Communications in Mathematical Physics

, Volume 39, Issue 2, pp 91–110 | Cite as

Markovian master equations

  • E. B. Davies


We give a rigorous proof that under certain technical conditions the memory effects in a quantum-mechanical master equation become negligible in the weak coupling limit. This is sufficient to show that a number of open systems obey an exponential decay law in the weak coupling limit for a rescaled time variable. The theory is applied to a fairly general finite dimensional system weakly coupled to an infinite free heat bath.


Neural Network Open System Complex System Time Variable Nonlinear Dynamics 
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.
    Prigogine, I.: The statistical interpretation of nonequilibrium entropy. In: Thirring, W., Cohen, E. G. D. (Eds.): The Boltzmann equation, theory and applications. Berlin-Heidelberg-New York: Springer 1973Google Scholar
  2. 2.
    Haake, F.: Statistical treatment of open systems by generalised master equations. Springer tracts in modern physics, Vol.66. Berlin-Heidelberg-New York: Springer 1973Google Scholar
  3. 3.
    Kato, T.: Perturbation theory for linear operators. Berlin-Heidelberg-New York: Springer 1966Google Scholar
  4. 4.
    Davies, E. B.: Commun. math. Phys.33, 171–186 (1973)Google Scholar
  5. 5.
    Sewell, G. L.: Relaxation, amplification and the K.M.S. conditions (to appear)Google Scholar
  6. 6.
    Davies, E. B.: Dynamics of a multi-level Wigner-Weisskopf atom. J. math. Phys. (To appear)Google Scholar
  7. 7.
    Papanicolaou, G. C., Varadhan, S. R. S.: Comm. Pure. Appl. Math.26, 497–524 (1973)Google Scholar
  8. 8.
    Bongaarts, P. J. M., Fannes, M., Verbeure, A.: A remark on ergodicity, dissipativity, return to equilibrium, (To appear)Google Scholar
  9. 9.
    Pulè, J. V.: The Bloch equation. Commun. Math. Phys. (To appear) (1974)Google Scholar
  10. 10.
    Balslev, E., Verbeure, A.: Commun. math. Phys.7, 55–76 (1968)Google Scholar
  11. 11.
    Presutti, E., Scacciatelli, E., Sewell, G. L., Wanderlingh, F.: J. Math. Phys.13, 1085–1098 (1972)Google Scholar
  12. 12.
    Chernoff, P. R.: J. Funct. Anal.2 (1968) 238–242Google Scholar
  13. 13.
    Davies, E. B.: Commun. math. Phys.19, 83–105 (1970)Google Scholar
  14. 14.
    Davies, E. B.: Z. Wahrscheinlichkeitsch.23, 261–273 (1972)Google Scholar
  15. 15.
    Hepp, K., Lieb, E. H.: Phase transitions in reservoir-driven open systems with applications to lasers and superconductors. Helv. Phys. Acta46, 573–603 (1973)Google Scholar

Copyright information

© Springer-Verlag 1974

Authors and Affiliations

  • E. B. Davies
    • 1
  1. 1.Mathematical InstituteOxfordEngland

Personalised recommendations