Equipartition theorem

The classical equipartition theorem connects temperature with the average energy stored in independent quadratic terms. It turns a count of active degrees of freedom into predictions for internal energy and heat capacity. It does not say that every particle or mode has equal energy at every instant.

One quadratic term

Suppose an independent coordinate \(q\) appears in the classical energy as

\[E_q=\frac12a q^2\]

At classical thermal equilibrium, its mean contribution is

\[\left\langle E_q\right\rangle=\frac12k_BT\]

where \(k_B\) is Boltzmann's constant and \(T\) is measured in kelvin. The positive coefficient \(a\) changes the width of the coordinate distribution, but not its mean energy.

For \(f\) independent, thermally active terms,

\[\left\langle E_{\mathrm{quadratic}}\right\rangle=\frac{f}{2}k_BT\]

The count must say whether it applies per particle, per molecule, per mode, or to the complete finite system.

Translational motion

A free classical point particle in three dimensions has

\[K=\frac{p_x^2}{2m}+\frac{p_y^2}{2m}+\frac{p_z^2}{2m}\]

The three independent momentum terms give

\[\langle K\rangle=\frac32k_BT\]

For \(N\) monatomic ideal-gas particles,

\[U=\frac32Nk_BT=\frac32nRT\]

where \(n\) and \(R=N_Ak_B\). If the degree count stays constant, the molar heat capacity at constant volume is

\[C_V=\frac{1}{n}\left(\frac{\partial U}{\partial T}\right)_V=\frac32R\]

Rotation and vibration

An ideal linear rigid molecule has two active rotational kinetic terms: rotation about the molecular axis has negligible moment of inertia in the simple model. At ordinary temperatures where both rotations are active, three translations plus two rotations give

\[\langle E\rangle=\frac52k_BT,\qquad C_V=\frac52R\]

A nonlinear rigid molecule generally has three rotational terms.

One harmonic vibrational normal mode contains a quadratic kinetic term and a quadratic potential term:

\[E_{\mathrm{vib}}=\frac{P^2}{2M_{\mathrm{eff}}}+\frac12M_{\mathrm{eff}}\omega^2Q^2\]

Each term contributes \(k_BT/2\), so one fully active vibration contributes \(k_BT\) and adds \(R\) to molar \(C_V\). Counting a vibration as only one half contribution forgets its conjugate momentum.

Classical solid limit

In a harmonic model of a three-dimensional solid, each atom has three kinetic and three potential quadratic terms. Equipartition predicts

\[\langle E\rangle=3k_BT,\qquad C_V\approx3R\]

This Dulong–Petit heat-capacity limit works well for many solids at sufficiently high temperature. Measured heat capacities decrease at low temperature, an important failure of classical equipartition.

Quantum freeze-out

A quantum harmonic mode has energy spacing \(\hbar\omega\). When \(k_BT\ll\hbar\omega\), thermal fluctuations rarely excite higher levels, so the mode does not supply its classical heat-capacity contribution. When the thermal scale becomes large relative to the spacing, classical behavior emerges.

Rotational and vibrational spacings differ, so their contributions turn on at different temperatures. The effective degree count is therefore not always a fixed molecular integer. At absolute zero, quantum zero-point energy may remain even though it does not produce the classical temperature-dependent heat capacity.

Heat capacity as an experimental test

Heat capacity measures how rapidly average internal energy changes as temperature changes. If the same set of quadratic terms remains active over a temperature interval, equipartition predicts a constant classical contribution from each term. A graph of internal energy against temperature is then approximately a straight line, and its slope reveals the effective number of active terms.

For a system containing \(N\) particles, the classical prediction can be written as

\[C_V=\left(\frac{\partial U}{\partial T}\right)_V=\frac{f}{2}Nk_B\]

Here \(f\) is the number of active quadratic terms per particle. This equation is a diagnostic, not a universal definition of heat capacity. It assumes that \(f\) does not change across the measured temperature range and that no phase transition or chemical change occurs.

Real measurements make the limitations visible. A monatomic ideal gas approaches the three-term prediction because translational motion dominates. A molecular gas can show larger heat capacity as rotational and then vibrational excitations become accessible. A solid approaches the Dulong–Petit value at high temperature but falls below it as quantum modes freeze out. The theorem therefore gives both a prediction and a way to notice when its assumptions have stopped working.

In a numerical calculation, estimate the slope from several equilibrated temperatures rather than differentiating two noisy points. Each temperature needs its own uncertainty estimate. A smooth change in slope can signal gradual activation of modes, while a sharp change may indicate a phase transition or a problem in the model.

