Equipartition Theorem
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In classical
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
, the equipartition theorem relates the
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
of a system to its average
energies In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat ...
. 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 Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in ...
, energy is shared equally among all of its various forms; for example, the average
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
per
degree of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
in
translational motion In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every ...
of a molecule should equal that in
rotational motion Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
. The equipartition theorem makes quantitative predictions. Like the
virial theorem In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by potential forces, with that of the total potential energy of the system. ...
, it gives the total average kinetic and potential energies for a system at a given temperature, from which the system's
heat capacity Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K). Heat capacity i ...
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 Spring(s) may refer to: Common uses * Spring (season) Spring, also known as springtime, is one of the four temperate seasons, succeeding winter and preceding summer. There are various technical definitions of spring, but local usage of ...
. For example, it predicts that every atom in a
monatomic In physics and chemistry, "monatomic" is a combination of the words "mono" and "atomic", and means "single atom". It is usually applied to gases: a monatomic gas is a gas in which atoms are not bound to each other. Examples at standard conditions ...
ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
has an average kinetic energy of in thermal equilibrium, where is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
and ''T'' is the (thermodynamic) temperature. More generally, equipartition can be applied to any classical system in
thermal equilibrium Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in ...
, no matter how complicated. It can be used to derive the
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stat ...
, and the
Dulong–Petit law The Dulong–Petit law, a thermodynamic law proposed by French physicists Pierre Louis Dulong and Alexis Thérèse Petit, states that the classical expression for the molar specific heat capacity of certain chemical elements is constant for tempe ...
for the specific heat capacities of solids. The equipartition theorem can also be used to predict the properties of
star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...
s, even
white dwarf A white dwarf is a stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very dense: its mass is comparable to the Sun's, while its volume is comparable to the Earth's. A white dwarf's faint luminosity comes fro ...
s and
neutron star A neutron star is the collapsed core of a massive supergiant star, which had a total mass of between 10 and 25 solar masses, possibly more if the star was especially metal-rich. Except for black holes and some hypothetical objects (e.g. w ...
s, since it holds even when relativistic effects are considered. Although the equipartition theorem makes accurate predictions in certain conditions, it is inaccurate when
quantum effects Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
are significant, such as at low temperatures. When the
thermal energy The term "thermal energy" is used loosely in various contexts in physics and engineering. It can refer to several different well-defined physical concepts. These include the internal energy or enthalpy of a body of matter and radiation; heat, d ...
is smaller than the quantum energy spacing in a particular
degree of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
, 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 heat capacity 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 heat capacity were among the first signs to physicists of the 19th century that classical physics was incorrect and that a new, more subtle, scientific model was required. Along with other evidence, equipartition's failure to model
black-body radiation Black-body radiation is the thermal electromagnetic radiation within, or surrounding, a body in thermodynamic equilibrium with its environment, emitted by a black body (an idealized opaque, non-reflective body). It has a specific, continuous spect ...
—also known as the
ultraviolet catastrophe The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was the prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium would emit an unbounded quantity of energy ...
—led
Max Planck Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918. Planck made many substantial contributions to theoretical p ...
to suggest that energy in the oscillators in an object, which emit light, were quantized, a revolutionary hypothesis that spurred the development of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
and
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
.


Basic concept and simple examples

The name "equipartition" means "equal division," as derived from the
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
''equi'' from the antecedent, æquus ("equal or even"), and partition from the noun, ''partitio'' ("division, portion"). The original concept of equipartition was that the total
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
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 an inert
noble gas The noble gases (historically also the inert gases; sometimes referred to as aerogens) make up a class of chemical elements with similar properties; under standard conditions, they are all odorless, colorless, monatomic gases with very low chemi ...
, in thermal equilibrium at temperature , has an average translational kinetic energy of , where is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
. As a consequence, since kinetic energy is equal to (mass)(velocity)2, the heavier atoms of
xenon Xenon is a chemical element with the symbol Xe and atomic number 54. It is a dense, colorless, odorless noble gas found in Earth's atmosphere in trace amounts. Although generally unreactive, it can undergo a few chemical reactions such as the ...
have a lower average speed than do the lighter atoms of
helium Helium (from el, ἥλιος, helios, lit=sun) is a chemical element with the symbol He and atomic number 2. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas and the first in the noble gas group in the periodic table. ...
at the same temperature. Figure 2 shows the
Maxwell–Boltzmann distribution In physics (in particular in statistical mechanics), the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann. It was first defined and use ...
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 Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
(such as a component of the position or velocity of a particle) which appears only quadratically in the energy has an average energy of and therefore contributes to the system's
heat capacity Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K). Heat capacity i ...
. This has many applications.


