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Maxwell-Boltzmann Statistics

Applies to ideal gases, providing the probability distribution of particle speeds based on temperature and mass.
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The statement of the theorem

Let ri∈R3\mathbf{r}_i \in \mathbb{R}^3 and \mathbf{p}_i \in \namespace{T}\mathbb{R}^3 be the position and momentum of the ii-th particle, respectively, for i=1,…,Ni=1, \dots, N. Define the Hamiltonian H(r,p)\mathcal{H}(\mathbf{r}, \mathbf{p}) for the ideal gas system as H=∑i=1Npi22m\mathcal{H} = \sum_{i=1}^{N} \frac{\mathbf{p}_i^2}{2m}. The canonical partition function is Z(β,V)=1N!h3N∫e−βHd3Npd3NrZ(\beta, V) = \frac{1}{N! h^{3N}} \int e^{-\beta \mathcal{H}} d^{3N}p d^{3N}r, where β=1/(kBT)\beta = 1/(k_B T). The probability density function P(v)P(\mathbf{v}) for the velocity vector v=(vx,vy,vz)\mathbf{v} = (v_x, v_y, v_z) of a single particle is given by the Maxwell-Boltzmann distribution:\n\nP(v)=(m2πkBT)3/2e−m(v)2/(2kBT)P(\mathbf{v}) = \left(\frac{m}{2\pi k_B T}\right)^{3/2} e^{-m(\mathbf{v})^2 / (2k_B T)}\n\nFurthermore, the expected value of the kinetic energy ⟨Ek⟩\langle E_k \rangle is derived from the equipartition theorem, yielding ⟨Ek⟩=32kBT\langle E_k \rangle = \frac{3}{2} k_B T.
Source: Wikipedia