Abstract
A model of an open system of fermion particles submitted to a constant magnetic field and immersed in a reservoir of phonons is considered within this article. The focus is set on the large time behavior of this system, the purpose being to illustrate the methods of Quantum Markov Semigroup Theory. After providing sufficient conditions to ensure the existence of the semigroup, this report goes through the construction of stationary states, the analysis of the equilibrium via detailed balance conditions. Ergodic behavior of the system is obtained as a byproduct of the above.
Keywords
Mathematics Subject Classification (2000). 81S25, 81S22, 82C22, 82C70, 60H30.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
L. Accardi, Y.G. Lu and I. Volovich,Quantum Theory and its Stochastic Limit, Springer-Verlag, 2002.
R. Alicki and K. Lendi, Quantum Dynamical Semigroups and Applications, Lect. Notes in Physics 286, Springer-Verlag, 1987.
S.T. Ali and et al. Modular structures on trace class operators and applications to Landau levels. Journal of Physics A:..., 2010.
G. Alli and G.L. Sewell: New methods and structures in the theory of the multimode Dicke laser model. J. Math. Phys. 36 (1995), no. 10, 5598-5626
Ph. Biane. Quelques propriétés du mouvement Brownien non-commutatif. Hommage a P.A. Meyer et J. Neveu, Astérisque, 236, 73-102, 1996.
J. Bellissard, R, Rebolledo, D. Spehner and W. von Waldenfels. The quantum flow of electronic transport I: the finite volume case. Mathematical Physics Preprint Archive mp_ arc 02-212.
Ph. Blanchard and R. Olkiewicz: Decoherence induced transition from quantum to classical dynamics, Rev. Math. Phys., 15, 217-243, 2003.
O. Bratteli and D.W. Robinson, Operator algebras and quantum statistical mechanics, 2nd. ed., vol. 1, Springer-Verlag, 1987.
O. Bratteli and D.W. Robinson, Operator algebras and quantum, statistical mechanics, 2nd. ed., vol. 2, Springer-Verlag, 1996.
M. Brune et al.: Phys. Rev. Lett., 77, 4887, 1996.
E. Christensen and D.E. Evans, Cohomology of operator algebras and quantum dynamical semigroups, J.Lon.Math.Soc. 20 (1979), 358-368.
A.M. Chebotarev and F. Fagnola. Sufficient conditions for conservativity of minimal quantum dynamical semigroups. J. Funct. Anal. 153, 382-404, 1998.
A.M. Chebotarev. Lectures on Quantum Probability. Aportaciones Matemáticas, Ser. Textos, 14, Mexico, 2000.
E.B. Davies: Quantum dynamical semigroups and the neutron diffusion equation. Rep. Math. Phys. 11, 169-188 (1977).
C. Dellacherie and P.-A. Meyer. Probabilités et potentiel. Chapitres XII-XVI, Second edition, Hermann, Paris, 1987.
G. Dell Antonio: On Decoherence, J.Math.Phys., 44, 4939-4956, 2003
A. Dhahri, F. Fagnola, and R. Rebolledo. The decoherence-free subalgebra of a quantum markov semigroup with unbounded generator. Infin. Dimens. Anal. Quantum, Probab. Relat. Top., 13:413-433, 2010.
F. Fagnola and R. Rebolledo. The approach to equilibrium of a class of quantum dynamical semigroups. Inf. Dim. Anal. Q. Prob. and Rel. Topics, 1(4):1-12, April 1998.
F. Fagnola and R. Rebolledo. On the existence of invariant states for quantum dynamical semigroups. J.Math.Phys., 42, 1296-1308, 2001.
F. Fagnola and R. Rebolledo. Algebraic conditions for convergence of a quantum Markov semigroup to a steady state. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 11:467-474, 2008.
Franco Fagnola and Rolando Rebolledo. Subharmonic projections for a quantum Markov semigroup. J. Math. Phys., 43(2):1074-1082, 2002.
Franco Fagnola and Rolando Rebolledo. Quantum Markov semigroups and their stationary states. In Stochastic analysis and mathematical physics II, Trends Math., pages 77-128. Birkhäuser, Basel, 2003.
Franco Fagnola and Rolando Rebolledo. Transience and recurrence of quantum Markov semigroups. Probab. Theory Related, Fields, 126(2):289-306, 2003.
Franco Fagnola and Rolando Rebolledo. Notes on the qualitative behaviour of quantum Markov semigroups. In Open quantum systems. III, volume 1882 of Lecture Notes in Math., pages 161-205. Springer, Berlin, 2006.
Franco Fagnola and Rolando Rebolledo. From Classical to Quantum Entropy Production, QP-PQ: Quantum Probab. White Noise Anal., World Sci. Publ., vol. 25, (2010), 245-261.
F. Fagnola and V. Umanità. Generators of KMS Symmetric Markov Semigroups on B(h) Symmetry and Quantum Detailed Balance. Commun. Math. Phys. 2010. DOI 10.1007/s00220-010-1011-1.
A. Frigerio and M. Verri. Long-time asymptotic properties of dynamical semigroups on w*-algebras. Math. Zeitschrift, (1982).
A. Kossakowski V. Gorini and E.C.G. Sudarshan. Completely positive dynamical semigroups of n-level systems. J. Math. Phys., 17:821-825, 1976.
Thomas M. Liggett, Interacting particle systems, Springer-Verlag, New York, 1985.
G. Lindblad. On the generators of quantum dynamical semigroups. Commun. Math. Phys., 48:119-130, 1976.
P.-A. Meyer. Quantum Probability for Probabilists, volume 1538 of Lect. Notes in Math. Springer-Verlag, Berlin, Heidelberg, New York, 1993.
K.R. Parthasarathy, An introduction to quantum stochastic calculus, Monographs in Mathematics, vol. 85, Birkhäuser Verlag, Basel-Boston-Berlin, 1992.
R. Rebolledo and D. Spehner. Adiabatic limits and quantum decoherence. In Stochastic analysis in mathematical physics, pages 94-108. World Sci. Publ., Hackensack, NJ, 2008.
Rolando Rebolledo. Complete positivity and open quantum systems. In Stochastic analysis and mathematical physics, volume 50 of Progr. Probab., pages 101-132. Birkhauser Boston, Boston, MA, 2001.
Rolando Rebolledo. Quantum interacting particles related to the exclusion process. In Stochastic models, II (Spanish) (Guanajuato, 2000), volume 16 of Aportaciones Mat. Investig., pages 271-293. Soc. Mat. Mexicana, México, 2001.
Rolando Rebolledo. Decoherence of quantum Markov semigroups. Ann. Inst. H. Poincaré Probab. Statist., 41(3):349-373, 2005.
Rolando Rebolledo. Unraveling Open Quantum Systems: Classical Reductions and Classical Dilations of Quantum Markov Semigroups, Confluentes Mathematici, vol. 1, (2009), 123-167.
W.F. Stinespring, Positive functions on C*-algebras, Proc.Amer.Math.Soc. 6, 211216, 1955.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Basel
About this paper
Cite this paper
Rebolledo, R. (2012). Equilibrium Analysis of a Dissipative Fermion System. In: Benguria, R., Friedman, E., Mantoiu, M. (eds) Spectral Analysis of Quantum Hamiltonians. Operator Theory: Advances and Applications, vol 224. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0414-1_13
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0414-1_13
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0413-4
Online ISBN: 978-3-0348-0414-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)