Classical Equipartition Theorem Derivation

Welcome to our next lesson. In our previous discussions, we explored how the canonical partition function $Z$ serves as a generating function for thermodynamic observables. We saw that first derivatives of $ln Z$ yield average quantities like internal energy, while second derivatives correspond to fluctuations, which we connected to response functions like the heat capacity. These fluctuations, we found, become negligible in the thermodynamic limit, justifying the equivalence of ensembles.

Today, we will delve into another cornerstone of classical statistical mechanics that emerges directly from the canonical formalism: the equipartition theorem. This theorem provides a remarkably simple and powerful rule for distributing the total average energy among the various degrees of freedom of a system in thermal equilibrium. To properly derive this classical result, we will first need to transition our description of the partition function from a sum over discrete quantum states to an integral over continuous classical phase space.

1. The Classical Partition Function

The equipartition theorem is fundamentally a result of classical statistical mechanics. Our definition of the partition function so far, $Z = \sum_{j} e^{- β E_{j}}$ , is inherently quantum, as it involves a sum over discrete energy eigenstates. The classical analogue involves integrating over all possible positions and momenta of the particles, which constitute the system's phase space.

For a single particle, the classical partition function is defined as an integral over its position $q$ and momentum $p$ :

Z_{1} = \frac{1}{h ^{3}} \int d^{3} q d^{3} p e^{- β H (q, p)}

where $H (q, p)$ is the classical Hamiltonian. The factor of $h^{3}$ , where $h$ is Planck's constant, might seem out of place. It essentially arises from the correspondence principle and ensures that the partition function remains dimensionless. It represents the "volume" of a single microstate in phase space, a concept rooted in the uncertainty principle ( $Δ q Δ p \sim ℏ$ ).

Given your background in quantum mechanics, you will find the formal derivation of this classical limit from the quantum partition function, $Z = Tr (e^{- β H^})$ , quite insightful. The following reading traces these steps.

2. Classical Gases

These notes by David Tong from the University of Cambridge begin by introducing the classical partition function and then provide a careful derivation showing how it emerges as the high-temperature (or classical) limit of the quantum trace formula.

First, read the introductory section "The Classical Partition Function" to see the definition. Then, focus on the subsection "From Quantum to Classical", which details the transition. The key steps involve inserting complete sets of position and momentum states and using the Baker-Campbell-Hausdorff formula, where terms of order \hbar are neglected.

2. The Equipartition Theorem: Statement and Proof

With the classical partition function established, we can now derive the theorem.

The Equipartition Theorem states that for a classical system in thermal equilibrium at temperature $T$ , if a degree of freedom $y$ (which can be any component of position or momentum) appears in the Hamiltonian only as a quadratic term of the form $a y^{2}$ , where $a$ is a constant, then the average contribution of that term to the total internal energy is $\frac{1}{2} k_{B} T$ .

⟨ a y^{2} ⟩ = \frac{1}{2} k_{B} T

This implies that every "quadratic" degree of freedom in the system gets an equal share of the thermal energy.

The proof of this theorem is a straightforward and elegant application of the phase-space integral for the canonical average. The average value of the term $a y^{2}$ is given by:

⟨ a y^{2} ⟩ = \frac{\int d Γ ( a y ^{2} ) e ^{- β H (Γ)}}{\int d Γ e ^{- β H (Γ)}}

where $d Γ$ represents the volume element in phase space. Since the Hamiltonian is separable in the variable $y$ , $H (Γ) = a y^{2} + H_{other}$ , the integrals factorize.

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These lecture notes by John McGreevy from UC San Diego provide a particularly clear and rigorous proof that you will appreciate.

Please read the section titled "Equipartition Theorem". The text first states the conditions and the result, and then provides a concise proof. The key is to notice how the integral for the average value separates, allowing the integrals over all other variables to cancel out, leaving a simple Gaussian integral to evaluate.

For a more visual, step-by-step walkthrough of the same derivation, the following video is an excellent supplement.

Equipartition Theorem

This video from the "Physical Chemistry" channel illustrates the generality of the theorem by focusing on the quadratic form of energy.

First, watch the segment identifying quadratic terms, where the narrator shows how kinetic and potential energies for translation, rotation, and vibration all have a quadratic dependence on some variable. Then, follow the general derivation from the partition function to the internal energy. Notice how the calculation shows that the average energy contribution is independent of the constant prefactor in the energy term.

3. Applications of the Equipartition Theorem

The power of the theorem lies in its broad applicability.

Monatomic Ideal Gas

For an ideal gas of $N$ non-interacting point particles, the Hamiltonian is purely kinetic:

H = i = 1 \sum N \frac{1}{2 m} (p_{i x}^{2} + p_{i y}^{2} + p_{i z}^{2})

There are $3 N$ quadratic terms in total (three momentum components for each of the $N$ particles). According to the equipartition theorem, the total average energy is:

E = ⟨ H ⟩ = 3 N \times (\frac{1}{2} k_{B} T) = \frac{3}{2} N k_{B} T

From this, we can immediately find the molar heat capacity at constant volume:

C_{V} = (\frac{\partial E}{\partial T})_{V} = \frac{3}{2} N k_{B}

This is a celebrated result that accurately describes noble gases at most temperatures.