Translational energy and ideal gases

The (Newtonian) kinetic energy of a particle of mass , velocity is given by :H_ = \tfrac 1 2 m , \mathbf, ^2 = \tfrac m\left( v_x^2 + v_y^2 + v_z^2 \right), where , and are the Cartesian components of the velocity . Here, is short for
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
, and used henceforth as a symbol for energy because the
Hamiltonian formalism Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
plays a central role in the most general form of the equipartition theorem. Since the kinetic energy is quadratic in the components of the velocity, by equipartition these three components each contribute to the average kinetic energy in thermal equilibrium. Thus the average kinetic energy of the particle is , as in the example of noble gases above. More generally, in a monatomic ideal gas the total energy consists purely of (translational) kinetic energy: by assumption, the particles have no internal degrees of freedom and move independently of one another. Equipartition therefore predicts that the total energy of an ideal gas of particles is . It follows that the
heat capacity Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K). Heat capacity i ...
of the gas is and hence, in particular, the heat capacity of a
mole Mole (or Molé) may refer to: Animals * Mole (animal) or "true mole", mammals in the family Talpidae, found in Eurasia and North America * Golden moles, southern African mammals in the family Chrysochloridae, similar to but unrelated to Talpida ...
of such gas particles is , where ''N''A is the
Avogadro constant The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining con ...
and ''R'' is the
gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
. Since ''R'' ≈ 2 cal/( mol· K), equipartition predicts that the
molar heat capacity The molar heat capacity of a chemical substance is the amount of energy that must be added, in the form of heat, to one mole of the substance in order to cause an increase of one unit in its temperature. Alternatively, it is the heat capacity of a ...
of an ideal gas is roughly 3 cal/(mol·K). This prediction is confirmed by experiment when compared to monatomic gases. The mean kinetic energy also allows the
root mean square speed In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often b ...
of the gas particles to be calculated: :v_ = \sqrt = \sqrt = \sqrt, where is the mass of a mole of gas particles. This result is useful for many applications such as Graham's law of
effusion In physics and chemistry, effusion is the process in which a gas escapes from a container through a hole of diameter considerably smaller than the mean free path of the molecules. Such a hole is often described as a ''pinhole'' and the escape ...
, which provides a method for enriching
uranium Uranium is a chemical element with the symbol U and atomic number 92. It is a silvery-grey metal in the actinide series of the periodic table. A uranium atom has 92 protons and 92 electrons, of which 6 are valence electrons. Uranium is weak ...
.


Rotational energy and molecular tumbling in solution

A similar example is provided by a rotating molecule with
principal moments of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...
, and . The rotational energy of such a molecule is given by :H_ = \tfrac ( I_1 \omega_1^2 + I_2 \omega_2^2 + I_3 \omega_3^2 ), where , , and are the principal components of the
angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
. By exactly the same reasoning as in the translational case, equipartition implies that in thermal equilibrium the average rotational energy of each particle is . Similarly, the equipartition theorem allows the average (more precisely, the root mean square) angular speed of the molecules to be calculated. The tumbling of rigid molecules—that is, the random rotations of molecules in solution—plays a key role in the relaxations observed by
nuclear magnetic resonance Nuclear magnetic resonance (NMR) is a physical phenomenon in which nuclei in a strong constant magnetic field are perturbed by a weak oscillating magnetic field (in the near field) and respond by producing an electromagnetic signal with a ...
, particularly
protein NMR Nuclear magnetic resonance spectroscopy of proteins (usually abbreviated protein NMR) is a field of structural biology in which NMR spectroscopy is used to obtain information about the structure and dynamics of proteins, and also nucleic acids, and ...
and
residual dipolar coupling The residual dipolar coupling between two spins in a molecule occurs if the molecules in solution exhibit a partial alignment leading to an incomplete averaging of spatially anisotropic dipolar couplings. Partial molecular alignment leads to an in ...
s. Rotational diffusion can also be observed by other biophysical probes such as
fluorescence anisotropy Fluorescence anisotropy or fluorescence polarization is the phenomenon where the light emitted by a fluorophore has unequal intensities along different axes of polarization. Early pioneers in the field include Aleksander Jablonski, Gregorio Web ...
,
flow birefringence In biochemistry, flow birefringence is a hydrodynamic technique for measuring the rotational diffusion constants (or, equivalently, the rotational drag coefficients). The birefringence of a solution sandwiched between two concentric cylinders is ...
and
dielectric spectroscopy Dielectric spectroscopy (which falls in a subcategory of impedance spectroscopy) measures the dielectric properties of a medium as a function of frequency.Kremer F., Schonhals A., Luck W. Broadband Dielectric Spectroscopy. – Springer-Verlag, 200 ...
.


Potential energy and harmonic oscillators

Equipartition applies to potential energies as well as kinetic energies: important examples include
harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'': \v ...
s such as a
spring Spring(s) may refer to: Common uses * Spring (season) Spring, also known as springtime, is one of the four temperate seasons, succeeding winter and preceding summer. There are various technical definitions of spring, but local usage of ...
, which has a quadratic potential energy :H_ = \tfrac 1 2 a q^2,\, where the constant describes the stiffness of the spring and is the deviation from equilibrium. If such a one-dimensional system has mass , then its kinetic energy is :H_ = \fracmv^2 = \frac, where and denote the velocity and momentum of the oscillator. Combining these terms yields the total energy :H = H_ + H_ = \frac + \frac a q^2. Equipartition therefore implies that in thermal equilibrium, the oscillator has average energy : \langle H \rangle = \langle H_ \rangle + \langle H_ \rangle = \tfrac k_\text T + \tfrac k_\text T = k_\text T, where the angular brackets \left\langle \ldots \right\rangle denote the average of the enclosed quantity, This result is valid for any type of harmonic oscillator, such as a
pendulum A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the ...
, a vibrating molecule or a passive
electronic oscillator An electronic oscillator is an electronic circuit that produces a periodic, oscillation, oscillating electronic signal, often a sine wave or a square wave or a triangle wave. Oscillation, Oscillators convert direct current (DC) from a power supp ...
. Systems of such oscillators arise in many situations; by equipartition, each such oscillator receives an average total energy and hence contributes to the system's
heat capacity Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K). Heat capacity i ...
. This can be used to derive the formula for
Johnson–Nyquist noise Johnson–Nyquist noise (thermal noise, Johnson noise, or Nyquist noise) is the electronic noise generated by the thermal agitation of the charge carriers (usually the electrons) inside an electrical conductor at equilibrium, which happens rega ...
and the
Dulong–Petit law The Dulong–Petit law, a thermodynamic law proposed by French physicists Pierre Louis Dulong and Alexis Thérèse Petit, states that the classical expression for the molar specific heat capacity of certain chemical elements is constant for tempe ...
of solid heat capacities. The latter application was particularly significant in the history of equipartition.


