The equipartition theorem states that in thermal equilibrium at temperature T, every quadratic term in a system's energy contributes on average exactly:
⟨E_i⟩ = (1/2) k_B T per quadratic degree of freedom
Each dumbbell here is a rigid-rotor-plus-spring model of a diatomic molecule with three energy channels, each simulated as an independent Langevin (heat-bath-coupled) degree of freedom:
Translation: m dv = -γv dt + √(2γk_BT dt)·ξ (3 components)
Rotation: I dω = -γω dt + √(2γk_BT dt)·ξ (2 components ⟂ bond)
Vibration: μ dv_b = -kx dt - γv_b dt + √(2γk_BT dt)·ξ, dx = v_b dt
The random force ξ and the drag coefficient γ are linked by the fluctuation-dissipation theorem — the same friction that damps a fast-moving mode also supplies exactly the noise needed to reheat it to (1/2)k_BT at equilibrium. Solve the resulting Fokker-Planck equation for any of these and the stationary velocity distribution is Maxwell-Boltzmann with variance k_BT/m (or k_BT/I), independent of m, I, k or γ — only T sets the scale. That's why the three measured bars converge to the same red line even though translation, rotation and the spring have completely different "masses" and stiffness.
- Temperature T — sets the bath energy scale; every bar's target moves to (1/2)k_BT.
- Bath coupling γ — how strongly each mode exchanges energy with the bath; higher γ reaches equilibrium faster but doesn't change the equilibrium value itself.
- Bond stiffness k — changes the vibration frequency and amplitude, but not its average energy: ⟨KE_vib⟩ = ⟨PE_vib⟩ = (1/2)k_BT regardless of k.
- Molecules N — more molecules average faster and with less statistical noise, exactly as a real macroscopic gas sample would.
Real-world relevance: classical equipartition is exactly why an ideal diatomic gas's classical heat capacity should be C_v = (7/2)R (3 translational + 2 rotational + 2 vibrational "half-kT" terms). Real gases measured only C_v ≈ (5/2)R at room temperature — the vibrational mode is quantum-mechanically frozen out because its energy-level spacing ħω greatly exceeds k_BT. That 19th-century mismatch between classical equipartition and measured heat capacities was one of the direct historical motivations for quantum theory.