, Quantum gravity, Scientific American 249 (Issue 6), 112-129 (1983) https://doi.org/10.1038/ scientificamerican1283-112 [5] L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1 (Third Edition, ButterworthHeinemann, Oxford, UK, 1980) [6] M. Le Bellac, Thermal Field Theory (Cambridge University Press, Cambridge, UK, 2000) [7] H. B. Callen, Thermodynamics and an Introduction to Thermostatistics (Second Edition, Wiley, New York, 1985)
4