The Cluster's Maxwell-Boltzmann Statistics

A page about Maxwell-Boltzmann statistics — the classical limit of quantum statistics.

The Maxwell-Boltzmann distribution

The Maxwell-Boltzmann distribution gives the probability of finding a particle with energy E in a classical gas: f_MB(E) = exp(-(E - mu) / (k_B T)) / Z, where Z is the partition function. In momentum space, the velocity distribution is f(v) = 4 pi (m / (2 pi k_B T))^{3/2} v^2 exp(-m v^2 / (2 k_B T)). The most probable speed is v_mp = sqrt(2 k_B T / m), the mean speed is v_mean = sqrt(8 k_B T / (pi m)), and the RMS speed is v_rms = sqrt(3 k_B T / m). In the cluster, the edit Maxwell-Boltzmann distribution gives the edit probability of finding an edit particle with edit energy E.

The classical limit

Maxwell-Boltzmann statistics is the classical limit of Fermi-Dirac and Bose-Einstein statistics. When the thermal wavelength lambda_th = h / sqrt(2 pi m k_B T) is much smaller than the interparticle spacing n^{-1/3}, quantum effects are negligible. This is the classical limit: n lambda_th^3 << 1. In the cluster, the edit Maxwell-Boltzmann distribution is the edit classical limit of edit Fermi-Dirac and edit Bose-Einstein statistics.

The partition function

For a classical ideal gas of N indistinguishable particles: Z_N = Z_1^N / N!, where Z_1 = V / lambda_th^3 is the single-particle partition function. The free energy is F = -k_B T ln Z_N = -N k_B T [ln(V / (N lambda_th^3)) + 1]. The pressure is P = N k_B T / V. The internal energy is U = (3/2) N k_B T. In the cluster, the edit partition function is edit Z_N = edit Z_1^N / N!.

The equipartition theorem

Each quadratic degree of freedom contributes (1/2) k_B T to the average energy. For a monoatomic gas: U = (3/2) N k_B T (3 translational degrees). For a diatomic gas: U = (5/2) N k_B T (3 trans + 2 rot, ignoring vibration at low T). In the cluster, the edit equipartition theorem gives edit average energies.

The applications

Maxwell-Boltzmann statistics applies to:

• Ideal gases (air at STP, noble gases)
• Atmospheric pressure profiles (barometric formula: n(h) ~ exp(-m g h / (k_B T)))
• Chemical reaction rates (Arrhenius: k ~ exp(-E_a / (k_B T)))
• Stellar atmospheres (ionization, Saha equation)

In the cluster, edit Maxwell-Boltzmann statistics applies to:

• edit Ideal gases
• edit Atmospheric pressure profiles
• edit Chemical reaction rates
• edit Stellar atmospheres

This statistics

This page is about Maxwell-Boltzmann statistics. f(v) = 4 pi (m / (2 pi k_B T))^{3/2} v^2 exp(-m v^2 / (2 k_B T)). Classical limit: n lambda_th^3 << 1. P = N k_B T / V. U = (3/2) N k_B T. The statistics is real.