The equipartition theorem

Equipment

• Energy-bar canvas for the degrees of freedom (drag to rotate the view)
• “Nhiệt độ” slider (0.3–3, normalized)
• “Tham số” slider (0.1–2, controls the number of active degrees of freedom)
• Quantitative-graph canvas with normalized axes
• Readout “T = … (normalized); control parameter = …”

Procedure

1. Watch energy bars vs temperature

   In section 1, each bar represents a degree of freedom and its height scales with $T$. Drag the “Nhiệt độ” slider from 0.3 up to 3 and watch all bars rise together; read “T = … (normalized)” on the readout and match the linear trend to $\langle H_x \rangle = \frac{1}{2}k_BT$.

2. Change the active degrees of freedom

   Move the “Tham số” slider through 0.5, 1.0, 1.5, 2.0; the number of lit bars (active degrees of freedom) changes accordingly. Note that total energy scales with the count — e.g., a monatomic gas molecule has 3 translational degrees so $\langle E \rangle = \frac{3}{2}k_BT$, while a harmonic oscillator has two quadratic terms so $\langle E \rangle = k_BT$.

3. Compare the quantitative view

   In section 2, the graph updates with both sliders on normalized axes; drag the canvas to change the angle. Hold “Nhiệt độ” fixed and vary “Tham số”, then do the reverse — confirm the energy changes linearly with $T$ and with the number of degrees of freedom, in the spirit of the virial–equipartition statement $\langle x\,\partial H/\partial x \rangle = k_BT$.

4. Predict the quantum limit

   Drag “Nhiệt độ” down to 0.3 — the model's lowest setting. Classically the bars still scale with $T$, but in a real system when $k_BT \ll \hbar\omega$ vibrational modes “freeze out” and contribute less than $\frac{1}{2}k_BT$; this is why solid heat capacities fall as $T \to 0$. Predict before reading: the classical heat capacity is $C_V = \frac{f}{2}k_B$ per particle for $f$ degrees of freedom.

Simulation

Experiment history

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