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## Bose gasAn ideal ## Additional recommended knowledge
## The Thomas-Fermi approximationThe thermodynamics of an ideal Bose gas is best calculated using the grand partition function. The grand partition function for a Bose gas is given by: where each term in the product corresponds to a particular energy ε *z*(β,μ) =*e*^{βμ}
and β defined as: where Following the procedure described in the gas in a box article, we can apply the Thomas-Fermi approximation which assumes that the average energy is large compared to the energy difference between levels so that the above sum may be replaced by an integral: The degeneracy where α is a constant, where Λ is the thermal wavelength. For a massive Bose gas in a harmonic trap we will have α=3 and the critical energy is given by: where E is a function of volume only.
_{c }We can solve the equation for the grand potential by integrating the Taylor series of the integrand term by term, or by realizing that it is proportional to the Mellin transform of the Li The problem with this continuum approximation for a Bose gas is that the ground state has been effectively ignored, giving a degeneracy of zero for zero energy. This inaccuracy becomes serious when dealing with the Bose-Einstein condensate and will be dealt with in the next section. ## Inclusion of the ground stateThe total number of particles is found from the grand potential by The polylogarithm term must remain real and positive, and the maximum value it can possibly have is at z=1 where it is equal to ζ(α) where ζ is the Riemann zeta function. For a fixed This corresponds to a critical temperature T For example, for α = 3 / 2 and using the above noted value of Again, we are presently unable to calculate results below the critical temperature, because the particle numbers using the above equation become negative. The problem here is that the Thomas-Fermi approximation has set the degeneracy of the ground state to zero, which is wrong. There is no ground state to accept the condensate and so the equation breaks down. It turns out, however, that the above equation gives a rather accurate estimate of the number of particles in the excited states, and it is not a bad approximation to simply "tack on" a ground state term: where
This equation can now be solved down to absolute zero in temperature. Figure 1 shows the results of the solution to this equation for α=3/2, with k=ε N =10,000 and the dotted black line is the solution for N =1000. The blue lines are the fraction of condensed particles N The red lines plot values of the
negative of the chemical potential μ and the green lines plot the corresponding values of _{0}/N z . The horizontal axis is the normalized temperature τ defined by
It can be seen that each of these parameters become linear in τ The equation for the number of particles can be written in terms of the normalized temperature as: For a given ## ThermodynamicsAdding the ground state to the equation for the particle number corresponds to adding the equivalent ground state term to the grand potential: All thermodynamic properties may now be computed from the grand potential. The following table lists various thermodynamic quantities calculated in the limit of low temperature and high temperature, and in the limit of infinite particle number. An equal sign (=) indicates an exact result, while an approximation symbol indicates that only the first few terms of a series in τ
It is seen that all quantities approach the values for a classical ideal gas in the limit of large temperature. The above values can be used to calculate other thermodynamic quantities. For example, the relationship between internal energy and the product of pressure and volume is the same as that for a classical ideal gas over all temperatures: A similar situation holds for the specific heat at constant volume The entropy is given by: Note that in the limit of high temperature, we have which, for α=3/2 is simply a restatement of the Sackur-Tetrode equation. ## See also## References- Huang, Kerson (1967).
*Statistical Mechanics*. New York: John Wiley and Sons. - Isihara, A. (1971).
*Statistical Physics*. New York: Academic Press. - Landau, L. D.; E. M. Lifshitz (1996).
*Statistical Physics, 3rd Edition Part 1*. Oxford: Butterworth-Heinemann. - Pethick, C. J.; H. Smith (2004).
*Bose-Einstein Condensation in Dilute Gases*. Cambridge: Cambridge University Press. - Yan, Zijun (2000). "General Thermal Wavelength and its Applications" (PDF).
*Eur. J. Phys***21**: 625-631.
Categories: Gases | Thermodynamics | Statistical mechanics |
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Bose_gas". A list of authors is available in Wikipedia. |