Skip to main content
SpringerLink
Log in

On fluctuations in statistical mechanics

  • Published:
Il Nuovo Cimento (1955-1965)

Summary

Closed formulas for small fluctuations are obtained for the case of macroscopic quantities which can be written as an integral inμ-space (one-particle phase-space) of the form ∫ϰ(x, p, ϱ(x, p))dxdp, whereχ is an arbitrary function of co-ordinates, momentum, and of particle density ϱ(x, p). The second-order approximation of Einstein’s expression is employed. Comparison is made with the fluctuations in energ and occupation numbers derived from Darwin-Fowler technique and from the Canonical and Grand Canonical ensembles. The physical meaning of the results obtained is discussed.

Riassunto

Si ottengono formule chiuse per le piccole fluttuazioni nel caso di quantità macroscopiche che possono essere scritte sotto forma di un integrale nello spazioμ (spazio delle fasi di una particella) della forma εχ(x, p, ϱ(x, p) dxdp, in cuiχ è una funzione arbitraria delle coordinate, dell’impulso e della densità delle particelle ϱ(x, p). Si usa l’approssimazione di secondo ordine dell’espressione di Einstein. Si fa il confronto con le fluttuazioni in energia e numero d’occupazione dedotte con la tecnica di Darwin-Fowler e dagli insiemi canonico e gran canonico. Si discute il significato fisico dei risultati ottenuti.

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

Literatur

  1. A. Einstein:Ann. d. Phys.,33, 1275 (1910).

    Article  ADS  Google Scholar 

  2. R. F. Greene andH. B. Callen:Phys. Rev.,83, 1231 (1951).

    Article  ADS  MATH  Google Scholar 

  3. Although expressions (4) and (5) can also be derived without the help of approximation (2) (see ref. (2)) we will limit ourselves to this simpler case, since a similar approximation will be employed in the explicit calculations of the next Section.

    Article  ADS  MATH  Google Scholar 

  4. L. Onsager:Phys. Rev.,38, 2265 (1931).

    Article  ADS  MATH  Google Scholar 

  5. D. Ter Haar:Elements of Statistical Mechanics (New York, 1960), p. 75.

  6. P. A. M. Dirac:The Principles of Quantum Mechanics, 4th Ed. (Oxford, 1958), p. 238.

  7. O. Bolza:Lectures on the Calculus of Variations (New York, 1946), p. 208.

  8. C. G. Darwin andR. H. Fowler:Phil. Mag.,44, 450 (1922).

    Article  Google Scholar 

  9. C. G. Darwin andR. H. Fowler:Proc. Camb. Phil. Soc.,21, 391 (1922).

    Google Scholar 

  10. R. H. Fowler:Statistical Mechanics (Cambridge, 1955), p. 748, 753, 755.

  11. R. C. Tolman:The Principle of Statistical Mechanics (Oxford, 1938), Section141.

  12. R. Courant andD. Hilbert:Methods of Mathematical Physics, vol.1 (New York, 1953), p. 61.

    ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Escola de Engenharia de S. Carlos, Universidade de S. Paulo, S. Paulo

    R. C. T. da Costa

Authors
  1. R. C. T. da Costa
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Most of this work was done during the author’s stay at Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro.

Rights and permissions

Reprints and permissions

About this article

Cite this article

da Costa, R.C.T. On fluctuations in statistical mechanics. Nuovo Cim 32, 654–678 (1964). https://doi.org/10.1007/BF02735889

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02735889

Search

Navigation