Equipartition and Where It Fails

The Classical Intuition

In classical mechanics, the energy of a system is a continuous function of its coordinates and momenta. If the Hamiltonian \(H\) can be written as a sum of independent quadratic terms, such as \(H = \sum_i a_i x_i^2 + \sum_j b_j p_j^2\), the probability of finding the system in a particular state is proportional to \(e^{-H/k_B T}\). Because the exponential of a quadratic term is a Gaussian, the average value of each quadratic term \(a_i x_i^2\) or \(b_j p_j^2\) is identical, regardless of the coefficients \(a_i\) or \(b_j\). This is the essence of equipartition: the "stiffness" of the spring or the mass of the particle does not matter; only the number of independent quadratic terms matters. Each term acts as a reservoir that shares the thermal energy equally. This leads to the prediction that the internal energy \(U\) is simply \(N f \frac{1}{2} k_B T\), where \(f\) is the number of quadratic degrees of freedom per particle. Consequently, the heat capacity \(C_V = \left(\frac{\partial U}{\partial T}\right)_V\) becomes a constant, independent of temperature. This is a powerful and simple result, but it relies entirely on the assumption that the energy levels are continuous and that the thermal energy \(k_B T\) is large compared to the spacing between energy levels.

Mathematical Statement of the Law

For a system in the canonical ensemble, the average energy associated with a degree of freedom \(q\) that appears quadratically in the Hamiltonian as \(\frac{1}{2} m \dot{q}^2\) or \(\frac{1}{2} k q^2\) is derived from the partition function. The result, derived in the embedded derivation below, is that the mean energy per quadratic degree of freedom is \(\frac{1}{2} k_B T\).

If a molecule has \(f\) such degrees of freedom, the total internal energy per mole is \(U = N_A f \frac{1}{2} k_B T = \frac{f}{2} R T\). The molar heat capacity at constant volume is then \(C_V = \frac{f}{2} R\). For a monatomic ideal gas, \(f=3\) (three translational degrees of freedom), so \(C_V = \frac{3}{2} R\). For a diatomic gas at moderate temperatures, we add two rotational degrees of freedom (rotation about the axis of symmetry is negligible due to small moment of inertia), giving \(f=5\) and \(C_V = \frac{5}{2} R\). If vibrational modes are active, they contribute two quadratic terms (kinetic and potential energy of the oscillator), adding \(k_B T\) to the energy and \(R\) to the heat capacity, resulting in \(C_V = \frac{7}{2} R\).

The Quantum Failure: Freezing Out

The failure of equipartition arises because quantum mechanics discretizes energy levels. A harmonic oscillator, for example, has energy levels \(E_n = \hbar \omega (n + \frac{1}{2})\). The average energy is not \(\frac{1}{2} k_B T\) but rather \(\langle E \rangle = \frac{\hbar \omega}{2} + \frac{\hbar \omega}{e^{\hbar \omega / k_B T} - 1}\). At high temperatures, where \(k_B T \gg \hbar \omega\), the exponential term can be expanded, and \(\langle E \rangle \approx k_B T\), recovering the classical result (including the zero-point energy which is often ignored in classical thermodynamics). However, at low temperatures, where \(k_B T \ll \hbar \omega\), the exponential term becomes very large, and \(\langle E \rangle \approx \frac{\hbar \omega}{2}\). The energy becomes nearly constant, meaning the heat capacity \(C_V = \frac{\partial \langle E \rangle}{\partial T}\) approaches zero. This phenomenon is known as "freezing out." The vibrational modes of a diatomic molecule, for instance, have characteristic temperatures \(\Theta_{\text{vib}} = \hbar \omega / k_B\) often in the thousands of Kelvin. At room temperature, \(k_B T\) is much smaller than \(\hbar \omega\), so the vibrational modes are frozen out and do not contribute to the heat capacity. This explains why the experimental \(C_V\) for diatomic gases is \(\frac{5}{2} R\) rather than \(\frac{7}{2} R\) at standard conditions.

Implications for Solids: The Dulong-Petit Law

The same principle applies to solids. In the classical model (Einstein or Debye model in the high-temperature limit), each atom in a solid lattice is treated as a 3D harmonic oscillator. There are \(6N\) quadratic terms (3 kinetic and 3 potential per atom), each contributing \(\tfrac12 k_BT\), so \(U = 3Nk_BT\). Equipartition predicts a molar heat capacity of \(C_V = 3R\), known as the Dulong-Petit law. This law holds well for many solids at room temperature because the characteristic Debye temperature \(\Theta_D\) is often comparable to or lower than room temperature. However, at low temperatures, the quantum nature of the lattice vibrations (phonons) becomes dominant. The heat capacity drops as \(T^3\) (Debye law) rather than remaining constant. This temperature dependence is a direct violation of the classical equipartition theorem and provides strong evidence for the quantization of vibrational energy in solids.