Skip to main content
SpringerLink
Log in

A Canonical Ensemble Approach to the Fermion/Boson Random Point Processes and Its Applications

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We introduce the boson and the fermion point processes from the elementary quantum mechanical point of view. That is, we consider quantum statistical mechanics of the canonical ensemble for a fixed number of particles which obey Bose-Einstein, Fermi-Dirac statistics, respectively, in a finite volume. Focusing on the distribution of positions of the particles, we have point processes of the fixed number of points in a bounded domain. By taking the thermodynamic limit such that the particle density converges to a finite value, the boson/fermion processes are obtained. This argument is a realization of the equivalence of ensembles, since resulting processes are considered to describe a grand canonical ensemble of points. Random point processes corresponding to para-particles of order two are discussed as an application of the formulation. Statistics of a system of composite particles at zero temperature are also considered as a model of determinantal random point processes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Benard, C., Macchi, O.: Detection and emission processes of quantum particles in a chaotic state. J. Math. Phys. 14, 155–167 (1973)

    Article  MathSciNet  Google Scholar 

  2. Chaturvedi, S.: Canonical partition functions for parastatistical systems of any order. Phys. Rev. E 54, 1378–1382 (1996)

    Article  ADS  Google Scholar 

  3. Chaturvedi, S., Srinivasan, V.: Grand canonical partition functions for multi-level para-Fermi systems of any order. Phys. Lett. A 224, 249–252 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  4. Doplicher, S., Haag, R., Roberts, J.E.: Local observables and particle statistics I. Commun. Math. Phys. 23, 199–230 (1971)

    Article  ADS  MathSciNet  Google Scholar 

  5. Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes. Berlin: Springer Verlag, 1988

  6. Fichtner, K.-H.: On the position distribution of the ideal Bose gas. Math. Nachr. 151, 59–67 (1991)

    MATH  MathSciNet  Google Scholar 

  7. Freudenberg, W.: Characterization of states of infinite boson systems. II: On the existence of the conditional reduced density matrix. Commun. Math. Phys. 137, 461–472 (1991)

    MATH  MathSciNet  Google Scholar 

  8. Fichtner, K.-H., Freudenberg, W.: Point processes and the position distribution of infinite boson systems. J. Stat. Phys. 47, 959–978 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  9. Fichtner, K.-H., Freudenberg, W.: Characterization of states of infinite boson systems. I: On the construction of states of boson systems. Commun. Math. Phys. 137, 315–357 (1991)

    MATH  MathSciNet  Google Scholar 

  10. Green, H.S.: A generalized method of field quantization. Phys. Rev. 90, 270–273 (1953)

    Article  ADS  MATH  Google Scholar 

  11. Goldin, G.A., Grodnik, J., Powers, R.T., Sharp, D.H.: Nonrelativistic current algebra in the N/V limit. J. Math. Phys. 15, 88–100 (1974)

    Article  MathSciNet  Google Scholar 

  12. Hartle, J.B., Taylor, J.R.: Quantum mechanics of paraparticles. Phys. Rev. 178, 2043–2051 (1969)

    Article  ADS  Google Scholar 

  13. James, G., Kerber, A.: The Representation Theory of the Symmetric Group. (Encyclopedia of mathematics and its applications vol. 16) London: Addison-Wesley Publishing, 1981

  14. Lenard, A.: States of classical statistical mechanical systems of infinitely many particles. I. Arch. Rat. Mech. Anal. 59, 219–139 (1975)

    MathSciNet  Google Scholar 

  15. Lytvynov, E.: Fermion and boson random point processes as particle distributions of infinite free Fermi and Bose gases of finite density. Rev. Math. Phys. 14, 1073–1098 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Macchi, O.: The coincidence approach to stochastic point processes. Adv. Appl. Prob. 7, 83–122 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  17. Macchi, O.: The fermion process–a model of stochastic point process with repulsive points. In: Transactions of the Seventh Prague Conference on Information Theory, Statistical Decision Functions, Random Processes and of the Eighth European Meeting of Statisticians Vol. A, Prague 1974. Dordrecht: Reidel Publishing, 1977, pp. 391–398

  18. Menikoff, R.: The Hamiltonian and generating functional for a nonrelativistic local current algebra. J. Math. Phys. 15, 1138–1152 (1974)

    Article  MathSciNet  Google Scholar 

  19. Messiah, A.M.L., Greenberg, O.W.: Symmetrization postulate and its experimental foundation. Phys. Rev. 136, B248–B267 (1964)

    Google Scholar 

  20. Ohnuki, Y., Kamefuchi, S.: Wavefunctions of identical particles. Ann. Phys. 51, 337–358 (1969)

    Article  Google Scholar 

  21. Ohnuki, Y., Kamefuchi, S.: Quantum field theory and parastatistics. Berlin: Springer-Verlag 1982

  22. Sagan, B.E.: The Symmetric Group. New York: Springer-Verlag, 1991

  23. Shirai, T., Takahashi, Y.: Random point fields associated with certain Fredholm determinants I: fermion, Poisson and boson point processes. J. Funct. Anal. 205, 414–463 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  24. Simon, B.: Trace ideals and their applications. London Mathematical Society Lecture Note Series, Vol. 35, Cambridge: Cambridge University Press, 1979

  25. Simon, B.: Representations of Finite and Compact Groups. Providence, RI: A.M.S, 1996

  26. Soshnikov, A.: Determinantal random point fields. Russ. Math. Surv. 55, 923–975 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  27. Stolt, R.H., Taylor, J.R.: Classification of paraparticles. Phys. Rev. D1, 2226–2228 (1970)

    Google Scholar 

  28. Suranyi, P.: Thermodynamics of parabosonic and parafermionic systems of order two. Phys. Rev. Lett. 65, 2329–2330 (1990)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  29. Vere-Jones, D.: A generalization of permanents and determinants. Linear Algebra Appl. 111, 119–124 (1988)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Kanazawa University, Kanazawa, 920-1192, Japan

    H. Tamura

  2. Department of Mathematics and Physics, Setsunan University, Neyagawa, Osaka, 572-8508, Japan

    K.R. Ito

Authors
  1. H. Tamura
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K.R. Ito
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. Tamura.

Additional information

Communicated by J.L. Lebowitz

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tamura, H., Ito, K. A Canonical Ensemble Approach to the Fermion/Boson Random Point Processes and Its Applications. Commun. Math. Phys. 263, 353–380 (2006). https://doi.org/10.1007/s00220-005-1507-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-005-1507-2

Keywords

Search

Navigation