Advertisement

Il Nuovo Cimento B (1971-1996)

, Volume 82, Issue 2, pp 169–191 | Cite as

Quantum thermodynamics. A new equation of motion for a single constituent of matter

  • G. P. Beretta
  • E. P. Gyftopoulos
  • J. L. Park
  • G. N. Hatsopoulos
Article

Summary

A novel nonlinear equation of motion is proposed for quantum systems consisting of a single elementary constituent of matter. It is satisfied by pure states and by a special class of mixed states evolving unitarily. But, in general, it generates a nonunitary evolution of the state operator. It keeps the energy invariant and causes the entropy to increase with time until the system reaches a state of equilibrium or a limit cycle.

PACS. 03.65. Quantum theory quantum mechanics 

Квантовая термодинамика. Новое уравнение движения для однокомпонентного вещества

Резюме

Для квантовых систем, состоящих из вещества, представляюего одну элементарную компоненту, предлагается новое нелинейное уравнение движения. Уравнение удовлетворяется для чистых состояний и для специального класса смешанных состояний. В общем случае, это уравнение генерирует неунитарную эволюцию оператора состояния. Это уравнение сохраняет энергию инвариантной, вызывает увелиыение энтропии со временем, пока система не достигнет состояния равновесия.

Riassunto

Si propone una nuova equazione di evoluzione per sistemi quantistici composti da un singolo costituente materiale elementare. L'equazione è soddisfatta dall'evoluzione unitaria degli stati puri e di una sottoclasse di stati misti. Ma, in generale, essa genera un'evoluzione non unitaria dell'operatore di stato. L'equazione mantiene costante il valor medio dell'energia e causa aumenti di entropia finché il sistema non raggiunge uno stato di equilibrio oppure un ciclo limite.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. (1).
    See,e.g.,J. von Neumann:Mathematical Foundations of Quantum Mechanics, English translation (Princeton University Press, Princeton, N. J., 1955);E. C. Kemble Phys. Rev.,56, 1013, 1146 (1939);U. Fano:Rev. Mod. Phys.,29, 74 (1957);E. T. Jaynes:Phys. Rev.,106, 620 (1957);108, 171 (1957);W. Band:Am. J. Phys.,26, 440, 540 (1958);E. C. G. Sudarshan, P. M. Mathews andJ. Ran:Phys. Rev.,121, 920 (1961);R. Jancel:Foundations of Classical and Quantum Statistical Mechanics, (Pergamon Press, Oxford, 1969);J. Mehra andE. C. G. Sudarshan:Nuovo Cimento B,11, 215 (1972).zbMATHGoogle Scholar
  2. (2).
    A. Wehrl:Rev. Mod. Phys.,50, 221 (1978).MathSciNetADSCrossRefGoogle Scholar
  3. (3).
    G. N. Hatsopoulos andE. P. Gyftopoulos:Found. Phys.,6, 15, 127, 439, 561, (1976).MathSciNetADSCrossRefGoogle Scholar
  4. (4).
    A recent review of these attempts is given byJ. L. Park andR. F. Simmons, jr:The knots of thermodynamics, inOld and New Questions in Physics, Cosmology, Philosophy and Theoretical Biology: Essays in Honor of Wolfgang Jourgrau, edited byA. van der Merwe (Plenum Press, New York, N. Y., 1983). See alsoJ. L. Park andW. Band:Found. Phys.,7, 813 (1977).Google Scholar
  5. (5).
    R. F. Simmons, jr. andJ. L. Park:Found. Phys.,11, 297 (1981).MathSciNetADSCrossRefGoogle Scholar
  6. (6).
    The nondynamical postulates are equivalent to those used in ref. (3).MathSciNetADSCrossRefGoogle Scholar
  7. (7).
    The concept of unambiguous preparation has been defined in ref. (3).MathSciNetADSCrossRefGoogle Scholar
  8. (9).
    G. P. Beretta: Thesis, Sc. D. M.I.T. (1981), unpublished.Google Scholar
  9. (11).
    See,e.g.,E. F. Beckenbach andR. Bellman,Inequalities (Springer-Verlag, New York, N. Y., 1965), p. 59–60; see alsoG. P. Beretta: inFrontiers of Nonequilibrium Statistical Physics, edited byG. T. Moore andM. O. Scully (Plenum Press, New York, N.Y., in press).CrossRefGoogle Scholar
  10. (12).
    Fora, b real scalars,A, B linear operators andA °,B ° their adjoints, we have (A 1+A2‖B)=(A1‖B)+(A2‖B), (A‖B1+B2)=(A‖B2)+(A‖B2), (aA‖B)=a(A‖B), (A‖bB)=b(A‖B), and(A‖A)=Tr(A °A)>0 forA≠0.Google Scholar
  11. (13).
    This requirement is generally nontrivial because one expects that some generators of the motion (in particularH) may be unbounded. For the definition of a well-defined (densely defined) operator and that of commutativity for unbounded operators see,e.g.,M. Reed andB. Simon:Methods of Modern Mathematical Physics, Vol.1:Functional Analysis (Academic Press, New York, N.Y., 1972), Chapt. I and VIII. For the extension of the definition of Tr(ϱA) to unbounded operatorsA, see,e.g.,A. Jamiolkowski:Rep. Math. Phys.,10, 267 (1976);H. Araki andE. H. Lieb:Commun. Math. Phys.,18, 160 (1970). For the additional technical conditions that must be imposed on the generators of the motion in order for theorem 10 to hold, seeW. Ochs andW. Bayer:Z. Naturforsch.,28a, 1571 (1973).Google Scholar
  12. (14).
    See,e.g.,E. B. Davies:Quantum Theory of Open Systems (Academic Press, New York, N.Y., 1976), p. 82–83.zbMATHGoogle Scholar
  13. (15).
    W. Ochs andW. Bayer: ref. (13); see alsoA. Katz:Principles of Statistical Mechanics (W. H. Freeman, San Francisco, Cal., 1967), p. 45–51.MathSciNetADSGoogle Scholar
  14. (16).
    See alsoG. N. Hatsopoulos andJ. H. Keenan:Principles of General Thermodynamics (Wiley and Sons, New York, N.Y., 1965), p. 30, 361.Google Scholar
  15. (17).
    A. Liapunoff:Problème Général de la Stabilité du Mouvement, inAnnals of Mathematics Studies, Vol.17 (Princeton University Press, Princeton, N.J., 1949), p. 210–213.Google Scholar
  16. (18).
    A referee suggested two other aspects of our theory that need further investigation and clarification. The first is to study the invariance properties of the new equation of motion under the usual symmetry groups. The second is to explore the implications of the nonlinear equation on the quantum theory of measurement.Google Scholar
  17. (19).
  18. (21).
    The continuity of operator ϱ ln ϱ is discussed,e.g., byA. Wehrl: ref. (2), p. 251.MathSciNetADSCrossRefGoogle Scholar
  19. (22).
    See,e.g.,E. L. Ince:Ordinary Differential Equations (Dover, New York, N.Y., 1956), p. 62–72.Google Scholar

Copyright information

© Società Italiana di Fisica 1984

Authors and Affiliations

  • G. P. Beretta
    • 1
  • E. P. Gyftopoulos
    • 1
  • J. L. Park
    • 2
  • G. N. Hatsopoulos
    • 3
  1. 1.Massachusetts Institute of TechnologyCambridge
  2. 2.Washington State UniversityPullman
  3. 3.Thermo Electron CorporationWaltham

Personalised recommendations