Equipartition theorem

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The equipartition theorem is a principle of classical (non-quantum) statistical mechanics which states that the internal energy of a system composed of a large number of particles at thermal equilibrium will distribute itself evenly among each of the quadratic degrees of freedom allowed to the particles of the system.

History

The equipartition principle was proposed initially in 1867 by James Clerk Maxwell who stated that the energy of a gas is equally divided between linear and rotational energy. Then, in 1868 and 1872, Ludwig Boltzmann, an enthusiastic follower of Maxwell’s, expanded on this principle by showing that energy could not only be divided equally between linear and rotational movements but among all the independent components of motion in the system.

Overview

As an example, in thermodynamics, the equipartition theorem says that the mean internal energy associated with each degree of freedom of a monatomic ideal gas is the same.

For a molecule of gas, each component of velocity has an associated kinetic energy. This kinetic energy is, on average,

<math>\frac{1}{2}k_BT</math>

where <math>k_B</math> is the Boltzmann constant, and T is the temperature of the molecule in kelvins. The components of velocity can be either linear or angular.

In general, for any system with a classical Hamiltonian of the form:

<math>H=\sum_i^m{a_ip_i^2}+\sum_j^n{b_jq_j^2}+U(p_{m+1}, p_{m+2}, \dots p_{M}, q_{n+1}, q_{n+2}, \dots q_{N})</math>

where <math>a_i</math> and <math>b_i</math> are constant with respect to all <math>q_{i<N}</math> and <math>p_{i<M}</math>,

<math>q_j</math> and <math>p_i</math> are spatial coordinates and their conjugate momenta,

each degree of freedom <math>q_i</math> and <math>p_j</math> will contribute a total of <math>\frac{1}{2}k_BT</math> to the system's total energy, resulting in a total of <math>\frac{1}{2}(m+n)k_BT</math> equipartition energy.

The equipartition theorem is valid only in the classical limit of an energy continuum. The equipartition theorem breaks down in the limit of large gaps between quantum energy levels, because it becomes more difficult to excite degrees of freedom which are highly quantized, such as electronic excitations in non-metals, vibrational modes with a large ratio of force constant to reduced mass, or rotational degrees of freedom about an axis with a low moment of inertia.

See also

• Degrees of freedom (physics and chemistry)
• Kinetic theory
• Heat capacity
• Statistical mechanics
• Quantum statistical mechanicsTemplate:Physics-stub

fr:Théorème d'équipartition nl:Equipartitiebeginsel ja:エネルギー等配分の法則 sl:Ekviparticijski izrek