Part II: Quantum Gases | Chapter 1

Ideal Quantum Gases

The grand potential approach, virial expansion, polylogarithm functions, and thermodynamics of ideal Bose and Fermi gases

Historical Context

The theory of ideal quantum gases was developed in the 1920s by Bose, Einstein, Fermi, and Dirac. The mathematical framework centers on the polylogarithm (or Fermi-Dirac and Bose-Einstein) functions, which were systematically studied by Dingle, Robinson, and others in the 1950s. These functions provide exact expressions for all thermodynamic quantities as power series in the fugacity \(z = e^{\beta\mu}\).

The virial expansion, relating pressure to density through virial coefficients, bridges quantum statistics and classical gas theory, revealing how quantum effects manifest as effective interactions even for non-interacting particles.

1. Grand Potential for Quantum Gases

Derivation 1: Pressure as a Polylogarithm

The grand potential \(\Phi_G = -PV\) for an ideal quantum gas is:

\[PV = \mp k_BT \sum_{\mathbf{k}} \ln\left(1 \mp ze^{-\beta\epsilon_k}\right)\]

where the upper sign is for bosons, lower for fermions. Converting to an integral using the 3D density of states \(g(\epsilon) = (4\pi V g_s/h^3)(2m)^{3/2}\epsilon^{1/2}\):

\[PV = \mp k_BT \int_0^{\infty} g(\epsilon)\ln(1 \mp ze^{-\beta\epsilon})\,d\epsilon\]

Expanding the logarithm and integrating term by term:

\[\frac{P}{k_BT} = \frac{g_s}{\lambda_{dB}^3} \sum_{l=1}^{\infty} \frac{(\pm z)^l}{l^{5/2}} = \frac{g_s}{\lambda_{dB}^3} f_{5/2}^{\pm}(z)\]

where \(f_{\nu}^+(z) = \text{Li}_{\nu}(-z) \cdot (-1)\) for fermions and\(f_{\nu}^-(z) = \text{Li}_{\nu}(z) = g_{\nu}(z)\) for bosons. The polylogarithm is:

\[\text{Li}_{\nu}(z) = \sum_{l=1}^{\infty} \frac{z^l}{l^{\nu}} = \frac{1}{\Gamma(\nu)}\int_0^{\infty}\frac{x^{\nu-1}}{z^{-1}e^x - 1}\,dx\]

2. Particle Number and Chemical Potential

Derivation 2: Implicit Equation for \(\mu(T, n)\)

The average particle number:

\[\frac{N}{V} = \frac{g_s}{\lambda_{dB}^3}\sum_{l=1}^{\infty}\frac{(\pm z)^l}{l^{3/2}} = \frac{g_s}{\lambda_{dB}^3} f_{3/2}^{\pm}(z)\]

This implicitly determines \(z(\beta, n)\) and hence \(\mu(T, n)\). For the ideal Bose gas, \(z \leq 1\), which limits\(f_{3/2}^-(z) \leq \zeta(3/2) \approx 2.612\). This bound leads to Bose-Einstein condensation below a critical temperature.

Summary of Key Functions

Pressure: \(P\lambda_{dB}^3/(g_s k_BT) = f_{5/2}^{\pm}(z)\)

Density: \(n\lambda_{dB}^3/g_s = f_{3/2}^{\pm}(z)\)

Energy: \(E/(Nk_BT) = \frac{3}{2}f_{5/2}^{\pm}(z)/f_{3/2}^{\pm}(z)\)

Entropy: \(S/(Nk_B) = \frac{5}{2}f_{5/2}^{\pm}(z)/f_{3/2}^{\pm}(z) - \ln z\)

3. The Virial Expansion

Derivation 3: Quantum Virial Coefficients

Eliminating the fugacity order by order, we obtain the equation of state as an expansion in density:

\[\frac{P}{nk_BT} = \sum_{l=1}^{\infty} a_l \left(\frac{n\lambda_{dB}^3}{g_s}\right)^{l-1}\]

The first few virial coefficients are:

\[a_1 = 1, \qquad a_2 = \mp\frac{1}{2^{5/2}}, \qquad a_3 = \frac{1}{3^{5/2}} - \frac{2}{2^{5/2} \cdot 2^{3/2}}\]

For fermions (\(a_2 = +1/4\sqrt{2} > 0\)), the effective interaction is repulsive (Pauli pressure). For bosons (\(a_2 = -1/4\sqrt{2} < 0\)), it is attractive (bunching tendency).