Specific heat capacity of solids

An important application of the equipartition theorem is to the specific heat capacity of a crystalline solid. Each atom in such a solid can oscillate in three independent directions, so the solid can be viewed as a system of independent
simple harmonic oscillator In mechanics and physics, simple harmonic motion (sometimes abbreviated ) is a special type of periodic motion of a body resulting from a dynamic equilibrium between an inertial force, proportional to the acceleration of the body away from the ...
s, where denotes the number of atoms in the lattice. Since each harmonic oscillator has average energy , the average total energy of the solid is , and its heat capacity is . By taking to be the
Avogadro constant The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining con ...
, and using the relation between the
gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
and the Boltzmann constant , this provides an explanation for the
Dulong–Petit law The Dulong–Petit law, a thermodynamic law proposed by French physicists Pierre Louis Dulong and Alexis Thérèse Petit, states that the classical expression for the molar specific heat capacity of certain chemical elements is constant for tempe ...
of specific heat capacities of solids, which stated that the specific heat capacity (per unit mass) of a solid element is inversely proportional to its
atomic weight Relative atomic mass (symbol: ''A''; sometimes abbreviated RAM or r.a.m.), also known by the deprecated synonym atomic weight, is a dimensionless physical quantity defined as the ratio of the average mass of atoms of a chemical element in a giv ...
. A modern version is that the molar heat capacity of a solid is ''3R'' ≈ 6 cal/(mol·K). However, this law is inaccurate at lower temperatures, due to quantum effects; it is also inconsistent with the experimentally derived
third law of thermodynamics The third law of thermodynamics states, regarding the properties of closed systems in thermodynamic equilibrium: This constant value cannot depend on any other parameters characterizing the closed system, such as pressure or applied magnetic fiel ...
, according to which the molar heat capacity of any substance must go to zero as the temperature goes to absolute zero. A more accurate theory, incorporating quantum effects, was developed by
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
(1907) and
Peter Debye Peter Joseph William Debye (; ; March 24, 1884 – November 2, 1966) was a Dutch-American physicist and physical chemist, and Nobel laureate in Chemistry. Biography Early life Born Petrus Josephus Wilhelmus Debije in Maastricht, Netherlands, D ...
(1911). Many other physical systems can be modeled as sets of
coupled oscillators Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
. The motions of such oscillators can be decomposed into
normal mode A normal mode of a dynamical system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencies. ...
s, like the vibrational modes of a
piano string Piano wire, or "music wire", is a specialized type of wire made for use in piano strings but also in other applications as springs. It is made from tempered high-carbon steel, also known as spring steel, which replaced iron as the material st ...
or the
resonance Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillatin ...
s of an
organ pipe An organ pipe is a sound-producing element of the pipe organ that resonates at a specific pitch when pressurized air (commonly referred to as ''wind'') is driven through it. Each pipe is tuned to a specific note of the musical scale. A set o ...
. On the other hand, equipartition often breaks down for such systems, because there is no exchange of energy between the normal modes. In an extreme situation, the modes are independent and so their energies are independently conserved. This shows that some sort of mixing of energies, formally called ''ergodicity'', is important for the law of equipartition to hold.


Sedimentation of particles

Potential energies are not always quadratic in the position. However, the equipartition theorem also shows that if a degree of freedom contributes only a multiple of (for a fixed real number ) to the energy, then in thermal equilibrium the average energy of that part is . There is a simple application of this extension to the
sedimentation Sedimentation is the deposition of sediments. It takes place when particles in suspension settle out of the fluid in which they are entrained and come to rest against a barrier. This is due to their motion through the fluid in response to the ...
of particles under
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
. For example, the haze sometimes seen in
beer Beer is one of the oldest and the most widely consumed type of alcoholic drink in the world, and the third most popular drink overall after water and tea. It is produced by the brewing and fermentation of starches, mainly derived from ce ...
can be caused by clumps of
protein Proteins are large biomolecules and macromolecules that comprise one or more long chains of amino acid residues. Proteins perform a vast array of functions within organisms, including catalysing metabolic reactions, DNA replication, respo ...
s that scatter light. Over time, these clumps settle downwards under the influence of gravity, causing more haze near the bottom of a bottle than near its top. However, in a process working in the opposite direction, the particles also
diffuse Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
back up towards the top of the bottle. Once equilibrium has been reached, the equipartition theorem may be used to determine the average position of a particular clump of
buoyant mass Buoyancy (), or upthrust, is an upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the pr ...
. For an infinitely tall bottle of beer, the
gravitational potential energy Gravitational energy or gravitational potential energy is the potential energy a massive object has in relation to another massive object due to gravity. It is the potential energy associated with the gravitational field, which is released (conver ...
is given by :H^ = m_\text g z where is the height of the protein clump in the bottle and '' g'' is the
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
due to gravity. Since , the average potential energy of a protein clump equals . Hence, a protein clump with a buoyant mass of 10 
MDa MDA, mda, or ''variation'', may refer to: Places * Moldova, a country in Europe with the ISO 3166-1 country code MDA Politics * Meghalaya Democratic Alliance (2018), ruling coalition government in the Indian State of Meghalaya led by National Pe ...
(roughly the size of a
virus A virus is a submicroscopic infectious agent that replicates only inside the living cells of an organism. Viruses infect all life forms, from animals and plants to microorganisms, including bacteria and archaea. Since Dmitri Ivanovsky's 1 ...
) would produce a haze with an average height of about 2 cm at equilibrium. The process of such sedimentation to equilibrium is described by the
Mason–Weaver equation The Mason–Weaver equation (named after Max Mason and Warren Weaver) describes the sedimentation and diffusion of solutes under a uniform force, usually a gravitational field. Assuming that the gravitational field is aligned in the ''z'' direction ...
.


