Abstract
Generalized master equations correspond to different kinds of cumulants. Here, we discuss the recently introduced non-crossing cumulants from a physical point of view and propose the corresponding integro-differential master equation as a new type of equation for a self-consistent treatment of memory effects. We prove the cluster property of the non-crossing cumulants, and show that the Gaussian approximation of our equation is given by a random matrix process. As an instructive example for our expansion formula we treat the random frequency-modulated oscillator.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson, P.W.: J. Phys. Soc. Jpn.9, 316 (1954)
Kubo, R.: J. Phys. Soc. Jpn.9, 935 (1954)
Mori, H.: Prog. Theor. Phys.33, 127 (1965)
Nakajima, S.: Prog. Theor. Phys.20, 948 (1958)
Zwanzig, R.: J. Chem. Phys.33 1338 (1960)
Tokuyama, M., Mori, H.: Prog. Theor. Phys.55, 411 (1976)
Hashitsume, N., Shibata, F., Shingu, M.: J. Stat. Phys.17, 155 (1977)
Shibata, F., Takahashi, Y., Hashitsume, N.: J. Stat. Phys.17, 171 (1977)
Hänggi, P., Thomas, H.: Z. Phys. B26, 85 (1977)
Grabert, H., Talkner, P., Hänggi, P.: Z. Phys. B26, 389 (1977)
Kubo, R.: J. Math. Phys.4, 174 (1963)
Kampen, N.G. van: Physica74, 215, 239 (1974)
Terwiel, R.H.: Physica74, 248 (1974)
Chaturvedi, S., Shibata, F.: Z. Phys. B35, 297 (1979)
Speicher, R.: Probab. Theory Relat. Fields84, 141 (1990)
Speicher, R.: SFB-preprint 654, Heidelberg 1991 (to appear in: Quantum Probability and Related Topics. Vol. VIII, Accardi, L. (ed.). Singapore: World Scientific 1993
Speicher, R.: SFB-preprint 706, Heidelberg 1992 (to be published)
Voiculescu, D.: In: Lecture Notes in Mathematics. Vol. 1132, Araki, H., Moore, C.C., Stratila, S.V., Voiculescu, D.V., Arsene, G. (eds.), pp. 556–588. Berlin, Heidelberg, New York: Springer 1985
Voiculescu, D.: J. Funct. Anal.66, 323 (1986)
Voiculescu, D.: Invent. Math.104, 201 (1991)
Kümmerer, B., Speicher, R.: J. Funct. Anal.103, 372 (1992)
Kraichnan, R.H.: J. Math. Phys.2, 124 (1961)
Frisch, U., Bourret, R.: J. Math. Phys.11, 364 (1970)
Kraichnan, R.H.: Phys. Rev.109, 1407 (1958); J. Fluid Mech.5, 497 (1959)
Kraichnan, R.H.: J. Math. Phys.3, 475, 496 (1962)
Shibata, F., Arimitsu, T.: J. Phys. Soc. Jpn.49, 891 (1980)
Hegerfeldt, G.C., Schulze, H.: J. Stat. Phys.51, 691 (1988)
Waldenfels, W. von: In: Lecture Notes in Mathematics. Vol. 296, pp. 19–69. Berlin, Heidelberg, New York: Springer 1973; Lecture Notes in Mathematics. Vol. 465, pp. 565–588. Berlin, Heidelberg, New York: Springer 1975
Maruyama, K., Shibata, F.: Physica A149, 447 (1988)
Wigner, E.P.: Ann. Math.62, 548 (1955)
Wigner, E.P.: SIAM Rev.9, 1 (1967)
Bożejko, M., Speicher, R.: Commun. Math. Phys.137, 519 (1991)
Greenberg, O.W.: Phys. Rev. D43, 4111 (1991)
Chvosta, P.: Physica A178, 168 (1991); J. Phys. B21, 3155 (1988)
Sompolinsky, H., Zippelius, A.: Phys. Rev. B,25, 6860 (1982)
Kirkpatrick, S., Sherrington, D.: Phys. Rev. B17, 4384 (1978)
Horner, H.: Z. Phys. B57, 29 (1984)
Kühn, R., Bös, S., Hemmen, J.L. van: Phys. Rev. A43, 2084 (1991)