Physical Interpretation

The quantum second virial coefficient \(a_2\) arises purely from exchange symmetry, not from any physical interaction. Bosons (minus sign) tend to cluster, making the gas easier to compress; fermions (plus sign) resist compression due to Pauli exclusion. This effective interaction has range \(\sim \lambda_{dB}\) and strength \(\sim k_BT\).

4. Energy and Specific Heat

Derivation 4: Energy from the Grand Potential

Using \(PV = \frac{2}{3}E\) (valid for non-relativistic ideal gases):

\[\frac{E}{N} = \frac{3}{2}k_BT\frac{f_{5/2}^{\pm}(z)}{f_{3/2}^{\pm}(z)}\]

The specific heat at constant volume:

\[\frac{C_V}{Nk_B} = \frac{15}{4}\frac{f_{5/2}^{\pm}(z)}{f_{3/2}^{\pm}(z)} - \frac{9}{4}\frac{f_{3/2}^{\pm}(z)}{f_{1/2}^{\pm}(z)}\]

In the classical limit (\(z \to 0\)), \(C_V \to \frac{3}{2}Nk_B\). For fermions at low \(T\), \(C_V \propto T\) (Sommerfeld expansion). For bosons below \(T_c\), \(C_V \propto T^{3/2}\).

5. Equation of State

Derivation 5: Compressibility and Bulk Modulus

The isothermal compressibility for quantum gases:

\[\kappa_T = \frac{1}{nk_BT}\frac{f_{3/2}^{\pm}(z)}{f_{1/2}^{\pm}(z)}\]

For an ideal Fermi gas at \(T = 0\):

\[\kappa_T^{-1} = \frac{2}{3}n\epsilon_F \quad \text{(bulk modulus = degeneracy pressure)}\]

For a Bose gas near \(T_c\), \(\kappa_T \to \infty\) as\(f_{1/2}^-(z \to 1) \to \infty\), signaling the instability associated with Bose-Einstein condensation.

The isothermal bulk modulus \(B_T = 1/\kappa_T\) measures the resistance to compression. For white dwarf stars, the electron degeneracy pressure provides\(B_T \sim n\epsilon_F \sim 10^{22}\) Pa, supporting the star against gravitational collapse.

6. Applications

Photon Gas Thermodynamics

For photons (\(\mu = 0, z = 1, g_s = 2\)), the energy density is:

\[\frac{E}{V} = \frac{3PV}{V} = 3 \cdot \frac{2k_BT}{\lambda_{\gamma}^3}\zeta(5/2) \cdot k_BT\]

More directly, using the photon dispersion \(\epsilon = \hbar ck\):\(E/V = (\pi^2/15)(k_BT)^4/(\hbar c)^3\).

Phonon Gas: Debye Model

Phonons are bosonic quasiparticles with \(\mu = 0\) and a linear dispersion\(\omega = c_s k\) up to a cutoff (Debye frequency). The specific heat:

\[C_V = 9Nk_B\left(\frac{T}{\Theta_D}\right)^3\int_0^{\Theta_D/T}\frac{x^4 e^x}{(e^x - 1)^2}\,dx\]

At low \(T\): \(C_V \propto T^3\) (Debye law). At high \(T\): \(C_V \to 3Nk_B\) (Dulong-Petit).

7. Computational Exploration

This simulation computes polylogarithm functions, equation of state for Bose and Fermi gases, quantum virial coefficients, and the Debye specific heat.