History

The equipartition of kinetic energy was proposed initially in 1843, and more correctly in 1845, by
John James Waterston John James Waterston (1811 – 18 June 1883) was a Scotland, Scottish physicist and a neglected pioneer of the kinetic theory of gases. Early life Waterston's father, George, was an Edinburgh sealing wax manufacturer and stationer, a relative of ...
. In 1859,
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
argued that the kinetic heat energy of a gas is equally divided between linear and rotational energy. In 1876,
Ludwig Boltzmann Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of thermodyn ...
expanded on this principle by showing that the average energy was divided equally among all the independent components of motion in a system. Boltzmann applied the equipartition theorem to provide a theoretical explanation of the
Dulong–Petit law The Dulong–Petit law, a thermodynamic law proposed by French physicists Pierre Louis Dulong and Alexis Thérèse Petit, states that the classical expression for the molar specific heat capacity of certain chemical elements is constant for tempe ...
for the specific heat capacities of solids. The history of the equipartition theorem is intertwined with that of
specific heat capacity In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
, both of which were studied in the 19th century. In 1819, the French physicists
Pierre Louis Dulong Pierre Louis Dulong FRS FRSE (; ; 12 February 1785 – 19 July 1838) was a French physicist and chemist. He is remembered today largely for the law of Dulong and Petit, although he was much-lauded by his contemporaries for his studies into ...
and
Alexis Thérèse Petit Alexis Thérèse Petit (; 2 October 1791, Vesoul, Haute-Saône – 21 June 1820, Paris) was a French physicist. Petit is known for his work on the efficiencies of air- and steam-engines, published in 1818 (''Mémoire sur l’emploi du principe ...
discovered that the specific heat capacities of solid elements at room temperature were inversely proportional to the atomic weight of the element. Their law was used for many years as a technique for measuring atomic weights. However, subsequent studies by
James Dewar Sir James Dewar (20 September 1842 – 27 March 1923) was a British chemist and physicist. He is best known for his invention of the vacuum flask, which he used in conjunction with research into the liquefaction of gases. He also studied a ...
and
Heinrich Friedrich Weber Heinrich Friedrich Weber (; ; 7 November 1843 – 24 May 1912) was a physicist born in the town of Magdala, near Weimar. Biography Around 1861 he entered the University of Jena, where Ernst Abbe became the first of two physicists who decisi ...
showed that this
Dulong–Petit law The Dulong–Petit law, a thermodynamic law proposed by French physicists Pierre Louis Dulong and Alexis Thérèse Petit, states that the classical expression for the molar specific heat capacity of certain chemical elements is constant for tempe ...
holds only at high
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
s; at lower temperatures, or for exceptionally hard solids such as
diamond Diamond is a Allotropes of carbon, solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. Another solid form of carbon known as graphite is the Chemical stability, chemically stable form of car ...
, the specific heat capacity was lower. Experimental observations of the specific heat capacities of gases also raised concerns about the validity of the equipartition theorem. The theorem predicts that the molar heat capacity of simple monatomic gases should be roughly 3 cal/(mol·K), whereas that of diatomic gases should be roughly 7 cal/(mol·K). Experiments confirmed the former prediction, but found that molar heat capacities of diatomic gases were typically about 5 cal/(mol·K), and fell to about 3 cal/(mol·K) at very low temperatures. Maxwell noted in 1875 that the disagreement between experiment and the equipartition theorem was much worse than even these numbers suggest; A lecture delivered by Prof. Maxwell at the Chemical Society on 18 February 1875. since atoms have internal parts, heat energy should go into the motion of these internal parts, making the predicted specific heats of monatomic and diatomic gases much higher than 3 cal/(mol·K) and 7 cal/(mol·K), respectively. A third discrepancy concerned the specific heat of metals. According to the classical Drude model, metallic electrons act as a nearly ideal gas, and so they should contribute to the heat capacity by the equipartition theorem, where ''N''e is the number of electrons. Experimentally, however, electrons contribute little to the heat capacity: the molar heat capacities of many conductors and insulators are nearly the same. Several explanations of equipartition's failure to account for molar heat capacities were proposed.
Boltzmann Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of thermodyn ...
defended the derivation of his equipartition theorem as correct, but suggested that gases might not be in
thermal equilibrium Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in ...
because of their interactions with the aether.
Lord Kelvin William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, Mathematical physics, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy (Glasgow), Professor of Natural Philoso ...
suggested that the derivation of the equipartition theorem must be incorrect, since it disagreed with experiment, but was unable to show how. In 1900
Lord Rayleigh John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. Am ...
instead put forward a more radical view that the equipartition theorem and the experimental assumption of thermal equilibrium were ''both'' correct; to reconcile them, he noted the need for a new principle that would provide an "escape from the destructive simplicity" of the equipartition theorem.
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
provided that escape, by showing in 1906 that these anomalies in the specific heat were due to quantum effects, specifically the quantization of energy in the elastic modes of the solid. Einstein used the failure of equipartition to argue for the need of a new quantum theory of matter. Nernst's 1910 measurements of specific heats at low temperatures supported Einstein's theory, and led to the widespread acceptance of
quantum theory Quantum theory may refer to: Science *Quantum mechanics, a major field of physics *Old quantum theory, predating modern quantum mechanics * Quantum field theory, an area of quantum mechanics that includes: ** Quantum electrodynamics ** Quantum ...
among physicists.


