Classical Limit and the Maxwell-Boltzmann Distribution

In the classical (dilute) limit, both Fermi-Dirac and Bose-Einstein distributions reduce to the Maxwell-Boltzmann distribution. The condition for the classical limit is

$e^{\beta(\varepsilon - \mu)} \gg 1$

For all relevant energies. This is equivalent to $n\lambda_{\mathrm{th}^3 \ll 1}$ (the thermal de Broglie wavelength is much smaller than the inter-particle spacing).

Theorem 7.1. In the classical limit:

$f_{\mathrm{FD}(\varepsilon) \approx f_{\mathrm{BE}(\varepsilon) \approx f_{\mathrm{MB}(\varepsilon) = e^{-\beta(\varepsilon - \mu)}}}}$

Proof. When $e^{\beta(\varepsilon - \mu)} \gg 1$The $+1$ or $-1$ in the denominator is negligible:

$\frac{1}{e^{\beta(\varepsilon - \mu)} \pm 1} \approx \frac{1}{e^{\beta(\varepsilon - \mu)}} = e^{-\beta(\varepsilon - \mu)}$

$\blacksquare$

For a classical ideal gas, the probability distribution of molecular speeds is

$f(v)\,dv = 4\pi\left(\frac{m}{2\pi k_BT}\right)^{3/2} v^2 e^{-mv^2/(2k_BT)}\,dv$

Characteristic speeds:

• Most probable: $v_p = \sqrt{2k_BT/m}$
• Mean: $\langle v \rangle = \sqrt{8k_BT/(\pi m)}$
• RMS: $v_{\mathrm{rms} = \sqrt{3k_BT/m}}$

The ordering is $v_p < \langle v \rangle < v_{\mathrm{rms}}$.

For a system of $N$ indistinguishable non-interacting particles, the canonical partition function factorises:

$Z_N = \frac{1}{N!} Z_1^N$

where $Z_1$ is the single-particle partition function. For a classical ideal gas in three dimensions:

$Z_1 = V \left(\frac{2\pi m k_B T}{h^2}\right)^{3/2} = \frac{V}{\lambda_{\mathrm{th}}^3}$

with thermal de Broglie wavelength $\lambda_{\mathrm{th}} = h/\sqrt{2\pi m k_B T}$.

The Helmholtz free energy is $F = -k_B T \ln Z_N$, from which all thermodynamic quantities follow:

$P = -\frac{\partial F}{\partial V} = \frac{N k_B T}{V}$

$S = -\frac{\partial F}{\partial T} = Nk_B\left[\ln\!\left(\frac{V}{N\lambda_{\mathrm{th}}^3}\right) + \frac{5}{2}\right]$

Theorem 7.2 (Equipartition). For a classical system in thermal equilibrium at temperature $T$, each quadratic degree of freedom in the Hamiltonian contributes $\frac{1}{2}k_B T$ to the mean energy.

For a monatomic ideal gas with 3 translational degrees of freedom: $\langle E \rangle = \frac{3}{2}Nk_B T$. For a diatomic gas with additional rotational degrees of freedom (at sufficiently high $T$): $\langle E \rangle = \frac{5}{2}Nk_B T$.

The equipartition theorem fails at low temperatures when quantum effects freeze out degrees of freedom (the equipartition theorem is a classical result valid only in the high-temperature limit).

The Maxwell-Boltzmann distribution can be derived by maximising the Boltzmann entropy $S = -k_B \sum_i p_i \ln p_i$ subject to constraints $\sum_i p_i = 1$ and $\sum_i p_i \varepsilon_i = \langle E \rangle$:

$\delta\left[-k_B \sum_i p_i \ln p_i - \alpha\left(\sum_i p_i - 1\right) - \beta\left(\sum_i p_i \varepsilon_i - \langle E \rangle\right)\right] = 0$

This yields $p_i = e^{-\alpha - 1} e^{-\beta \varepsilon_i}$, and normalisation gives:

$p_i = \frac{e^{-\beta\varepsilon_i}}{\sum_j e^{-\beta\varepsilon_j}} = \frac{e^{-\beta\varepsilon_i}}{Z_1}$

With $\beta = 1/(k_B T)$, this is the Maxwell-Boltzmann distribution for discrete energy states.

Problem. Find the density of an ideal gas at height $z$ in a uniform gravitational field, assuming constant temperature $T$. This is the barometric formula.

Problem. A gas of molecular mass $m$ at temperature $T$ effuses through a small hole. Find the distribution of speeds of the effusing molecules and the mean kinetic energy per effusing molecule.

Problem. Estimate the mean free path of nitrogen molecules in air at STP ($T = 273$ K, $P = 1$ atm). The molecular diameter of N$_2$ is approximately $0.37$ nm.

The Maxwell-Boltzmann distribution fails when quantum effects become significant:

• Degenerate Fermi gases (high density, low temperature): Fermi-Dirac statistics must be used; the Pauli exclusion principle prevents multiple occupancy of quantum states.
• Bose-Einstein condensation occurs when $n\lambda_{\mathrm{th}}^3 \gtrsim 2.612$; the classical approximation breaks down as bosons accumulate in the ground state.
• Equipartition failure at low temperatures: rotational and vibrational degrees of freedom freeze out when $k_B T \ll \hbar\omega$, violating the classical prediction.