Ideal Quantum Gases: Polylogarithms, Equation of State, and Debye Model

Python

script.py162 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ============================================================
# Ideal Quantum Gases: Polylogarithms and Thermodynamics
# ============================================================

def polylog(nu, z, nterms=200):
    """Compute polylogarithm Li_nu(z) by series summation (overflow-safe)."""
    result = 0.0
    log_z = np.log(np.abs(z)) if np.abs(z) > 0 else -np.inf
    sign_z = np.sign(z)
    for l in range(1, nterms + 1):
        log_term = l * log_z - nu * np.log(l)
        if log_term > 700:  # prevent overflow
            break
        term = (sign_z ** l) * np.exp(log_term)
        result += term
        if abs(term) < 1e-15 * max(abs(result), 1e-30):
            break
    return result

def fermi_f(nu, z, nterms=200):
    """Fermi function f_nu(z) = -Li_nu(-z)."""
    return -polylog(nu, -z, nterms)

fig, axes = plt.subplots(2, 2, figsize=(12, 10))
fig.suptitle('Ideal Quantum Gases', fontsize=16, fontweight='bold', color='white')
fig.patch.set_facecolor('#0a0a0a')

for ax in axes.flat:
    ax.set_facecolor('#1a0a0a')
    ax.tick_params(colors='white')
    ax.xaxis.label.set_color('white')
    ax.yaxis.label.set_color('white')
    ax.title.set_color('#fca5a5')
    for spine in ax.spines.values():
        spine.set_color('#333')

# --- Panel 1: Polylogarithm functions ---
ax1 = axes[0, 0]
z_vals = np.linspace(0.01, 0.99, 200)
for nu in [0.5, 1.0, 1.5, 2.0, 2.5]:
    bose_vals = [polylog(nu, z) for z in z_vals]
    ax1.plot(z_vals, bose_vals, linewidth=1.5, label='Li_' + str(nu) + '(z)')
ax1.set_xlabel('Fugacity z')
ax1.set_ylabel('Li_nu(z)')
ax1.set_title('Bose Polylogarithm Functions')
ax1.legend(fontsize=8, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 2: P/(nkT) vs n*lambda^3 for both statistics ---
ax2 = axes[0, 1]
nlam3 = np.linspace(0.01, 3.0, 100)
# Bose gas
P_over_nkT_bose = []
P_over_nkT_fermi = []
for nl in nlam3:
    # Find z such that f_{3/2}(z) = nl (Bose)
    z_lo, z_hi = 1e-10, 0.9999
    for _ in range(100):
        z_mid = (z_lo + z_hi) / 2
        if polylog(1.5, z_mid) < nl:
            z_lo = z_mid
        else:
            z_hi = z_mid
    z_b = (z_lo + z_hi) / 2
    P_over_nkT_bose.append(polylog(2.5, z_b) / nl)

# Find z such that fermi_f_{3/2}(z) = nl (Fermi)
    z_lo, z_hi = 1e-10, 1000.0
    for _ in range(100):
        z_mid = (z_lo + z_hi) / 2
        if fermi_f(1.5, z_mid) < nl:
            z_lo = z_mid
        else:
            z_hi = z_mid
    z_f = (z_lo + z_hi) / 2
    P_over_nkT_fermi.append(fermi_f(2.5, z_f) / nl)

ax2.plot(nlam3, P_over_nkT_bose, color='#f87171', linewidth=2, label='Bose')
ax2.plot(nlam3, P_over_nkT_fermi, color='#60a5fa', linewidth=2, label='Fermi')
ax2.axhline(y=1, color='white', linestyle=':', alpha=0.3, label='Classical')
ax2.set_xlabel('n * lambda_dB^3')
ax2.set_ylabel('P / (n k_B T)')
ax2.set_title('Equation of State')
ax2.legend(fontsize=8, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 3: Virial expansion ---
ax3 = axes[1, 0]
nl_virial = np.linspace(0, 2.0, 200)
# Classical
P_classical = np.ones_like(nl_virial)
# Virial to 2nd order
a2_bose = -1.0 / (2**2.5)
a2_fermi = 1.0 / (2**2.5)
P_bose_v2 = 1 + a2_bose * nl_virial
P_fermi_v2 = 1 + a2_fermi * nl_virial