General formulation of the equipartition theorem

The most general form of the equipartition theorem states that under suitable assumptions (discussed below), for a physical system with
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
energy function and degrees of freedom , the following equipartition formula holds in thermal equilibrium for all indices and : :\left\langle x_ \frac \right\rangle = \delta_ k_\text T. Here is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
, which is equal to one if and is zero otherwise. The averaging brackets \left\langle \ldots \right\rangle is assumed to be an
ensemble average In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents ...
over phase space or, under an assumption of
ergodic In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies tha ...
ity, a time average of a single system. The general equipartition theorem holds in both the
microcanonical ensemble In statistical mechanics, the microcanonical ensemble is a statistical ensemble that represents the possible states of a mechanical system whose total energy is exactly specified. The system is assumed to be isolated in the sense that it canno ...
, when the total energy of the system is constant, and also in the
canonical ensemble In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the heat b ...
, when the system is coupled to a
heat bath In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is al ...
with which it can exchange energy. Derivations of the general formula are given later in the article. The general formula is equivalent to the following two: # \left\langle x_n \frac \right\rangle = k_\text T \quad \text n # \left\langle x_m \frac \right\rangle = 0 \quad \text m \neq n. If a degree of freedom ''xn'' appears only as a quadratic term ''anxn''2 in the Hamiltonian ''H'', then the first of these formulae implies that :k_\text T = \left\langle x_n \frac\right\rangle = 2\left\langle a_n x_n^2 \right\rangle, which is twice the contribution that this degree of freedom makes to the average energy \langle H\rangle. Thus the equipartition theorem for systems with quadratic energies follows easily from the general formula. A similar argument, with 2 replaced by ''s'', applies to energies of the form ''anxns''. The degrees of freedom ''xn'' are coordinates on the
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
of the system and are therefore commonly subdivided into generalized position coordinates ''qk'' and generalized momentum coordinates ''pk'', where ''pk'' is the
conjugate momentum In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of ...
to ''qk''. In this situation, formula 1 means that for all ''k'', : \left\langle p_ \frac \right\rangle = \left\langle q_k \frac \right\rangle = k_\text T. Using the equations of
Hamiltonian mechanics Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
, these formulae may also be written :\left\langle p_k \frac \right\rangle = -\left\langle q_k \frac \right\rangle = k_\text T. Similarly, one can show using formula 2 that : \left\langle q_j \frac \right\rangle = \left\langle p_j \frac \right\rangle = 0 \quad \text \, j,k and : \left\langle q_j \frac \right\rangle = \left\langle p_j \frac \right\rangle = 0 \quad \text \, j \neq k.


Relation to the virial theorem

The general equipartition theorem is an extension of the
virial theorem In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by potential forces, with that of the total potential energy of the system. ...
(proposed in 1870), which states that : \left\langle \sum_k q_k \frac \right\rangle = \left\langle \sum_k p_k \frac \right\rangle = \left\langle \sum_k p_k \frac \right\rangle = -\left\langle \sum_k q_k \frac \right\rangle, where ''t'' denotes
time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
. Two key differences are that the virial theorem relates ''summed'' rather than ''individual'' averages to each other, and it does not connect them to the
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
''T''. Another difference is that traditional derivations of the virial theorem use averages over time, whereas those of the equipartition theorem use averages over
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
.


