Journal of Statistical Physics

, Volume 161, Issue 4, pp 942–964 | Cite as

Grand Canonical Versus Canonical Ensemble: Universal Structure of Statistics and Thermodynamics in a Critical Region of Bose–Einstein Condensation of an Ideal Gas in Arbitrary Trap

  • S. V. Tarasov
  • Vl. V. KocharovskyEmail author
  • V. V. Kocharovsky


We find a self-similar analytical solution for the grand-canonical-ensemble (GCE) statistics and thermodynamics in the critical region of Bose–Einstein condensation. It is valid for an arbitrary trap, loaded with an ideal gas, in the thermodynamic limit. We show that for the quantities, changing by a finite amount across the critical region, the exact GCE result differs from the corresponding canonical-ensemble result by a factor on the order of unity even in the thermodynamic limit. Thus, a widely used GCE approach does not describe correctly the critical phenomena at the phase transition for the actual systems with a fixed number of particles and yields only an asymptotics far outside the critical region.


Bose–Einstein condensation Critical phenomena Critical region Grand canonical ensemble Canonical ensemble Mesoscopic system 



A support from RFBR (Grant 12-02-00855-a), Program of fundamental research of the Physical Science Branch of the Russian Academy of Science (section III.7) and the Council on grants of the President of the Russian Federation for support of the leading scientifc schools of the Russian Federation (Grant HIII-1041.2014.2) is acknowledged.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • S. V. Tarasov
    • 1
    • 2
  • Vl. V. Kocharovsky
    • 1
    • 2
    Email author
  • V. V. Kocharovsky
    • 1
    • 3
  1. 1.Institute of Applied PhysicsRussian Academy of ScienceNizhny NovgorodRussia
  2. 2.Lobachevsky State University of Nizhny NovgorodNizhny NovgorodRussia
  3. 3.Department of Physics and AstronomyTexas A&M UniversityCollege StationUSA

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