Equipartition theorem Totally Explained

Everything about Equipartition Theorem totally explained

In classical statistical mechanics, the equipartition theorem is a general formula that relates the temperature of a system with its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. The original idea of equipartition was that, in thermal equilibrium, energy is shared equally among its various forms; for example, the average kinetic energy in the translational motion of a molecule should equal the average kinetic energy in its rotational motion.
The equipartition theorem makes quantitative predictions. Like the virial theorem, it gives the total average kinetic and potential energies for a system at a given temperature, from which the system's heat capacity can be computed. However, equipartition also gives the average values of individual components of the energy, such as the kinetic energy of a particular particle or the potential energy of a single spring. For example, it predicts that every molecule in an ideal gas has an average kinetic energy of (3/2)k_BT in thermal equilibrium, where k_B is the Boltzmann constant and T is the temperature. More generally, it can be applied to any classical system in thermal equilibrium, no matter how complicated. The equipartition theorem can be used to derive the classical ideal gas law, and the Dulong–Petit law for the specific heat capacities of solids. It can also be used to predict the properties of stars, even white dwarfs and neutron stars, since it holds even when relativistic effects are considered. Although the equipartition theorem makes very accurate predictions in certain conditions, it becomes inaccurate when quantum effects are significant, namely at low enough temperatures. When the thermal energy k_BT is smaller than the quantum energy spacing in a particular degree of freedom, the average energy and heat capacity of this degree of freedom are less than the values predicted by equipartition. Such a degree of freedom is said to be "frozen out" when the thermal energy is much smaller than this spacing. For example, the specific heat of a solid decreases at low temperatures as various types of motion become frozen out, rather than remaining constant as predicted by equipartition. Such decreases in specific heat were the first sign to physicists of the 19th century that classical physics was incorrect and that new physics was needed. Along with other evidence, equipartition's failure for electromagnetic radiation — also known as the ultraviolet catastrophe — led Albert Einstein to suggest that light itself was quantized into photons, a revolutionary hypothesis that spurred the development of quantum mechanics and quantum field theory.

Basic concept and simple examples

kinetic energy of a system is shared equally among all of its independent parts, on the average, once the system has reached thermal equilibrium. Equipartition also makes quantitative predictions for these energies. For example, it predicts that every atom of a noble gas, in thermal equilibrium at temperature T, has an average translational kinetic energy of (3/2)k_BT, where k_B is the Boltzmann constant. As a consequence, the heavier atoms of xenon have a lower average speed than do the lighter atoms of helium at the same temperature. Figure 2 shows the Maxwell–Boltzmann distribution for the speeds of the atoms in four noble gases.
In this example, the key point is that the kinetic energy is quadratic in the velocity. The equipartition theorem shows that in thermal equilibrium, any degree of freedom (such as a component of the position or velocity of a particle) which appears only quadratically in the energy has an average energy of ½k_BT and therefore contributes ½k_B to the system's heat capacity. This has many applications.

Translational energy and ideal gases

m, velocity v is given by ť

H^ T

as stated by the equipartition theorem.

General proofs

General derivations of the equipartition theorem can be found in many statistical mechanics textbooks, both for the microcanonical ensemble The requirements for isolated systems to ensure ergodicity — and, thus equipartition — have been studied, and provided motivation for the modern chaos theory of dynamical systems. A chaotic Hamiltonian system need not be ergodic, although that's usually a good assumption. If the system is isolated from the rest of the world, the energy in each normal mode is constant; energy isn't transferred from one mode to another. Hence, equipartition doesn't hold for such a system; the amount of energy in each normal mode is fixed at its initial value. If sufficiently strong nonlinear terms are present in the energy function, energy may be transferred between the normal modes, leading to ergodicity and rendering the law of equipartition valid. However, the Kolmogorov–Arnold–Moser theorem states that energy won't be exchanged unless the nonlinear perturbations are strong enough; if they're too small, the energy will remain trapped in at least some of the modes.

Failure due to quantum effects

k_BT is significantly smaller than the spacing between energy levels. Equipartition no longer holds because it's a poor approximation to assume that the energy levels form a smooth continuum, which is required in the derivations of the equipartition theorem above. The paradox arises because there are an infinite number of independent modes of the electromagnetic field in a closed container, each of which may be treated as a harmonic oscillator. If each electromagnetic mode were to have an average energy k_BT, there would be an infinite amount of energy in the container. However, by the reasoning above, the average energy in the higher-ω modes goes to zero as ω goes to infinity; moreover, Planck's law of black body radiation, which describes the experimental distribution of energy in the modes, follows from the same reasoning. Other, more subtle quantum effects can lead to corrections to equipartition, such as identical particles and continuous symmetries. The effects of identical particles can be dominant at very high densities and low temperatures. For example, the valence electrons in a metal can have a mean kinetic energy of a few electronvolts, which would normally correspond to a temperature of tens of thousands of kelvins. Such a state, in which the density is high enough that the Pauli exclusion principle invalidates the classical approach, is called a degenerate fermion gas. Such gases are important for the structure of white dwarf and nuetron stars. At low temperatures, a fermionic analogue of the Bose–Einstein condensate (in which a large number of identical particles occupy the lowest-energy state) can form; such superfluid electrons are responsible for superconductivity.

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