Applications


Ideal gas law

Ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
es provide an important application of the equipartition theorem. As well as providing the formula : \begin \langle H^ \rangle &= \frac \langle p_^ + p_^ + p_^ \rangle\\ &= \frac \left( \left\langle p_ \frac \right\rangle + \left\langle p_ \frac \right\rangle + \left\langle p_ \frac \right\rangle \right) = \frac k_\text T \end for the average kinetic energy per particle, the equipartition theorem can be used to derive the
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stat ...
from classical mechanics. If q = (''qx'', ''qy'', ''qz'') and p = (''px'', ''py'', ''pz'') denote the position vector and momentum of a particle in the gas, and F is the net force on that particle, then : \begin \langle \mathbf \cdot \mathbf \rangle &= \left\langle q_x \frac \right\rangle + \left\langle q_y \frac \right\rangle + \left\langle q_z \frac \right\rangle\\ &=-\left\langle q_x \frac \right\rangle - \left\langle q_y \frac \right\rangle - \left\langle q_z \frac \right\rangle = -3k_\text T, \end where the first equality is
Newton's second law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...
, and the second line uses
Hamilton's equations Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
and the equipartition formula. Summing over a system of ''N'' particles yields : 3Nk_\text T = - \left\langle \sum_^ \mathbf_ \cdot \mathbf_ \right\rangle. By
Newton's third law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in moti ...
and the ideal gas assumption, the net force on the system is the force applied by the walls of their container, and this force is given by the pressure ''P'' of the gas. Hence : -\left\langle\sum_^ \mathbf_ \cdot \mathbf_\right\rangle = P \oint_ \mathbf \cdot d\mathbf, where is the infinitesimal area element along the walls of the container. Since the
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
of the position vector is : \boldsymbol\nabla \cdot \mathbf = \frac + \frac + \frac = 3, the
divergence theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
implies that :P \oint_ \mathbf \cdot \mathbf = P \int_ \left( \boldsymbol\nabla \cdot \mathbf \right) \, dV = 3PV, where is an infinitesimal volume within the container and is the total volume of the container. Putting these equalities together yields :3Nk_\text T = -\left\langle \sum_^N \mathbf_k \cdot \mathbf_k \right\rangle = 3PV, which immediately implies the
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stat ...
for ''N'' particles: :PV = N k_\text T = nRT, where is the number of moles of gas and is the
gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
. Although equipartition provides a simple derivation of the ideal-gas law and the internal energy, the same results can be obtained by an alternative method using the partition function. Vu-Quoc, L.
Configuration integral (statistical mechanics)
2008. this wiki site is down; se
this article in the web archive on 2012 April 28


Diatomic gases

A diatomic gas can be modelled as two masses, and , joined by a
spring Spring(s) may refer to: Common uses * Spring (season) Spring, also known as springtime, is one of the four temperate seasons, succeeding winter and preceding summer. There are various technical definitions of spring, but local usage of ...
of
stiffness Stiffness is the extent to which an object resists deformation in response to an applied force. The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is. Calculations The stiffness, k, of a b ...
, which is called the ''rigid rotor-harmonic oscillator approximation''. The classical energy of this system is :H = \frac + \frac + \frac a q^2, where and are the momenta of the two atoms, and is the deviation of the inter-atomic separation from its equilibrium value. Every degree of freedom in the energy is quadratic and, thus, should contribute to the total average energy, and to the heat capacity. Therefore, the heat capacity of a gas of ''N'' diatomic molecules is predicted to be : the momenta and contribute three degrees of freedom each, and the extension contributes the seventh. It follows that the heat capacity of a mole of diatomic molecules with no other degrees of freedom should be and, thus, the predicted molar heat capacity should be roughly 7 cal/(mol·K). However, the experimental values for molar heat capacities of diatomic gases are typically about 5 cal/(mol·K) and fall to 3 cal/(mol·K) at very low temperatures. This disagreement between the equipartition prediction and the experimental value of the molar heat capacity cannot be explained by using a more complex model of the molecule, since adding more degrees of freedom can only ''increase'' the predicted specific heat, not decrease it. This discrepancy was a key piece of evidence showing the need for a
quantum theory Quantum theory may refer to: Science *Quantum mechanics, a major field of physics *Old quantum theory, predating modern quantum mechanics * Quantum field theory, an area of quantum mechanics that includes: ** Quantum electrodynamics ** Quantum ...
of matter.


Extreme relativistic ideal gases

Equipartition was used above to derive the classical
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stat ...
from
Newtonian mechanics Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...
. However,
relativistic effects Relativistic quantum chemistry combines relativistic mechanics with quantum chemistry to calculate elemental properties and structure, especially for the heavier elements of the periodic table. A prominent example is an explanation for the color of ...
become dominant in some systems, such as
white dwarf A white dwarf is a stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very dense: its mass is comparable to the Sun's, while its volume is comparable to the Earth's. A white dwarf's faint luminosity comes fro ...
s and
neutron star A neutron star is the collapsed core of a massive supergiant star, which had a total mass of between 10 and 25 solar masses, possibly more if the star was especially metal-rich. Except for black holes and some hypothetical objects (e.g. w ...
s, and the ideal gas equations must be modified. The equipartition theorem provides a convenient way to derive the corresponding laws for an extreme relativistic
ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
. In such cases, the kinetic energy of a single particle is given by the formula :H_ \approx cp = c \sqrt. Taking the derivative of with respect to the momentum component gives the formula :p_x \frac = c \frac and similarly for the and components. Adding the three components together gives :\begin \langle H_ \rangle &= \left\langle c \frac \right\rangle\\ &= \left\langle p_x \frac \right\rangle + \left\langle p_y \frac \right\rangle + \left\langle p_z \frac \right\rangle \\ &= 3 k_\text T \end where the last equality follows from the equipartition formula. Thus, the average total energy of an extreme relativistic gas is twice that of the non-relativistic case: for particles, it is .


