Abstract.
The development of efficient strategies for the treatment of the dynamics of relevant observables in complex quantum systems plays a decisive role in the theory of quantum relaxation and transport behavior. Here we discuss the most important tools that are based on the projection operator techniques of nonequilibrium statistical mechanics. For both the Nakajima-Zwanzig and the time-convolutionless projection operator technique we derive the equations of motion for a set of relevant observables and develop explicit expressions in second and fourth order of the corresponding perturbation expansions. We also discuss the Hilbert space average method which is based on the idea of a best guess of conditional quantum expectations determined by an average over a suitable region in the underlying Hilbert space, and relate this method to the projection operator technique.
Similar content being viewed by others
References
H.-P. Breuer, F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2007)
M. Michel, G. Mahler, J. Gemmer, Phys. Rev. Lett. 95, 180602 (2005)
J. Gemmer, M. Michel, G. Mahler, Quantum Thermodynamics, Lect. Notes Phys. 657 (Springer, Berlin, Heidelberg, 2004)
J. Gemmer, M. Michel, Europhys. Lett. 73, 1 (2006)
M. Michel, R. Steinigeweg, H. Wimer, Correlated projection superoperators in relaxation and transport investigations, Eur. Phys. J. Special Topics 151, 13 (2007)
H.-P. Breuer, J. Gemmer, M. Michel, Phys. Rev. E 73, 016139 (2006)
S. Nakajima, Prog. Theor. Phys. 20, 948 (1958)
R. Zwanzig, J. Chem. Phys. 33, 1338 (1960)
F. Shibata, Y. Takahashi, N. Hashitsume, J. Stat. Phys. 17, 171 (1977)
S. Chaturvedi, F. Shibata, Z. Phys. B 35, 297 (1979)
F. Shibata, T. Arimitsu, J. Phys. Soc. Jpn. 49, 891 (1980)
C. Uchiyama, F. Shibata, Phys. Rev. E 60, 2636 (1999)
H. Mori, Prog. Theor. Phys. 34, 433 (1965)
D.N. Zubarev, Statistische Thermodynamik für das Nichtgleichgewicht (Akademie-Verlag, Berlin, 1976)
R. Kubo, M. Toda, N. Hashitsume, Statistical Physics II. Nonequilibrium Statistical Mechanics (Springer, Berlin, 1991)
H. Grabert, Projection Operator Techniques in Nonequilibrium Statistical Mechanics, in Springer Tracts in Modern Physics, Vol. 95 (Springer, Berlin, 1982)
H.-P. Breuer, Phys. Rev. A 75, 022103 (2007)
A.A. Budini, Phys. Rev. A 74, 053815 (2006)
H.-P. Breuer, Non-Markovian quantum dynamics and the method of correlated projection superoperators, in Lecture Notes in Physics (Springer-Verlag), arXiv:quant-ph/0707.0172 (to appear)
V. Romero-Rochin, I. Oppenheim, Physica A 155, 52 (1989)
V. Romero-Rochin, A. Orsky, I. Oppenheim, Physica A 156, 244 (1989)
V. Gorini, M. Verri, A. Frigerio, Physica A 161, 357 (1989)
R. Kubo, J. Math. Phys. 4, 174 (1963)
A. Royer, Phys. Rev. A 6, 1741 (1972)
N.G. van Kampen, Physica 74, 215 (1974)
N.G. van Kampen, Physica 74, 239 (1974)
H.-P. Breuer, D. Burgarth, F. Petruccione, Phys. Rev. B 70, 045323 (2004)
H.-P. Breuer, B. Kappler, F. Petruccione, Ann. Phys. (NY) 291, 36 (2001)
H.-P. Breuer, B. Kappler, F. Petruccione, Phys. Rev. A 59, 1633 (1999)
H.-P. Breuer, D. Faller, B. Kappler, F. Petruccione, Europhys. Lett. 54, 14 (2001)
S. Tasaki et al., Ann. Phys. 322, 631 (2007)
K. Yuasa et al., Ann. Phys. 322, 657 (2007)
A. Royer, Aspects of Open Quantum Dynamics, Lect. Notes Phys. 622, 47 (2003)
H.-P. Breuer, Phys. Rev. A 70, 012106 (2004)
J. Gemmer, M. Michel, Eur. Phys. J. B 53, 517 (2006)
R. Steinigeweg, J. Gemmer, M. Michel, Phys. Rev. B 73, 104302 (2006)
J. Gemmer, G. Mahler, Eur. Phys. J. B 31, 249 (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gemmer, J., Breuer, HP. Projection operator techniques and Hilbert space averaging in the quantum theory of nonequilibrium systems. Eur. Phys. J. Spec. Top. 151, 1–12 (2007). https://doi.org/10.1140/epjst/e2007-00357-6
Issue Date:
DOI: https://doi.org/10.1140/epjst/e2007-00357-6