Il Nuovo Cimento B (1971-1996)

, Volume 87, Issue 1, pp 77–97 | Cite as

Quantum thermodynamics. A new equation of motion for a general quantum system

  • G. P. Beretta
  • E. P. Gyftopoulos
  • J. L. Park


A novel nonlinear equation of motion is proposed for a general quantum system consisting of more than one distinguishable elementary constituent of matter. In the domain of idempotent quantummechanical state operators, it is satisfied by all unitary evolutions generated by the Schrödinger equation. But, in the broader domain of nonidempotent state operators not contemplated by conventional quantum mechanics, it generates a generally nonunitary evolution, 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 общей квантовой системы


Предлагается новое нелинейное уравнение движения для общей квантовой системе, состоящей из более чем одной элементарной составляющей вещества. В области идемпотентных квантовомеханических операторов состояний удовлетворяются все унитарные эволюции, генерированные уравнением Шредингера. В более широкой области неидемпотентных операторов состояний, не рассматриваемой в обычной квантовой механике, возникает неунитарная эволюция, при этом энергия является инвариантной, а энтропия увеличивается со временем, пока система достигает состояния равновесия или предельного цикла.


Si propone una nuova equazione non lineare di evoluzione per un sistema quantistico generale costituito da piú costituenti materiali elementari distinguibili. Nel dominio degli operatori di stato idempotenti della meccanica quantistica, l'equazione è soddisfatta da tutte le evoluzioni unitarie generate dall'equazione di Schrödinger. Ma, nel dominio piú ampio degli operatori di stato non idempotenti, non contemplati dalla meccanica quantistica convenzionale, l'equazione genera un'evoluzione generalmente non unitaria, mantiene costante il valor medio dell'energia e causa aumenti di entropia finchè il sistema non raggiunge uno stato di equilibrio oppure un ciclo limite.


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  1. (1).
    G. P. Beretta, E. P. Gyftopoulos, J. L. Park andG. N. Hatsopoulos:Nuovo Cimento B,82, 169 (1984).MathSciNetADSCrossRefGoogle Scholar
  2. (2).
    R. Jancel:Foundations of Classical and Quantum Statistical Mechanics (Pergamon Press, Oxford, 1969).Google Scholar
  3. (3).
    G. N. Hatsopoulos andE. P. Gyftopoulos:Found. Phys.,6, 15, 127, 439, 561 (1976).MathSciNetADSCrossRefGoogle Scholar
  4. (4).
    If |ϕ> is an eigenvector of the idempotent quantum-mechanical state operatorP (P 2=P), such thatP|ϕ>=|ϕ> and <ϕ|ϕ>=1, thenP=|ϕ><ϕ| and |ϕ> is the quantum-mechanical state vectorGoogle Scholar
  5. (5).
    See,e. g.,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);R. S. Ingarden andA. Kossakowski:Ann. Phys.,89, 451 (1975);J. L. Park andW. Band:Found. Phys.,8, 239 (1978);A. Wehrl:Rev. Mod. Phys.,50, 221 (1978), and references therein.zbMATHGoogle Scholar
  6. (6).
    See,e.g.,W. H. Louisell:Quantum Statistical Properties of Radiation (Wiley, New York, N. Y., 1973);E. B. Davies:Commun. Math. Phys.,39, 91 (1974);P. Pearle:Phys. Rev. D,13, 857 (1976);I. Bialynicki-Birula andJ. Myclelski:Ann. Phys. (N. Y.),100, 62 (1976);G. Lindblad:Commun. Math. Phys.,48, 119 (1976);V. Gorini, A. Kossakowski andE. C. G. Sudarshan:J. Math. Phys. (N. Y.),17, 821 (1976);V. Gorini, A. Frigerio, M. Verri, A. Kossakowski andE. C. G. Sudarshan:Rep. Math. Phys.,13, 149 (1978);R. F. Simmons jr. andJ. L. Park:Found. Phys.,11, 297 (1981);N. Gisin andC. Piron:Lett. Math. Phys.,5, 379 (1981);P. Caldirola andL. A. Lugiato:Physica A,116, 248 (1982), and references therein.Google Scholar
  7. (7).
    G. P. Beretta: Sc. D. Thesis, M.I.T. (1981), unpublished.Google Scholar
  8. (8).
    See alsoG. P. Beretta: inFrontiers of Nonequilibrium Statistical Physics, edited byG. T. Moore andM. O. Scully (Plenum Press, New York, N. Y., 1985), in press.Google Scholar
  9. (9).
    See footnote (10), inG. P. Beretta, E. P. Gyftopoulos, J. L. Park andG. N. Hatsopoulos:Nuovo Cimento B,82, 168 (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  10. (10).
    See alsoG. P. Beretta:Int. J. Theor. Phys.,24, 119 (1985), where existence and uniqueness of solutions in both forward and backward time are proved rigorously for a single isolated two-level system.MathSciNetCrossRefGoogle Scholar
  11. (12).
    A. Katz:Principles of Statistical Mechanics (W. H. Freeman, San Francisco, Cal., 1967), p. 45.Google Scholar
  12. (13).
    A system is said to be strictly isolated if and only if it interacts with no other system, and is at some time and, hence, at all times in an independent state when viewed as a subsystem of any conceivable composite system containing it.Google Scholar
  13. (14).
    H. Araki andE. Lieb:Commun. Math. Phys.,18, 160 (1970). See alsoA. Wehrl:Rev. Mod. Phys.,50, 242 (1978).MathSciNetADSCrossRefzbMATHGoogle Scholar

Copyright information

© Società Italiana di Fisica 1985

Authors and Affiliations

  • G. P. Beretta
    • 1
  • E. P. Gyftopoulos
    • 1
  • J. L. Park
    • 2
  1. 1.Massachusetts Institute of TechnologyCambridge
  2. 2.Washington State UniversityPullman

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