Law of Equipartition of Energy Simulation

Mathematical Foundations of the Equipartition Theorem

• Classical Statement: In thermal equilibrium at absolute temperature $T$, every independent quadratic variable appearing in the Hamiltonian system contributes an identical average energy of: $$\langle \epsilon \rangle = \frac{1}{2} k_B T$$
• Translational Degrees of Freedom ($f_{\text{trans}} = 3$): Kinetic motion along $x, y, z$ contributes $\frac{1}{2}mv_x^2 + \frac{1}{2}mv_y^2 + \frac{1}{2}mv_z^2 = \frac{3}{2}k_B T$.
• Rotational Degrees of Freedom ($f_{\text{rot}}$): Linear molecules have 2 rotation axes with non-negligible moment of inertia ($\frac{1}{2}I\omega_1^2 + \frac{1}{2}I\omega_2^2 = 2 \times \frac{1}{2}k_B T = k_B T$). Non-linear molecules possess 3 rotational axes contributing $\frac{3}{2}k_B T$.
• Vibrational Coordinates ($2 \times \text{Modes}$): Each normal vibrational mode comprises two quadratic energy terms: kinetic energy ($\frac{1}{2}\mu v^2$) and potential energy ($\frac{1}{2}kx^2$). Thus, each active vibrational mode absorbs $2 \times \frac{1}{2}k_B T = k_B T$.