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SUMMARY:Calculation of thermodynamic potential for Bose system near conden
sate point
DTSTART;VALUE=DATE-TIME:20181204T100500Z
DTEND;VALUE=DATE-TIME:20181204T102500Z
DTSTAMP;VALUE=DATE-TIME:20230330T012834Z
UID:indico-contribution-25@indico.bitp.kiev.ua
DESCRIPTION:Speakers: Kseniia Haponenko (Oles Honchar Dnipro National Univ
ersity)\nBose system in the presence of the condensate is investigated in
the Bogolyubov model of the separated condensate (see about in [1]). In th
is state occupation number ${{n}_{0}}$ of one-particle state with momentu
m ${p}=0$ is macroscopic one. In this model the system is described by the
statistical operator\n\n$\nw({{n}_{0}})={{e}^{\\beta [\\Omega ({{n}_{0}})
-\\hat{H}({{n}_{0}})+\\mu \\hat{N}({{n}_{0}})]}}\, \\quad \\mathrm{S
p}w({{n}_{0}})=1\n $\n\nwhere operators $\\hat{H}({{n}_{0}})$\, $\\hat{N}(
{{n}_{0}})$ are given by Hamiltonian of the system $\\hat{H}$ and operator
of particle number $\\hat{N}$ after substitution $n_{0}^{1/2}$ instead of
the Bose operators ${{a}_{0}}$\, ${{a}_{0}}^{+}$ ($\\beta $\,$\\mu $ are
the reverse temperature and the chemical potential). According to Bogolyub
ov equilibrium value $n_{0}^{o}$ of the occupation number $n_{0}^{{}}$ can
be found from the minimum condition of the thermodynamic potential $\\Ome
ga ({{n}_{0}})$ i.e. it is the non-equilibrium one of the system. Near tra
nsition point from normal state to the state with the condensate occupatio
n number $n_{0}^{{}}$ is small in comparison with the total number of part
icles and the potential $\\Omega ({{n}_{0}})$ can be calculated in a pertu
rbation theory in powers of $n_{0}^{1/2}$.\nThe purpose of this paper is c
alculating of the potential $\\Omega ({{n}_{0}})$ in a modified thermodyna
mic perturbation theory with small parameter $n_{0}^{{}}$. The obtained ex
pression for $\\Omega ({{n}_{0}})$ can be used as the potential Landau in
his theory of the phase transitions of the second kind for the system unde
r consideration. The statistical operator $w({{n}_{0}})$ can be written in
the form $w({{n}_{0}})=\\exp \\beta \\\,[F-({{\\mathbf{\\hat{H}}}_{0}}+{{
{\\hat{U}}}_{1}}+{{{\\hat{U}}}_{2}})]$ where the operators ${{{\\hat{U}}}_
{1}}\\sim n_{0}^{1/2}\, {{{\\hat{U}}}_{2}}\\sim{{n}_{0}}$. The calculation
is substantially simplified by the fact that the independent on ${{n}_{0}
}$ operator ${{\\mathbf{\\hat{H}}}_{0}}$ commutes with the operator of the
number of particles. It is shown that only integer powers of this number
are present in expansion of the potential. Relatively compact expressions
for the coefficients of this series are obtained because operators describ
ing the interaction in the system commutate under T-product. For a Bose ga
s they are calculated in an additional perturbation theory in interaction
between particles. An analyze of the results with connection to the Landau
theory of phase transitions is given. \n\n[1] Akhiezer\, A.I. Methods of
Statistical Physics [Text] / A.I. Akhiezer\, A.I. and S.V. Peletminskii.
– Oxford: Pergamon Press\, 1981. – 450 p.\n\nhttps://indico.bitp.kiev.
ua/event/2/contributions/25/
LOCATION:
URL:https://indico.bitp.kiev.ua/event/2/contributions/25/
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