Time Reversal in Dissipative Systems


An explicit formulation of the time reversal operator K with the property
$$K{\text{ }}q{\text{ }}{K^{ - 1}}{\text{ = q; }}K{\text{ }}p{\text{ }}{K^{ - 1}}{\text{ = - p; K }}\vec \sigma {\text{ }}{{\text{K}}^{{\text{ - 1}}}}{\text{ = - }}\vec \sigma $$
that it preserve the sign of the position operator q, and reverse the signs of the momentum operator p and the spin \(\vec{\sigma }\) was made by Wigner (1932, 1959).1


Master Equation Time Reversal Dissipative System Detailed Balance Phase Fluctuation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Wigner, E. P. (1932), “The Time Reversal Operation in Quantum Mechanics,” Nachr. Ges. Wiss. Gottingen Math-Phys. Klasse (546–559).Google Scholar
  2. Wigner, E. P. (1959) Group Theory and Its Applications to Quantum Mechanics of Atomic Spectra, Academic Press.Google Scholar
  3. 2.
    Lax, M. (1962), “The Influence of Time Reversal On Selection Rules Connecting Different Points in the Brillouin Zone”. Pp. 395–402 of Proceedings of International Conference on Physics of Semiconductors, Institute of Physics and Physical Society, London.Google Scholar
  4. 3.
    Lax, M. (1965), “Subgroup Techniques in Crystal and Molecular Physics”, Phys. Rev. 138, A793 - A802.MathSciNetADSCrossRefGoogle Scholar
  5. 4.
    Lax, M. (1974), Symmetry Principles in Solid State and Molecular Physics, Wiley.Google Scholar
  6. 5.
    Onsager, Lars (1931), “Reciprocal Relations in Irreversible Processes”, Phys. Rev. 37, (405–426); 38, (2265–2279).Google Scholar
  7. 6.
    Lax, M. (1966), Classical Noise III: “Nonlinear Markoff Processes”, Revs. Modern Physics 38, 359–379 hereafter referred to as I II.Google Scholar
  8. Lax, M. (1968), “Fluctuation and Coherence Phenomena” 1966 Brandeis Summer Institute in Theoretical Physics, in Statistical Physics, Phase Transitions and Superfluidity, Vol. 2, edited by M. Chretien, E. P. Gross and S. Deser, Gordon and Breach, pp. 269–478.Google Scholar
  9. 7.
    Graham, R. and Haken, H. (1971), “Generalized Thermodynamic Potential for Markoff Systems in Detailed Balance and Far from Thermal Equilibrium”, Z. Physik 243, 289–302.MathSciNetADSCrossRefGoogle Scholar
  10. 8.
    Risken, H. (1972), “Solutions of the Fokker-Planck Equation in Detailed Balance”, Z. Physik 251, 231–243.ADSCrossRefGoogle Scholar
  11. 9.
    Lax, M. and Zwanziger, M. (1973), “Exact Photocount Statistics: Lasers Near Threshold”, Phys. Rev. A7, 750–771. See Appendix A.Google Scholar
  12. 10.
    Pawula, R. F. (1967), “Approximation of the Linear Boltzmann Equation by the Fokker-Planck Equation”, Phys. Rev. 162, 186–188.Google Scholar
  13. 11.
    Morse, P. M. and Feshbach, H. (1953), Methods of Theoretical Physics, McGraw-Hill.Google Scholar
  14. 12.
    Lax, M. and Louisell, W. H. (1967), “Quantum Noise IX: Quantum Fokker-Planck Solution For Laser Noise”. IEEE J. Quant. Electronics QE-3, 47–58 Appendix A.Google Scholar
  15. 13.
    Haken, H. (1970), “Laser Theory” Encyclopedia of Physics vol. XXV/2C, Springer, Berlin.Google Scholar
  16. 14.
    Risken, H. (1970), “Statistical Properties of Laser Light” In Progress in Optics Vol. VIII, pp. 239–294, ed. E. Wolf, North Holland, Amsterdam.Google Scholar
  17. 15.
    Risken, H. and Vollmer, H. D. (1967), “Correlation Function of the Amplitude and of the Intensity Fluctuation Near Threshold”, Z. Physik 191, 301–312.Google Scholar
