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Root-Mean-Square Speed

The average speed of particles in a gas, calculated as \sqrt{\frac{3k_BT}{m}}, a central concept in kinetic theory.
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The statement of the theorem

Let S\mathcal{S} be a system of NN particles of mass mm confined to a volume VV at temperature TT. The phase space is Γ={(q,p)∈R3N×R3N}\Gamma = \{({\mathbf{q}}, \mathbf{p}) \in \mathbb{R}^{3N} \times \mathbb{R}^{3N}\}. The Hamiltonian is H(q,p)=∑i=1Npi22m+U(q)\mathcal{H}({\mathbf{q}}, \mathbf{p}) = \sum_{i=1}^{N} \frac{\mathbf{p}_i^2}{2m} + U(\mathbf{q}). Assuming the system is in thermal equilibrium, the probability density function is the canonical ensemble distribution ρ∝e−H/kBT\rho \propto e^{-\mathcal{H}/k_B T}. The average kinetic energy ⟨K⟩\langle K \rangle is calculated as ⟨K⟩=∫Γ(∑i=1Npi22m)ρdqdp/Z\langle K \rangle = \int_{\Gamma} \left(\sum_{i=1}^{N} \frac{\mathbf{p}_i^2}{2m}\right) \rho d\mathbf{q} d\mathbf{p} / Z. By the equipartition theorem, ⟨K⟩=32NkBT\langle K \rangle = \frac{3}{2} N k_B T. For a single particle, the average kinetic energy is ⟨K1⟩=12m⟨v2⟩=32kBT\langle K_1 \rangle = \frac{1}{2} m \langle v^2 \rangle = \frac{3}{2} k_B T. The Root-Mean-Square speed, vrmsv_{rms}, is defined by the expectation value of the squared speed: vrms=⟨v2⟩v_{rms} = \sqrt{\langle v^2 \rangle}. Therefore, vrms=3kBTmv_{rms} = \sqrt{\frac{3 k_B T}{m}}.
Source: Wikipedia