Classical Harmonic Oscillator

Consider a 1D harmonic oscillator with the Hamiltonian:

H = \frac{p ^{2}}{2 m} + \frac{1}{2} κ x^{2}

This system has two quadratic degrees of freedom: one from kinetic energy ( $p^{2}$ ) and one from potential energy ( $x^{2}$ ). The average energy is therefore:

E = ⟨ H ⟩ = 2 \times (\frac{1}{2} k_{B} T) = k_{B} T

For a solid crystal of $N$ atoms, modeling each atom as a 3D harmonic oscillator (vibrating independently in three directions), we have $3 N$ such oscillators. Each 1D oscillator has two quadratic terms, so the total number of quadratic degrees of freedom is $6 N$ . This gives a total energy of $E = 3 N k_{B} T$ and a heat capacity of $C_{V} = 3 N k_{B}$ . This is the Dulong-Petit law, an early success of statistical mechanics in describing the heat capacity of solids at high temperatures.

4. The Limits of a Classical Idea: Quantum Freeze-Out

Despite its successes, the equipartition theorem also leads to famous failures, which historically pointed the way toward quantum mechanics.

Consider a diatomic gas. Classically, we can model it as having:

3 translational degrees of freedom (center of mass momentum).
2 rotational degrees of freedom (for a linear molecule rotating about axes perpendicular to its bond).
2 vibrational degrees of freedom (kinetic and potential energy of a 1D harmonic oscillator).

Equipartition would predict a total average energy of $(\frac{3}{2} + \frac{2}{2} + \frac{2}{2}) N k_{B} T = \frac{7}{2} N k_{B} T$ , giving a heat capacity of $C_{V} = \frac{7}{2} N k_{B}$ . However, experiments show a different story.

2. Classical Gases

The heat capacity of hydrogen gas (H₂). The value plateaus at different multiples of k_B/2 as the temperature changes, contradicting the classical prediction of a constant C_V. At low temperatures, only translation contributes. As T increases, rotational modes become active, and at very high T, vibrational modes finally contribute.

This "freezing out" of degrees of freedom is a quantum mechanical effect. The equipartition theorem assumes that degrees of freedom can absorb any amount of energy, which is only true if the energy spectrum is continuous. In quantum mechanics, rotational and vibrational energies are quantized. If the thermal energy $k_{B} T$ is much smaller than the energy gap $Δ E$ to the first excited state of a particular mode (e.g., vibration), that mode cannot be excited and will not contribute to the heat capacity. As the temperature rises and $k_{B} T$ becomes comparable to $Δ E$ , the mode "unfreezes" and begins to contribute.

The classical result is recovered only in the high-temperature limit, $k_{B} T ≫ Δ E$ .

These notes elegantly demonstrate the quantum-to-classical transition.

First, read the subsection "High T: kBT ≫ ℏω". It shows explicitly how the quantum mechanical expression for the average energy of a harmonic oscillator reduces to the classical equipartition result \langle \epsilon \rangle = k_B T in the high-temperature limit. Then, read the section on diatomic molecules starting from the introduction, which applies this concept by defining characteristic temperatures \Theta_V and \Theta_R for vibration and rotation.

This failure was most catastrophic in the context of black-body radiation (the "ultraviolet catastrophe"), where classical equipartition predicted that an electromagnetic field (an infinite collection of harmonic oscillators) in a cavity would have infinite heat capacity. The resolution, of course, was Planck's hypothesis of quantized energy, which laid the foundation for your studies in quantum mechanics and QFT.

Conclusion

In this lesson, we have derived and explored the classical equipartition theorem, one of the most important and illustrative results of statistical mechanics.

Key Takeaways:

The equipartition theorem states that every quadratic degree of freedom in a classical Hamiltonian contributes an average energy of $\frac{1}{2} k_{B} T$ to the system.
The theorem is derived from the canonical ensemble formalism using the classical partition function, which is an integral over phase space.
It provides accurate predictions for systems like ideal gases and solids in the high-temperature limit (e.g., the ideal gas law and the Dulong-Petit law).
The theorem's failure at low temperatures is a direct consequence of the quantization of energy. Degrees of freedom are "frozen out" when the thermal energy $k_{B} T$ is insufficient to bridge the gap to the first excited quantum state.

In the previous lesson, we established the $1/ N$ scaling of fluctuations, providing a microscopic justification for the stability of macroscopic systems. This lesson has given us a powerful classical tool for calculating average energies. We are now well-equipped to begin the next module, where we will formalize the concept of thermodynamic equivalence. We will use ideas from large deviation theory—a topic you expressed interest in—to show precisely how the different ensembles become indistinguishable in the thermodynamic limit.

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