ax3.plot(nl_virial, P_classical, 'w--', linewidth=1, alpha=0.5, label='Classical')
ax3.plot(nl_virial, P_bose_v2, color='#f87171', linewidth=2, linestyle='--', label='Bose (2nd virial)')
ax3.plot(nl_virial, P_fermi_v2, color='#60a5fa', linewidth=2, linestyle='--', label='Fermi (2nd virial)')
ax3.plot(nlam3, P_over_nkT_bose, color='#f87171', linewidth=1, alpha=0.5, label='Bose (exact)')
ax3.plot(nlam3, P_over_nkT_fermi, color='#60a5fa', linewidth=1, alpha=0.5, label='Fermi (exact)')
ax3.set_xlabel('n * lambda_dB^3')
ax3.set_ylabel('P / (n k_B T)')
ax3.set_title('Virial Expansion vs Exact')
ax3.set_ylim(0.5, 2.0)
ax3.legend(fontsize=7, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 4: Debye specific heat ---
ax4 = axes[1, 1]
T_over_Td = np.linspace(0.01, 2.5, 200)
Cv_debye = []
for t in T_over_Td:
    xmax = 1.0 / t
    x = np.linspace(0.001, xmax, 1000)
    integrand = x**4 * np.exp(x) / (np.exp(x) - 1)**2
    integral = np.trapezoid(integrand, x)
    Cv_debye.append(9.0 * t**3 * integral)
Cv_debye = np.array(Cv_debye)

ax4.plot(T_over_Td, Cv_debye / 3, color='#fb923c', linewidth=2, label='Debye model')
ax4.axhline(y=1, color='white', linestyle=':', alpha=0.3, label='Dulong-Petit (3Nk)')
# T^3 law
T_low = T_over_Td[T_over_Td < 0.3]
ax4.plot(T_low, (4 * np.pi**4 / 5) * T_low**3 / 3, 'w--', linewidth=1, alpha=0.5, label='T^3 law')
ax4.set_xlabel('T / Theta_D')
ax4.set_ylabel('C_V / (3Nk_B)')
ax4.set_title('Debye Specific Heat')
ax4.legend(fontsize=8, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.close()
print("Plot saved to output.png")

print("\n" + "=" * 60)
print("IDEAL QUANTUM GASES: NUMERICAL RESULTS")
print("=" * 60)

print("\n--- Polylogarithm values at z = 1 (zeta functions) ---")
for nu in [1.5, 2.5, 3.5]:
    val = polylog(nu, 0.9999)
    print("  zeta(" + str(nu) + ") = Li_" + str(nu) + "(1) = " + str(round(val, 4)))

print("\n--- Quantum virial coefficients ---")
print("  a_2 (Bose)  = " + str(round(a2_bose, 6)))
print("  a_2 (Fermi) = " + str(round(a2_fermi, 6)))

print("\n--- Equation of state at n*lam^3 = 1 ---")
idx = np.argmin(np.abs(nlam3 - 1.0))
print("  P/(nkT) Bose  = " + str(round(P_over_nkT_bose[idx], 4)))
print("  P/(nkT) Fermi = " + str(round(P_over_nkT_fermi[idx], 4)))

print("\n--- Debye specific heat ---")
for t_val in [0.1, 0.5, 1.0, 2.0]:
    idx_t = np.argmin(np.abs(T_over_Td - t_val))
    print("  T/Theta_D = " + str(t_val) + ": C_V/(3Nk) = " + str(round(Cv_debye[idx_t] / 3, 4)))
print("\n" + "=" * 60)

Click Run to execute the Python code

Code will be executed with Python 3 on the server

8. Summary

Core Formulas

Pressure: \(P\lambda^3/(k_BT) = g_s f_{5/2}(z)\)
Density: \(n\lambda^3 = g_s f_{3/2}(z)\)
Virial: \(a_2 = \mp 1/4\sqrt{2}\)
Debye: \(C_V \propto T^3\) at low T

Physical Insights

Bosons lower pressure, fermions raise it
Quantum effects = effective interactions from exchange
Polylogarithms encode all thermodynamics
Phonon gas recovers Debye T-cubed law

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