Non-ideal gases

In an ideal gas the particles are assumed to interact only through collisions. The equipartition theorem may also be used to derive the energy and pressure of "non-ideal gases" in which the particles also interact with one another through
conservative force In physics, a conservative force is a force with the property that the total work done in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the total work done (the sum ...
s whose potential depends only on the distance between the particles. This situation can be described by first restricting attention to a single gas particle, and approximating the rest of the gas by a
spherically symmetric In geometry, circular symmetry is a type of continuous symmetry for a planar object that can be rotated by any arbitrary angle and map onto itself. Rotational circular symmetry is isomorphic with the circle group in the complex plane, or the ...
distribution. It is then customary to introduce a
radial distribution function In statistical mechanics, the radial distribution function, (or pair correlation function) g(r) in a system of particles (atoms, molecules, colloids, etc.), describes how density varies as a function of distance from a reference particle. If ...
such that the
probability density In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
of finding another particle at a distance from the given particle is equal to , where is the mean
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
of the gas. It follows that the mean potential energy associated to the interaction of the given particle with the rest of the gas is :\langle h_ \rangle = \int_0^\infty 4\pi r^2 \rho U(r) g(r)\, dr. The total mean potential energy of the gas is therefore \langle H_\text \rangle = \tfrac12 N \langle h_ \rangle , where is the number of particles in the gas, and the factor is needed because summation over all the particles counts each interaction twice. Adding kinetic and potential energies, then applying equipartition, yields the ''energy equation'' :H = \langle H_ \rangle + \langle H_ \rangle = \frac Nk_\textT + 2\pi N \rho \int_0^\infty r^2 U(r) g(r) \, dr. A similar argument, can be used to derive the ''pressure equation'' :3Nk_\textT = 3PV + 2\pi N \rho \int_0^\infty r^3 U'(r) g(r)\, dr.


Anharmonic oscillators

An anharmonic oscillator (in contrast to a simple harmonic oscillator) is one in which the potential energy is not quadratic in the extension (the generalized position which measures the deviation of the system from equilibrium). Such oscillators provide a complementary point of view on the equipartition theorem. Simple examples are provided by potential energy functions of the form :H_ = C q^,\, where and are arbitrary real constants. In these cases, the law of equipartition predicts that : k_\text T = \left\langle q \frac \right\rangle = \langle q \cdot s C q^ \rangle = \langle s C q^ \rangle = s \langle H_ \rangle. Thus, the average potential energy equals , not as for the quadratic harmonic oscillator (where ). More generally, a typical energy function of a one-dimensional system has a
Taylor expansion In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor seri ...
in the extension : :H_ = \sum_^\infty C_n q^n for non-negative
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s . There is no term, because at the equilibrium point, there is no net force and so the first derivative of the energy is zero. The term need not be included, since the energy at the equilibrium position may be set to zero by convention. In this case, the law of equipartition predicts that : k_\text T = \left\langle q \frac \right\rangle = \sum_^ \langle q \cdot n C_ q^ \rangle = \sum_^ n C_ \langle q^ \rangle. In contrast to the other examples cited here, the equipartition formula : \langle H_ \rangle = \frac k_\text T - \sum_^ \left( \frac \right) C_ \langle q^ \rangle does ''not'' allow the average potential energy to be written in terms of known constants.


Brownian motion

The equipartition theorem can be used to derive the
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
of a particle from the Langevin equation. According to that equation, the motion of a particle of mass with velocity is governed by
Newton's second law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...
:\frac = \frac \mathbf = -\frac + \frac \mathbf_, where is a random force representing the random collisions of the particle and the surrounding molecules, and where the
time constant In physics and engineering, the time constant, usually denoted by the Greek letter (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system.Concretely, a first-order LTI system is a sy ...
τ reflects the drag (physics), drag force that opposes the particle's motion through the solution. The drag force is often written ; therefore, the time constant equals . The dot product of this equation with the position vector , after averaging, yields the equation : \left\langle \mathbf \cdot \frac \right\rangle + \frac \langle \mathbf \cdot \mathbf \rangle = 0 for Brownian motion (since the random force is uncorrelated with the position ). Using the mathematical identities : \frac \left( \mathbf \cdot \mathbf \right) = \frac \left( r^ \right) = 2 \left( \mathbf \cdot \mathbf \right) and :\frac \left( \mathbf \cdot \mathbf \right) = v^ + \mathbf \cdot \frac, the basic equation for Brownian motion can be transformed into : \frac \langle r^ \rangle + \frac \frac \langle r^ \rangle = 2 \langle v^ \rangle = \frac k_\text T, where the last equality follows from the equipartition theorem for translational kinetic energy: : \langle H_ \rangle = \left\langle \frac \right\rangle = \langle \tfrac m v^ \rangle = \tfrac k_\text T. The above differential equation for \langle r^2\rangle (with suitable initial conditions) may be solved exactly: :\langle r^ \rangle = \frac \left( e^ - 1 + \frac \right). On small time scales, with , the particle acts as a freely moving particle: by the Taylor series of the exponential function, the squared distance grows approximately ''quadratically'': :\langle r^ \rangle \approx \frac t^2 = \langle v^ \rangle t^. However, on long time scales, with , the exponential and constant terms are negligible, and the squared distance grows only ''linearly'': :\langle r^ \rangle \approx \frac t = \frac. This describes the diffusion of the particle over time. An analogous equation for the rotational diffusion of a rigid molecule can be derived in a similar way.


