How Does Pressure Fluctuate in Equilibrium?

  • Ken Hiura
  • Shin-ichi Sasa


We study fluctuations of pressure in equilibrium for classical particle systems. In equilibrium statistical mechanics, pressure for a microscopic state is defined by the derivative of a thermodynamic function or, more mechanically, through the momentum current. We show that although the two expectation values converge to the same equilibrium value in the thermodynamic limit, the variance of the mechanical pressure is in general greater than that of the pressure defined through the thermodynamic relation. We also present a condition for experimentally detecting the difference between them in an idealized measurement of momentum transfer.


Thermodynamic fluctuation theory in equilibrium Pressure fluctuation Landau-Lifshitz fluctuation theory Classical statistical mechanics 



The authors thank Yoshi Oono for his useful comments. The present work was supported by KAKENHI Nos. 25103002 and 17H01148.


  1. 1.
    Einstein, A.: Theorie der Opaleszenz von homogenen Flüssigkeiten und Flüssigkeitsgemischen in der Nähe des kritischen Zustandes. Ann. Phys. 33, 1275–1298 (1910)CrossRefMATHGoogle Scholar
  2. 2.
    Callen, C.H.: Thermodynamics and an Introduction to Thermostatistics, 2nd edn. Willey, New York (1985)MATHGoogle Scholar
  3. 3.
    Landau, L.D., Lifshitz, E.M.: Statistical Physics, 3rd edn. Butterworth-Heinmann, Oxford (1980)MATHGoogle Scholar
  4. 4.
    Mishin, Y.: Thermodynamic theory of equilibrium fluctuations. Ann. Phys. 363, 48–97 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Ernst, M.H., Hauge, E.H., Van Leeuwen, J.M.J.: Asymptotic time behavior of correlation functions. II. Kinetic and potential terms. J. Stat. Phys. 15, 7–22 (1976)ADSCrossRefGoogle Scholar
  6. 6.
    Kawasaki, K.: Mode Coupling and Critical Dynamics. In: Domb, C., Green, M.S. (eds.) Phase Transition and Critical Phenomena, vol. 5A, pp. 165–403. Academic Press, New York (1976)Google Scholar
  7. 7.
    Münster, A.: The theory of fluctuation. In: de Groot, S.R. (eds.) Proc. Enrico Fermi Intl. School of Physics, pp. 23–130. Varenna, Italy (1959)Google Scholar
  8. 8.
    Wallace, D.C.: Theory of stress fluctuations. Phys. Rev. E 62, 3077–3082 (2000)ADSCrossRefGoogle Scholar
  9. 9.
    Kittel, C.: Temperature fluctuation: an oxymoron. Phys. Today 41, 93 (1988)ADSCrossRefGoogle Scholar
  10. 10.
    Gibbs, J.W.: Elementary Principles in Statistical Mechanics. Scribner’s, New York (1902)MATHGoogle Scholar
  11. 11.
    Presutti, E.: A mechanical definition of the thermodynamic pressure. J. Stat. Phys. 13, 301–314 (1975)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Itami, M., Sasa, S.: Derivation of Stokes’ Law from Kirkwood’s formula and the Green-Kubo formula via large deviation theory. J. Stat. Phys. 161, 532–552 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of PhysicsKyoto UniversityKyotoJapan

Personalised recommendations