Macroscopic limit, result for uncondensed fraction

Pressure vs temperature curves of classical and quantum ideal gases (Fermi gas, Bose gas) in three dimensions. The Bose gas pressure is lower than an equivalent classical gas, especially below the critical temperature (marked with ★) where particles begin moving en masse into the zero-pressure condensed phase.

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. This replacement gives the macroscopic grand potential function ${\displaystyle \Omega _{m}}$, which is close to ${\displaystyle \Omega }$:

${\displaystyle \Omega _{\rm {m}}=\int _{0}^{\infty }\ln \left(1-ze^{-\beta E}\right)\,dg\approx \Omega .}$

The degeneracy dg  may be expressed for many different situations by the general formula:

${\displaystyle dg={\frac {1}{\Gamma (\alpha )}}\,{\frac {E^{\,\alpha -1}}{E_{\rm {c}}^{\alpha }}}~dE}$

where α is a constant, E_c is a critical energy, and Γ is the Gamma function. For example, for a massive Bose gas in a box, α=3/2 and the critical energy is given by:

${\displaystyle {\frac {1}{(\beta E_{\rm {c}})^{\alpha }}}={\frac {Vf}{\Lambda ^{3}}}}$

where Λ is the thermal wavelength, and f is a degeneracy factor (f=1 for simple spinless bosons). For a massive Bose gas in a harmonic trap we will have α=3 and the critical energy is given by:

${\displaystyle {\frac {1}{(\beta E_{c})^{\alpha }}}={\frac {f}{(\hbar \omega \beta )^{3}}}}$

where V(r)=mω²r²/2  is the harmonic potential. It is seen that E_c  is a function of volume only.

This integral expression for the grand potential evaluates to:

${\displaystyle \Omega _{\rm {m}}=-{\frac {{\textrm {Li}}_{\alpha +1}(z)}{\left(\beta E_{c}\right)^{\alpha }}},}$

where Li_s(x) is the polylogarithm function.

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 sections. As will be seen, even at low temperatures the above result is still useful for accurately describing the thermodynamics of just the un-condensed portion of the gas.

Limit on number of particles in uncondensed phase, critical temperature

The total number of particles is found from the grand potential by

${\displaystyle N_{\rm {m}}=-z{\frac {\partial \Omega _{m}}{\partial z}}={\frac {{\textrm {Li}}_{\alpha }(z)}{(\beta E_{c})^{\alpha }}}.}$

This increases monotonically with z (up to the maximum z = +1). The behaviour when approaching z = 1 is however crucially dependent on the value of α (i.e., dependent on whether the gas is 1D, 2D, 3D, whether it is in a flat or harmonic potential well).

For α > 1, the number of particles only increases up to a finite maximum value, i.e., ${\displaystyle N_{\rm {m}}}$ is finite at z = 1:

${\displaystyle N_{\rm {m,max}}={\frac {\zeta (\alpha )}{(\beta E_{\rm {c}})^{\alpha }}},}$

where ζ(α) is the Riemann zeta function (using Li_α(1) = ζ(α)). Thus, for a fixed number of particles ${\displaystyle N_{\rm {m}}}$, the largest possible value that β can have is a critical value β_c. This corresponds to a critical temperature T_c=1/k_Bβ_c, below which the Thomas–Fermi approximation breaks down (the continuum of states simply can no longer support this many particles, at lower temperatures). The above equation can be solved for the critical temperature:

${\displaystyle T_{\rm {c}}=\left({\frac {N}{\zeta (\alpha )}}\right)^{1/\alpha }{\frac {E_{\rm {c}}}{k_{\rm {B}}}}}$

For example, for the three-dimensional Bose gas in a box (${\displaystyle \alpha =3/2}$ and using the above noted value of ${\textstyle E_{\rm {c}}}$) we get:

${\displaystyle T_{\rm {c}}=\left({\frac {N}{Vf\zeta (3/2)}}\right)^{2/3}{\frac {h^{2}}{2\pi mk_{\rm {B}}}}}$

For α ≤ 1, there is no upper limit on the number of particles (${\displaystyle N_{\rm {m}}}$ diverges as z approaches 1), and thus for example for a gas in a one- or two-dimensional box (${\displaystyle \alpha =1/2}$ and ${\displaystyle \alpha =1}$ respectively) there is no critical temperature.

Inclusion of the ground state

The above problem raises the question for α > 1: if a Bose gas with a fixed number of particles is lowered down below the critical temperature, what happens? 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 particles simply 'disappear' from the continuum of states. It turns out, however, that the macroscopic equation gives an 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 to accept the particles that fall out of the continuum:

${\displaystyle N=N_{0}+N_{\rm {m}}=N_{0}+{\frac {{\textrm {Li}}_{\alpha }(z)}{(\beta E_{\rm {c}})^{\alpha }}}}$

where N₀ is the number of particles in the ground state condensate.

Thus in the macroscopic limit, when T < T_c, the value of z is pinned to 1 and N₀ takes up the remainder of particles. For T > T_c there is the normal behaviour, with N₀ = 0. This approach gives the fraction of condensed particles in the macroscopic limit:

${\displaystyle {\frac {N_{0}}{N}}={\begin{cases}1-\left({\frac {T}{T_{\rm {c}}}}\right)^{\alpha }&{\mbox{if }}\alpha >1{\mbox{ and }}T<T_{\rm {c}},\\0&{\mbox{otherwise}}.\end{cases}}}$

Limitations of the macroscopic Bose gas model

The above standard treatment of a macroscopic Bose gas is straightforward, but the inclusion of the ground state is somewhat inelegant. Another approach is to include the ground state explicitly (contributing a term in the grand potential, as in the section below), this gives rise to an unrealistic fluctuation catastrophe: the number of particles in any given state follow a geometric distribution, meaning that when condensation happens at T < T_c and most particles are in one state, there is a huge uncertainty in the total number of particles. This is related to the fact that the compressibility becomes unbounded for T < T_c. Calculations can instead be performed in the canonical ensemble, which fixes the total particle number, however the calculations are not as easy.^[2]

Practically however, the aforementioned theoretical flaw is a minor issue, as the most unrealistic assumption is that of non-interaction between bosons. Experimental realizations of boson gases always have significant interactions, i.e., they are non-ideal gases. The interactions significantly change the physics of how a condensate of bosons behaves: the ground state spreads out, the chemical potential saturates to a positive value even at zero temperature, and the fluctuation problem disappears (the compressibility becomes finite).^[3] See the article Bose–Einstein condensate.

Thermodynamics

Expanded out, the grand potential is:

${\displaystyle \Omega =g_{0}\ln(1-z)-{\frac {{\textrm {Li}}_{\alpha +1}(z)}{\left(\beta E_{\rm {c}}\right)^{\alpha }}}}$

All thermodynamic properties can be computed from this 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 ${\displaystyle \tau ^{\alpha }}$ is shown.

Quantity	General	${\displaystyle T\ll T_{c}\,}$	${\displaystyle T\gg T_{c}\,}$
${\displaystyle z}$		${\displaystyle \approx {\frac {\zeta (\alpha )}{\tau ^{\alpha }}}-{\frac {\zeta ^{2}(\alpha )}{2^{\alpha }\tau ^{2\alpha }}}}$	${\displaystyle =1\,}$
Vapor fraction ${\displaystyle 1-{\frac {N_{0}}{N}}\,}$	${\displaystyle ={\frac {{\textrm {Li}}_{\alpha }(z)}{\zeta (\alpha )}}\,\tau ^{\alpha }}$	${\displaystyle =\tau ^{\alpha }\,}$	${\displaystyle =1\,}$
Equation of state ${\displaystyle {\frac {PV\beta }{N}}=-{\frac {\Omega }{N}}\,}$	${\displaystyle ={\frac {{\textrm {Li}}_{\alpha \!+\!1}(z)}{\zeta (\alpha )}}\,\tau ^{\alpha }}$	${\displaystyle ={\frac {\zeta (\alpha \!+\!1)}{\zeta (\alpha )}}\,\tau ^{\alpha }}$	${\displaystyle \approx 1-{\frac {\zeta (\alpha )}{2^{\alpha \!+\!1}\tau ^{\alpha }}}}$
Gibbs Free Energy ${\displaystyle G=\ln(z)\,}$	${\displaystyle =\ln(z)\,}$	${\displaystyle =0\,}$	${\displaystyle \approx \ln \left({\frac {\zeta (\alpha )}{\tau ^{\alpha }}}\right)-{\frac {\zeta (\alpha )}{2^{\alpha }\tau ^{\alpha }}}}$

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:

${\displaystyle U={\frac {\partial \Omega }{\partial \beta }}=\alpha PV}$

A similar situation holds for the specific heat at constant volume

${\displaystyle C_{V}={\frac {\partial U}{\partial T}}=k_{\rm {B}}(\alpha +1)\,U\beta }$

The entropy is given by:

${\displaystyle TS=U+PV-G\,}$

Note that in the limit of high temperature, we have

${\displaystyle TS=(\alpha +1)+\ln \left({\frac {\tau ^{\alpha }}{\zeta (\alpha )}}\right)}$

which, for α=3/2 is simply a restatement of the Sackur–Tetrode equation. In one dimension bosons with delta interaction behave as fermions, they obey Pauli exclusion principle. In one dimension Bose gas with delta interaction can be solved exactly by Bethe ansatz. The bulk free energy and thermodynamic potentials were calculated by Chen-Ning Yang. In one dimensional case correlation functions also were evaluated.^[5] In one dimension Bose gas is equivalent to quantum non-linear Schrödinger equation.

General 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 (6): 625–631. Bibcode:2000EJPh...21..625Y. doi:10.1088/0143-0807/21/6/314. S2CID 250870934.