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Ideal Bose condensation gas in gravitational field, Statistical Physic

  1. Feb 20, 2013 #1
    1. The problem statement, all variables and given/known data
    We have an ideal Bose gas in gravitational field. Show that the critical temperature( the temperature in which the condensation is starting is:
    [itex] T_c = T^0_c( 1 + \frac{8}{9} \frac{1}{ζ(3/2)} (\frac{\pi mgh}{kT^0_c})^{1/2} ) [/itex]

    Attempt
    Lege artis; Firstly I want to estimate the number [itex] N [/itex] of particles in that [itex] T_c [/itex] temperature. As we know at that state the chemical potential is zero so [itex] \mu = 0 [/itex], so the average number of particles will be given by the Bose - Einstein distribution
    [itex] n_{mean} = \frac{1}{e^{ε/kT_c}-1} [/itex]
    I want to find all that particles in such state, so I form an integral over the volume in which the gas is( we assume it is a container with axial symmetry, with area of the circle [itex] S [/itex] and hight [itex] L [/itex] ) and over all possible energy states. I will skip the mathematical formulation and give the integral I need to evaluate:
    [itex] N = 4 \pi g_0 S h^{-3} \int_0^{∞} dp \int_0^L dz \frac{p^2}{e^{ \frac{p^2}{2 m k T_c} - \frac{mgz}{kT_c} } - 1 } [/itex]
    where [itex] S [/itex] is area of the circle for the cylinder, [itex] g_0 = 2s + 1 [/itex] is the number of states and [itex] h [/itex] is the Planck constant
    After changing the variables I get [itex] \frac{ 4 \pi g_0 S (2mkT_c)^{3/2}kT_c}{h^3mg} \int^{∞}_0 dp' \int^{\frac{mgL}{kT_c}}_0 \frac{1}{e^{p'^2}e^{-z'}-1} [/itex]
    I have no idea how to handle this integral:
    I just get the [itex] e^{z'} [/itex] out and get [itex] \int \frac{e^{z'}}{e^{p'^2}-e^{z'}} [/itex] which leads to something like [itex] \int p'^2 ln(\frac{e^{p'^2} - e^{CL}}{e^{p'^2} - 1 } dp' [/itex] on which my best attempt was [itex] \int ln(\frac{e^{p'^2} - e^{CL}}{e^{p'^2} - 1 } \frac{dp'^3}{3} = \frac{p'^3}{3}.ln(...) - ....[/itex] and the first is in range of [itex] 0 [/itex] which is a problematic dot and [itex] ∞ [/itex] which doesn't seem pretty too.
    I hope for my post to be accurate enough. Thanks in advance.

    PS: I found the same problem but it can't help me to solve the integral
    https://www.physicsforums.com/showthread.php?t=455803
     
    Last edited: Feb 20, 2013
  2. jcsd
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