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\sect{Bose gas in a uniform gravitational field}

% orirginal version R.K. Pathria and P.D. Beale, "Statistical Mechanics", Elsevier Ltd., Chapter 7; problem 7.10. (2011)

\Dn

Consider an ideal Bose gas of particles of mass $m$ in a uniform gravitational field of acceleration $g$.

\Dn

(1) Show that the phenomenon of Bose-Einstein condensation in this gas sets in at a temperature $ T_c $ given by 
%
\[
T_c \approx T_c^0 \left[ 1+ \dfrac{8}{9} \dfrac{1}{\zeta(3/2)}  \sqrt{\dfrac{\pi mgL}{kT_c^0}} \right]
\]
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where $L$ is the height of the tank and $mgL \ll kT_c^0 $, where $T_c^0\equiv T_c^0(g=0)$.

\Dn

(2) Show that the condensation is accompanied by a discontinuity in the specific heat of the gas: 
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\[
(\Delta C_V)_{T=T_c}\approx -\dfrac{9}{8\pi}\zeta(3/2)Nk\sqrt{\dfrac{\pi mgL}{kT_c^0}} 
\]

\Dn

\textbf{Hint:} note the following expansion of the polylogarithmic function:
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\[
L_{\nu}(\eexp{-\alpha})=\dfrac{\Gamma(1-\nu)}{\alpha^{1-\nu}}+\sum_{i=0}^{\infty}{\dfrac{(-1)^i}{i}\zeta(\nu-i)\alpha^i}
\]


