Journal of Statistical Physics

, Volume 128, Issue 6, pp 1391–1414 | Cite as

Analysis of Path Integrals at Low Temperature: Box Formula, Occupation Time and Ergodic Approximation

  • Sébastien Paulin
  • Angel Alastuey
  • Thierry Dauxois


We study the low temperature behavior of path integrals for a simple one-dimensional model. Starting from the Feynman–Kac formula, we derive a new functional representation of the density matrix at finite temperature, in terms of the occupation times for Brownian motions constrained to stay within boxes with finite sizes. From that representation, we infer a kind of ergodic approximation, which only involves double ordinary integrals. As shown by its applications to different potentials, the ergodic approximation turns out to be quite efficient, especially in the low-temperature regime where other usual approximations fail.


Density Matrix Path Integral Stat Phys Occupation Time Brownian Bridge 
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  1. 1.
    Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Path Integrals. McGraw–Hill, New York (1965) MATHGoogle Scholar
  2. 2.
    Simon, B.: Functional Integration and Quantum Physics. Academic, New York (1979) MATHGoogle Scholar
  3. 3.
    Schulman, L.S.: Techniques and Applications of Path Integrals. Wiley, New York (1981) Google Scholar
  4. 4.
    Roepstorff, G.: Path Integral Approach to Quantum Physics. Springer, Berlin (1994) MATHGoogle Scholar
  5. 5.
    Kleinert, H.: Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics, and Financial Markets. World Scientific, Singapore (2004) Google Scholar
  6. 6.
    Wigner, E.P.: Phys. Rev. 40, 749 (1932) MATHCrossRefADSGoogle Scholar
  7. 7.
    Kirkwood, J.G.: Phys. Rev. 44, 31 (1933). MATHCrossRefADSGoogle Scholar
  8. 8.
    Kirkwood, J.G.: Phys. Rev. 45, 116 (1934) MATHCrossRefADSGoogle Scholar
  9. 9.
    Landau, L., Lifschitz, E.: Statistical Physics. Pergamon, Elmsford (1980) Google Scholar
  10. 10.
    Feynman, R., Kleinert, H.: Phys. Rev. A 34, 5080 (1986) CrossRefADSGoogle Scholar
  11. 11.
    Dashen, R., Hasslacher, B., Neveu, A.: Phys. Rev. D 10, 4114 (1974) CrossRefADSGoogle Scholar
  12. 12.
    Bogojevic̀, A., Balaz̀, A., Belic̀, A.: Phys. Lett. A 345, 258 (2005) CrossRefADSGoogle Scholar
  13. 13.
    Raffarin, J.P.: General communication (2004) Google Scholar
  14. 14.
    Bogojevic̀, A., Balaz̀, A., Belic̀, A.: Phys. Lett. A 344, 84 (2005) CrossRefADSGoogle Scholar
  15. 15.
    Krauth, W.: Introduction to Monte Carlo algorithms. In: Kertesz, J., Kondor, I. (eds.) Advances in Computer Simulation. Lecture Notes in Physics. Springer, Berlin (1998) Google Scholar
  16. 16.
    Krauth, W.: Statistical Mechanics: Algorithms and Computations. Oxford University Press, Oxford (2006) MATHGoogle Scholar
  17. 17.
    Ceperley, D.M.: Rev. Mod. Phys. 67, 279 (1995) CrossRefADSGoogle Scholar
  18. 18.
    Militzer, B., Ceperley, D.M.: Phys. Rev. E 63, 066404 (2001) CrossRefADSGoogle Scholar
  19. 19.
    Predescu, C., Sabo, D., Doll, J.D.: J. Chem. Phys. 119, 4641 (2003) CrossRefADSGoogle Scholar
  20. 20.
    Ginibre, J.: Some applications of functional integration in statistical mechanics. In: DeWitt, C., Stora, R. (eds.) Statistical Mechanics and Quantum Field Theory. Gordon and Breach, Les Houches (1971) Google Scholar
  21. 21.
    Cornu, F.: Phys. Rev. E 53, 4562 (1996) CrossRefADSGoogle Scholar
  22. 22.
    Brydges, D.C., Martin, Ph.A.: J. Stat. Phys. 96, 1163 (1999) MATHCrossRefGoogle Scholar
  23. 23.
    Martin, Ph.A.: Acta Phys. Pol. B 34, 3629 (2003) ADSGoogle Scholar
  24. 24.
    Kihara, T., Midzuno, Y., Shizume, T.: J. Phys. Soc. Jpn. 10, 249 (1955) CrossRefADSGoogle Scholar
  25. 25.
    Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals Series and Products, 5th edn. Academic Press, New York (2000) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Sébastien Paulin
    • 1
  • Angel Alastuey
    • 1
  • Thierry Dauxois
    • 1
  1. 1.Laboratoire de PhysiqueCNRS, ENS LyonLyonFrance

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