Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Convergence to equilibrium for classical and quantum spin systems
Download PDF
Download PDF
  • Published: December 1995

Convergence to equilibrium for classical and quantum spin systems

  • G. Da Prato1 &
  • J. Zabczyk2 

Probability Theory and Related Fields volume 103, pages 529–552 (1995)Cite this article

  • 151 Accesses

  • 10 Citations

  • Metrics details

Summary

The paper is devoted to stochastic equations describing the evolution of classical and quantum unbounded spin systems on discrete lattices and on Euclidean spaces. Existence and asymptotic properties of the corresponding transition semigroups are studied in a unified way using the theory of dissipative operators on weighted Hilbert and Banach spaces. This paper is an enlarged and rewritten version of the paper [7].

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Albeverio, S., Kondratiev, Y.G., Tsycalenko, T.V.: Stochastic dynamics for quantum lattice systems and Stochastic quantization 1: Ergodicity, BiBos, Universität Bielefeld, Nr. 601/11/93 1993

  2. Bakry, D., Émery, M.: Diffusions Hypercontractives. In Edité par Azéma, J. et Yor, M. Séminaire de Probabilitées XIX, 1983/84, LNiM 1123, 177–206 (1984)

  3. Blumenthal, R.M., Getoor, R.K.: Markov processes and potential theory. New York: Academic Press 1968

    Google Scholar 

  4. Da Prato, G., Zabczyk, J.: Non explosion, boundedness and ergodicity for stochastic semilinear equations. J. Differential Equations98(1), 181–195 (1992)

    Google Scholar 

  5. Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions. In Encylopedia of Mathematics and its Applications. Cambridge: Cambridge University Press 1992

    Google Scholar 

  6. Da Prato, G., Zabczyk, J.: Evolution equations with White-noise boundary conditions. Stochastics Stochastic Rep.42, 167–182 (1993)

    Google Scholar 

  7. Da Prato, G., Zabczyk, J.: Convergence to equilibrium for spin systems. Scuola Normale Superiore di Pisa. Preprint di Matematica n. 12 1994

  8. Dawson, D.A., Salehi, H.: Spatially homogenous random evolutions. J. Multivariate Anal.10, 141–180 (1980)

    Google Scholar 

  9. Doss, H., Royer, G.: Processus de diffusions associées aux mesures de Gibbs sur\(\mathbb{R}^{z^d } \). Z. Wahrscheinlichkeitstheorie verw. Gebiete46, 107–124 (1978)

    Google Scholar 

  10. Feller, W.: An introduction to probability theory and its applications. Wiley, New York 1971

    Google Scholar 

  11. Fritz, J.: Infinite lattice systems of interacting diffusion processes, existence and regularity properties. Z. Wahrscheinlichkeitstheorie verw. Gebiete59, 291–309 (1982)

    Google Scholar 

  12. Fritz, J.: Stationary measures of stochastic gradient systems. Infinite lattices models. Z. Wahrscheinlichkeitstheorie verw. Gebiete59, 479–490 (1982)

    Google Scholar 

  13. Halmos, P.R.: A Hilbert space problem book. New York: Van Nostrand 1967

    Google Scholar 

  14. Holley, R., Stroock, D.: Diffusions in inifinite dimensional torus. J. Funct. Anal.42, 29–53 (1981)

    Google Scholar 

  15. Lebowitz, J., Presutti, E.: Statistical mechanics of systems of unbounded spins. Comm. Math. Phys.50, 125 (1976)

    Google Scholar 

  16. Leha, G., Ritter, G.: On diffusion processes and their semigroups in Hillbert spaces with an application to interacting stochastic systems. Ann. Probab.12, 1077–1112 (1984)

    Google Scholar 

  17. Leha, G., Ritter, G.: On solutions to stochastic differential equations with discontinuous drift in Hilbert spaces. Math. Ann.270, 109–123 (1985)

    Google Scholar 

  18. Leha, G., Ritter, G.: Stationary distributions of diffusion processes in complete regular spaces. Stochastics Stochastic Rep., to appear.

  19. Manthey, R.: On the Cauchy problem for reaction-diffusion equations with white-noise. Math. Nachr.135, 209–228 (1988)

    Google Scholar 

  20. Marcus, R.: Parabolic Ito equations with monotone non-linearities. J. Funct. Anal.29, 275–286 (1978)

    Google Scholar 

  21. Marcus, R.: Stochastic diffusions on an unbounded domain. Pacific J. Math.84(1), 143–153 (1979)

    Google Scholar 

  22. Royer, G.: Processus de diffusions associées à certain models d'Ising à spin continu. Z. Wahrscheinlichkeitstheorie verw. Gebiete,59, 479–490 (1979)

    Google Scholar 

  23. Ruelle, D.: Probability estimates for continuous spin systems. Comm. Math. Phys.50, 189 (1976)

    Google Scholar 

  24. Schwartz, L.: Mathematics for the physical sciences. Paris: Hermann, 1966

    Google Scholar 

  25. Stroock, D., Zegarlinski, B.: The logarithmic Sobolev inequality for discrete spin systema on a lattice. Comm. Math. Phys.149, 175–193 (1992)

    Google Scholar 

  26. Zegarlinski, B.: The strong exponential decay to equilibrium for the stochastic dynamics associated to the unbounded spin systems on a lattice. Preprint

Download references

Author information

Authors and Affiliations

  1. Scuola Normale Superiore di Pisa, Piazza dei Cavalieri, I-56126, Pisa, Italy

    G. Da Prato

  2. Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland

    J. Zabczyk

Authors
  1. G. Da Prato
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Zabczyk
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Partially supported by the Italian National Project MURST “Problemi nonlinearinell' Analisi...” and by DRET under contract 901636/A000/DRET/DSISR.

Partially sponsored by the KBN grant 2 2003 91 02 and by the KBN grant 2PO3A 082 08

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Da Prato, G., Zabczyk, J. Convergence to equilibrium for classical and quantum spin systems. Probab. Th. Rel. Fields 103, 529–552 (1995). https://doi.org/10.1007/BF01246338

Download citation

  • Received: 12 January 1995

  • Revised: 01 June 1995

  • Issue Date: December 1995

  • DOI: https://doi.org/10.1007/BF01246338

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Mathematics Subject Classification

  • 60H15
  • 60K35
  • 35K55
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature