Thermodynamic behavior of an ideal Bose gas

1. When Does a Gas Become "Quantum"?

The key parameter determining quantum behavior is

\[ n \lambda^3 \equiv \frac{n h^3}{(2\pi m kT)^{3/2}}, \]

where

• \(n = N/V\) is number density,
• \(m\) is particle mass,

• \(\lambda = h / \sqrt{2\pi m kT}\) is the thermal de Broglie wavelength.

• If \(n\lambda^3 \ll 1\): wave packets hardly overlap → classical Maxwell–Boltzmann statistics is fine.

• If \(n\lambda^3 \sim 1\): quantum statistics is essential.

This chapter deals with ideal Bose systems: particles are non-interacting but indistinguishable and obey Bose–Einstein statistics.

2. Grand Canonical Description of an Ideal Bose Gas

Using the grand canonical ensemble, for a Bose gas we have

\[ \frac{PV}{kT} = \ln \mathcal{Q} = -\sum_\varepsilon \ln\left(1 - z e^{-\beta \varepsilon}\right), \]

\[ N = \sum_\varepsilon \langle n_\varepsilon \rangle = \sum_\varepsilon \frac{1}{z^{-1} e^{\beta \varepsilon} - 1}, \]

where

• \(\beta = 1/kT\),
• \(z = e^{\mu/kT}\) is the fugacity, \(\mu\) the chemical potential,
• the sums run over single-particle energy levels \(\varepsilon\).

For a nonrelativistic particle in a box, the density of one-particle states with energy between \(\varepsilon\) and \(\varepsilon + d\varepsilon\) is

\[ a(\varepsilon) d\varepsilon = \frac{(2\pi V) (2m)^{3/2}}{h^3} \varepsilon^{1/2} d\varepsilon \propto V \varepsilon^{1/2} d\varepsilon. \]

The sums over levels can be replaced by integrals over \(\varepsilon^{1/2}\) except for the ground state \(\varepsilon = 0\). Because in quantum statistics each nondegenerate level has weight 1, the ground state must be treated separately (if included in the integral it would effectively have zero measure).

After this step we obtain

\[ \frac{P}{kT} = \frac{2\pi}{h^3}(2m)^{3/2}\int_0^\infty \varepsilon^{1/2} \ln\left(1 - z e^{-\beta\varepsilon}\right) d\varepsilon - \frac{1}{V}\ln(1-z), \]

\[ \frac{N}{V} = \frac{2\pi}{h^3}(2m)^{3/2}\int_0^\infty \frac{\varepsilon^{1/2} d\varepsilon}{z^{-1} e^{\beta\varepsilon} - 1} + \frac{1}{V}\frac{z}{1-z}. \]

The \(1/V\) terms are exactly the contributions from the single ground state.

3. Bose–Einstein Functions and the Equation of State

Changing variables \(\beta\varepsilon = x\) and introducing the Bose–Einstein functions

\[ g_\nu(z) = \frac{1}{\Gamma(\nu)} \int_0^\infty \frac{x^{\nu-1} dx}{z^{-1} e^x - 1} = z + \frac{z^2}{2^\nu} + \frac{z^3}{3^\nu} + \cdots \]

and identifying the thermal wavelength \(\lambda = h/\sqrt{2\pi m kT}\), we obtain the clean forms

\[ \frac{P}{kT} = \frac{1}{\lambda^3} g_{5/2}(z), \]

\[ \frac{N - N_0}{V} = \frac{1}{\lambda^3} g_{3/2}(z), \]

with \(N_0\) the number in the ground state \(\varepsilon = 0\).

These are the basic equations of state of the ideal Bose gas.

4. High-Temperature (Classical) Limit and Virial Expansion

For small fugacity \(z \ll 1\) (non-degenerate gas, \(n\lambda^3 \ll 1\)), we use the series

\[ g_{3/2}(z) = z + \frac{z^2}{2^{3/2}} + \frac{z^3}{3^{3/2}} + \dots \]

\[ g_{5/2}(z) = z + \frac{z^2}{2^{5/2}} + \frac{z^3}{3^{5/2}} + \dots \]

Eliminating \(z\) between the two equations and expanding in powers of \(n\lambda^3\) gives the virial expansion

\[ \frac{PV}{NkT} = \sum_{\ell=1}^{\infty} a_\ell \left(\frac{\lambda^3}{v}\right)^{\ell-1}, \quad v = \frac{V}{N} \]

with virial coefficients

\[ a_1=1,\quad a_2 = -\frac{1}{4\sqrt{2}},\quad a_3 = -0.00330,\dots \]

For bosons \(a_2<0\): at fixed \(n\) and \(T\), pressure is lower than classical — quantum bunching makes the gas "softer".

A similar expansion gives the specific heat approaching the classical value \(C_V = \frac{3}{2}Nk\) from above as \(T\to\infty\).

5. Upper Bound on Excited Particles and Bose–Einstein Condensation

From

\[ \frac{N_e}{V} = \frac{1}{\lambda^3} g_{3/2}(z), \]

with \(N_e\) the number of particles in all excited states \(\varepsilon \neq 0\):

• \(g_{3/2}(z)\) is monotonically increasing for \(0<z<1\).
• As \(z\to 1\), \(g_{3/2}(z)\) approaches \(g_{3/2}(1) = \zeta(3/2) \approx 2.612\).

For given \(T\) and \(V\), there is a maximum number of particles in excited states:

\[ N_e \le V \frac{1}{\lambda^3} \zeta(3/2) = V \left(\frac{2\pi m kT}{h^2}\right)^{3/2} \zeta(3/2). \]

• If \(N < N_e^{\text{max}}\): everything is fine, \(N_e\simeq N\) and \(N_0\) is negligible.
• If \(N > N_e^{\text{max}}\): excited states are "saturated"; extra particles accumulate in the ground state.

That macroscopic occupation of the ground state is Bose–Einstein condensation.

For fixed \(N\), the critical temperature \(T_c\) is defined by

\[ N = V \frac{1}{\lambda_c^3} \zeta(3/2) \quad\Rightarrow\quad T_c = \frac{h^2}{2\pi m k} \left(\frac{n}{\zeta(3/2)}\right)^{2/3}. \]

For \(T < T_c\):

\[ N_e = N \left(\frac{T}{T_c}\right)^{3/2}, \qquad N_0 = N - N_e = N\left[1 - \left(\frac{T}{T_c}\right)^{3/2}\right]. \]

The condition for condensation can also be written as

\[ n > \frac{1}{\lambda^3} \zeta(3/2), \]

or, in terms of volume per particle \(v\),

\[ \frac{v}{\lambda^3} < \frac{1}{\zeta(3/2)} \simeq 0.382. \]

6. Behavior of the Fugacity

Using \(g_{3/2}(z) = (\lambda^3/v)^{-1}\) for \(T\le T_c\):

• For \(T < T_c\), \(v/\lambda^3 \le 1/\zeta(3/2)\), the only consistent solution is \(z \approx 1\). Below \(T_c\) the fugacity is essentially pinned at 1.
• For \(T > T_c\), \(v/\lambda^3 > 1/\zeta(3/2)\), we have \(z<1\), decreasing toward 0 as \(T\to\infty\).

7. Pressure and the \(P\)-\(T\) Diagram

General expression:

\[ P = \frac{kT}{\lambda^3} g_{5/2}(z). \]

• For \(T < T_c\): \(z\simeq 1\), so

\[ P(T) = \frac{kT}{\lambda^3} \zeta(5/2) \propto T^{5/2}. \]

Notice: no dependence on density \(n\) → infinite isothermal compressibility in the condensed phase.

• At \(T=T_c\):

\[ P(T_c) = \frac{\zeta(5/2)}{\zeta(3/2)} n k T_c \approx 0.5134 \, n k T_c. \]

At the transition, an ideal Bose gas exerts about half the pressure of a classical ideal gas.

• For \(T > T_c\):

\[ P = \frac{N}{V} kT \frac{g_{5/2}(z)}{g_{3/2}(z)}, \]

with \(z(T)\) determined implicitly by \(g_{3/2}(z) = \lambda^3 n\).

8. Internal Energy and Specific Heat

From thermodynamics,

\[ U = -\left(\frac{\partial}{\partial \beta} \ln\mathcal{Q} \right)_{z,V} = \frac{3}{2} \frac{V k T}{\lambda^3} g_{5/2}(z). \]

For all \(T\):

\[ U = \frac{3}{2} NkT \frac{g_{5/2}(z)}{g_{3/2}(z)}. \]

Then

\[ C_V = \left(\frac{\partial U}{\partial T}\right)_{V,N}. \]

For \(T < T_c\): \(z=1\), and \(N_e = N(T/T_c)^{3/2}\), giving

\[ \frac{C_V}{Nk} = \frac{15}{4} \left(\frac{T}{T_c}\right)^{3/2} \frac{\zeta(5/2)}{\zeta(3/2)}, \]

which rises as \(T^{3/2}\).

At \(T=T_c\):

\[ \frac{C_V(T_c)}{Nk} = \frac{15}{4} \frac{\zeta(5/2)}{\zeta(3/2)} \approx 1.925, \]

higher than the classical \(3/2\).

For \(T > T_c\): using the relation

\[ \frac{1}{z} \left(\frac{\partial z}{\partial T}\right)_v = -\frac{3}{2T} \frac{g_{3/2}(z)}{g_{1/2}(z)}, \]

we arrive at

\[ \frac{C_V}{Nk} = \frac{15}{4} \frac{g_{5/2}(z)}{g_{3/2}(z)} - \frac{9}{4} \frac{g_{3/2}(z)}{g_{1/2}(z)}. \]

As \(z\to 1\) from above, the second term vanishes, giving the same 1.925 value at \(T_c^+\).

• \(C_V\) itself is continuous at \(T_c\),
• but its derivative has a jump (a cusp).

This produces the characteristic "λ-shaped" curve: \(C_V\) grows from 0 at \(T=0\), peaks at \(T_c\), and decreases to the classical limit.

9. Isotherms and the \(P\)-\(v\) Diagram

Condensation sets in at a critical volume

\[ v_c = \frac{\lambda^3}{\zeta(3/2)} \propto T^{-3/2}. \]

• For \(v > v_c\): normal gas, \(z<1\), standard equation of state.
• For \(v < v_c\): gas is partly condensed and pressure is independent of \(v\):

\[ P_0(T) = \frac{kT}{\lambda^3} \zeta(5/2). \]

The region \(v<v_c\) at fixed \(T\) is a horizontal line in the \(P\)-\(v\) isotherm: the mixed phase, analogous to liquid–gas coexistence. The boundary transition line satisfies

\[ P_0 v_c^{5/3} = \frac{h^2}{2\pi m} \frac{\zeta(5/2)}{\big[\zeta(3/2)\big]^{5/3}} = \text{const}. \]

10. Adiabats and the Polytropic Law

The entropy is derived using

\[ U - TS + PV = \mu N. \]

For \(T > T_c\):

\[ \frac{S}{Nk} = \frac{5}{2} \frac{g_{5/2}(z)}{g_{3/2}(z)} - \ln z. \]

For \(T \le T_c\) (where \(z=1\), \(\ln z = 0\)):

\[ \frac{S}{Nk} = \frac{5}{2} \frac{v}{\lambda^3} \zeta(5/2) = \frac{5}{2} \left(\frac{T}{T_c}\right)^{3/2} \frac{\zeta(5/2)}{\zeta(3/2)}. \]

A reversible adiabatic process has constant \(S\) and \(N\). In both regimes this yields

\[ v T^{3/2} = \text{const}, \]

and using the equation of state,

\[ P T^{-5/2} = \text{const}. \]

The adiabats of an ideal Bose gas obey the polytropic law

\[ P v^{5/3} = \text{const}, \]

the same exponent \(5/3\) as a classical ideal gas.

11. Ratio of Specific Heats

Even though the adiabatic exponent is \(5/3\), the actual ratio \(C_P / C_V\) is not equal to \(5/3\) except at very high temperatures.

From thermodynamic identities:

\[ \gamma = \frac{C_P}{C_V} = 1 + \frac{4}{9} \frac{C_V}{Nk} \frac{g_{1/2}(z)}{g_{3/2}(z)} = \frac{5}{3} \frac{g_{5/2}(z) g_{1/2}(z)}{\big[g_{3/2}(z)\big]^2}. \]

• For \(T\gg T_c\) (small \(z\)): \(\gamma \to 5/3\), classical result.
• Near \(T_c\): \(\gamma > 5/3\) with nontrivial temperature dependence.

The exponent in the adiabatic law is fixed by the microscopic equation of state, but the ratio of specific heats depends on how \(U\) varies with \(T\).

12. Entropy in the Mixed Phase

In the condensed regime \(T<T_c\), the condensate particles \(N_0\) are all in the same single-particle state and do not carry entropy. Only the excited particles \(N_e\) contribute:

\[ S = N_e \cdot \frac{5}{2} k \frac{\zeta(5/2)}{\zeta(3/2)} \propto N_e. \]

Entropy is proportional to the normal component. This supports a two-fluid model: a superfluid component (condensate) with zero entropy and a normal component with finite entropy and viscosity.

Big Picture

This analysis builds a complete thermodynamic picture of an ideal Bose gas:

• Quantum degeneracy sets in when \(n\lambda^3 \gtrsim 1\).
• At fixed \(N,V\), cooling through \(T_c\) causes a macroscopic fraction to fall into the ground state → Bose–Einstein condensation.
• Below \(T_c\): pressure depends only on \(T\), internal energy and entropy come only from excited particles, and specific heat shows a λ-type cusp.
• Isotherms exhibit a flat mixed-phase region; adiabats obey \(P v^{5/3} = \text{const}\).
• The model provides a qualitative framework for understanding superfluid \(^4\)He and other Bose-condensed systems.