  18. 16.
    Hempstead, R. D. and Lax, M. (1967), Classical Noise VI, Noise in Self-Sustained Oscillators Near Threshold“, Phys. Rev. 161, 350–366.Google Scholar
  19. 17.
    Stratanovich, R. L. (1963), “Topics in the theory of Random Noise”, Vol. 1, pp. 78, 79, Gordon and Breach.Google Scholar
  20. 18.
    Gamo, H., Grace, R. E. and Walter, T. J. (1968), “Statistical Analysis of Intensity Fluctuations in Single Mode Laser Radiation near the Oscillation Threshold”, IEEE J. Quantum Electron, QE-4, 344.Google Scholar
  21. 19.
    Davidson, F. and Mandel, L. (1967), “Correlation Measurements of Laser Beam Fluctuations Near Threshold”, Phys. Lett. A25, 700–701.Google Scholar
  22. 20.
    Arecchi, F. T., Rodari, G. S., and Sona, A. (1967), “Statistics of The Laser Radiation At Threshold”, Phys. Lett. A25, 59–60.Google Scholar
  23. Arecchi, F. T., Giglio, M. and Sona, A. (1967), “Dynamics of the Laser Radiation At Threshold”, Phys. Lett. A25, 341–342.Google Scholar
  24. 21.
    Lax, M. (1966b), Tokyo Summer Lectures in Theoretical Physics, Part I Dynamical Process in Solid State Optics, edited by R. Kubo and H. Kamimura. W. A. Benjamin Inc. Section 22.Google Scholar
  25. 22.
    Pound, R. V. (1957), “Spontaneous Emission and the Noise Figure of Maser Amplifiers”, Ann. Phys. ( N.Y. ) 1, 24–32.Google Scholar
  26. 23.
    Schawlow, A. L. and Townes, C. H. (1958), “Infrared and Optical Masers”, Phys. Rev. 112, 1940–1949.Google Scholar
  27. 24.
    Weber, J. (1959), “Masers”, Revs. Modern Phys. 31, 681–710; erratum 32, 1033 (1960).ADSGoogle Scholar
  28. 25.
    Haken, H. (1966), “Theory of Intensity and Phase Fluctuations of a Homogeneously Broadened Laser” Z. Physik 190, 327–356.MathSciNetADSCrossRefGoogle Scholar
  29. 26.
    Sauermann, H. (1965), “Dissipation and Fluctuations in einem Zwei-Niveau-System”, Z. Physik 188, 480–505.ADSCrossRefGoogle Scholar
  30. 27.
    Risken, H., Schmid, C. and Weidlich, W. (1966), “Fokker-Planck Equation for atoms and light mode in a laser model with quantum-mechanically determined dissipation and fluctuation coefficients,” Phys. Letters 20, 489–491.ADSCrossRefGoogle Scholar
  31. 28.
    Lax, M. (1965b), Classical Noise V: “Noise in Self-Sustained Oscillators”, Phys. Rev. 160, 290–307 referred to later as V.Google Scholar
  32. 29.
    Gerhardt, H., Welling, H. and Güttner, A. (1972), “Measurements of the laser line width due to quantum phase and quantum amplitude noise above and below threshold,” Z. Physik 253, 113–126.ADSCrossRefGoogle Scholar
  33. 30.
    Grossman, S. and Richter, P. H. (1971), “Laser Threshold and Nonlinear Landau Fluctuation Theory of Phase Transitions”, Z. Physik 242, 458–475.ADSCrossRefGoogle Scholar
  34. 31.
    Risken, H. and Seybold, R. (1972), “Linewidth of a Detuned Single Mode Laser Near Threshold”, Physics Letters 38A, 63–64.CrossRefGoogle Scholar
  35. 32.
    Lax, M. (1966c), “Quantum Noise V: Phase Noise in a Homogeneously Broadened Maser,” in Physics of Quantum Electronics, P. L. Kelley, B. Lax and P. E. Tanenwald, eds. ( McGraw-Hill Book Co., Inc., N.Y. ) pp. 735–747.Google Scholar

Copyright information

© Plenum Press, New York 1980

Authors and Affiliations

  • M. Lax
    • 1
    • 2
  1. 1.Physics DepartmentCity College of the City University of New YorkNew YorkUSA
  2. 2.Bell LaboratoriesMurray HillUSA

Personalised recommendations