Constraints change the count

For \(N\) free particles in three dimensions, stored momenta contain \(3N\) components. If total center-of-mass momentum is constrained, three collective components are removed, leaving \(3N-3\) for a kinetic-temperature estimate.

Rigid bonds, fixed particles, conserved angular momentum, and geometric constraints may remove more independent modes. Count the rank of independent constraints rather than the number of constraint records; redundant constraints must not be subtracted twice.

k_b, temperature = 1.380649e-23, 300.0
# Classical, thermally active quadratic terms per molecule.
terms = {"monatomic translation": 3, "linear rigid molecule": 5, "one vibration": 2}
for description, count in terms.items():
    mean_energy = 0.5 * count * k_b * temperature
    print(description, f"{mean_energy:.6g} J per molecule")

The registry should record why each term is included or excluded. This makes a mismatch diagnosable rather than encouraging a guessed correction factor.

Equality is statistical

Equipartition is an ensemble or sufficiently long-time average. Energy moves among degrees of freedom. At one instant, one particle may be fast and another slow; a harmonic mode may hold mostly potential energy and later mostly kinetic energy.

Small systems fluctuate. Consecutive simulation samples are correlated, so uncertainty should come from independent runs or time blocks longer than the energy-exchange correlation time. A correct total average can also hide failure to equilibrate between directions or subsystems. Compare mode-by-mode and directional estimates.

What the theorem does not cover

The simple half-\(k_BT\) count applies to independent quadratic terms in classical equilibrium. Anharmonic potentials, hard constraints, strong interactions, and driven systems need more careful analysis. Assigning one half contribution to every stored program variable is wrong.

A sheared gas, two counter-streaming beams, or two weakly coupled groups at different temperatures can have a plausible total mean energy without one equilibrium temperature. Equipartition is a prediction to test after equilibration, not an algorithm that creates equilibrium.

Verification

For independent Gaussian momenta, verify that each Cartesian kinetic contribution approaches \(k_BT/2\). Add one independent harmonic coordinate and momentum; expected energy should rise by \(k_BT\), not half that.

Remove center-of-mass momentum and confirm the finite-system estimator uses the reduced count. Constrain one coordinate and ensure both the dynamics and term registry remove it. Divide a trajectory into blocks and report variation, not only one combined mean.

Finally compare energy at several temperatures. A systematic mismatch can indicate a wrong constraint count, incomplete equilibration, numerical bias, anharmonic behavior, or quantum freeze-out. The theorem helps identify the question; it does not decide among these causes by itself.

A worked Python notebook

Count both the kinetic and potential quadratic terms of one classical harmonic vibration. The question for this calculation is precise: how does mean energy of one active vibration change when we change temperature, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

Quantum freeze-out makes the classical result unreliable when temperature is too low for the mode. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 100 to 1000 K. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for temperature, expressed in K. The output is mean energy of one active vibration, expressed in J. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = 1.380649 \cdot 10^{-23} q\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 300 K, the expression gives 4.14195e-21 J. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("1.380649e-23*q", locals={"q": q})
reference_input = 300.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "K")
print("Predicted mean energy of one active vibration:", reference_output, "J")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 1.38065e-21 J. The midpoint, at 550 K, gives 7.59357e-21 J. The upper endpoint gives 1.38065e-20 J. Read these together with the assumptions: they describe different choices of temperature within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([100.0, 550.0, 1000.0])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} K -> {output_value:.6g} J")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows temperature; the vertical axis shows mean energy of one active vibration. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(100.0, 1000.0, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Temperature (K)")
ax.set_ylabel("Mean energy of one active vibration (J)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does mean energy of one active vibration change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 300 K is 1.38065e-23 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = 1.380649 \cdot 10^{-23}\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 304.5
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "J")
print("Full expression:", full_prediction, "J")
print("Approximation difference:", full_prediction - linear_prediction, "J")

Here the changed input is 304.5 K. The tangent estimate is 4.20408e-21 J, while the full expression gives 4.20408e-21 J. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 900.0
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of temperature inside the stated interval, predict whether mean energy of one active vibration will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: Quantum freeze-out makes the classical result unreliable when temperature is too low for the mode. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of J: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(1000.0) - evaluate(100.0))
for points in [101, 201, 401]:
    grid = np.linspace(100.0, 1000.0, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

• OpenStax, University Physics Volume 2, Section 2.3, “Heat Capacity and Equipartition of Energy”.
• MIT OpenCourseWare 8.01SC, Chapter 29, “Kinetic Theory of Gases: Equipartition of Energy and the Ideal Gas Law”.
• Benjamin Crowell, Light and Matter, temperature, molecular degrees of freedom, and heat capacity.
• The Physics Hypertextbook, “Kinetic-Molecular Theory”.