Stellar physics

The equipartition theorem and the related
virial theorem In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by potential forces, with that of the total potential energy of the system. ...
have long been used as a tool in astrophysics. As examples, the virial theorem may be used to estimate stellar temperatures or the Chandrasekhar limit on the mass of
white dwarf A white dwarf is a stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very dense: its mass is comparable to the Sun's, while its volume is comparable to the Earth's. A white dwarf's faint luminosity comes fro ...
stars. The average temperature of a star can be estimated from the equipartition theorem. Since most stars are spherically symmetric, the total Newton's law of universal gravitation, gravitational Potential energy#Gravitational potential energy, potential energy can be estimated by integration :H_ = -\int_0^R \frac M(r)\, \rho(r)\, dr, where is the mass within a radius and is the stellar density at radius ; represents the gravitational constant and the total radius of the star. Assuming a constant density throughout the star, this integration yields the formula :H_ = - \frac, where is the star's total mass. Hence, the average potential energy of a single particle is :\langle H_ \rangle = \frac = - \frac, where is the number of particles in the star. Since most
star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...
s are composed mainly of ionized hydrogen, equals roughly , where is the mass of one proton. Application of the equipartition theorem gives an estimate of the star's temperature :\left\langle r \frac \right\rangle = \langle -H_ \rangle = k_\text T = \frac. Substitution of the mass and radius of the Sun yields an estimated solar temperature of ''T'' = 14 million kelvins, very close to its core temperature of 15 million kelvins. However, the Sun is much more complex than assumed by this model—both its temperature and density vary strongly with radius—and such excellent agreement (≈7% approximation error, relative error) is partly fortuitous.


Star formation

The same formulae may be applied to determining the conditions for star formation in giant molecular clouds. A local fluctuation in the density of such a cloud can lead to a runaway condition in which the cloud collapses inwards under its own gravity. Such a collapse occurs when the equipartition theorem—or, equivalently, the
virial theorem In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by potential forces, with that of the total potential energy of the system. ...
—is no longer valid, i.e., when the gravitational potential energy exceeds twice the kinetic energy :\frac > 3 N k_\text T. Assuming a constant density for the cloud :M = \frac \pi R^ \rho yields a minimum mass for stellar contraction, the Jeans mass :M_\text^ = \left( \frac \right)^ \left( \frac \right). Substituting the values typically observed in such clouds (, ) gives an estimated minimum mass of 17 solar masses, which is consistent with observed star formation. This effect is also known as the Jeans instability, after the British physicist James Hopwood Jeans who published it in 1902.


Derivations


Kinetic energies and the Maxwell–Boltzmann distribution

The original formulation of the equipartition theorem states that, in any physical system in
thermal equilibrium Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in ...
, every particle has exactly the same average translational
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
, . However, this is true only for
ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
, and the same result can be derived from the
Maxwell–Boltzmann distribution In physics (in particular in statistical mechanics), the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann. It was first defined and use ...
. First, we choose to consider only the Maxwell–Boltzmann distribution of velocity of the z-component :f (v_z) = \sqrt\;e^ with this equation, we can calculate the mean square velocity of the -component :\langle ^2 \rangle = \int_^ f (v_z)^2 dv_z = \dfrac Since different components of velocity are independent of each other, the average translational kinetic energy is given by :\langle E_k \rangle = \dfrac 3 2 m \langle ^2 \rangle = \dfrac 3 2 k_\textT Notice, the
Maxwell–Boltzmann distribution In physics (in particular in statistical mechanics), the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann. It was first defined and use ...
should not be confused with the Boltzmann distribution, which the former can be derived from the latter by assuming the energy of a particle is equal to its translational kinetic energy. As stated by the equipartition theorem. The same result can also be obtained by averaging the particle energy using the probability of finding the particle in certain quantum energy state.


Quadratic energies and the partition function

More generally, the equipartition theorem states that any
degree of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
which appears in the total energy only as a simple quadratic term , where is a constant, has an average energy of in thermal equilibrium. In this case the equipartition theorem may be derived from the Partition function (statistical mechanics), partition function , where is the canonical inverse temperature. Integration over the variable yields a factor :Z_ = \int_^ dx \ e^ = \sqrt, in the formula for . The mean energy associated with this factor is given by :\langle H_ \rangle = - \frac = \frac = \frac k_\text T as stated by the equipartition theorem.


General proofs

General derivations of the equipartition theorem can be found in many
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
textbooks, both for the
microcanonical ensemble In statistical mechanics, the microcanonical ensemble is a statistical ensemble that represents the possible states of a mechanical system whose total energy is exactly specified. The system is assumed to be isolated in the sense that it canno ...
and for the
canonical ensemble In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the heat b ...
. They involve taking averages over the
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
of the system, which is a symplectic manifold. To explain these derivations, the following notation is introduced. First, the phase space is described in terms of canonical coordinates, generalized position coordinates together with their conjugate momentum, conjugate momenta . The quantities completely describe the Configuration space (physics), configuration of the system, while the quantities together completely describe its Classical mechanics, state. Secondly, the infinitesimal volume :d\Gamma = \prod_i dq_i \, dp_i \, of the phase space is introduced and used to define the volume of the portion of phase space where the energy of the system lies between two limits, and : :\Sigma (E, \Delta E) = \int_ d\Gamma . In this expression, is assumed to be very small, . Similarly, is defined to be the total volume of phase space where the energy is less than : :\Omega (E) = \int_ d\Gamma. Since is very small, the following integrations are equivalent :\int_ \ldots d\Gamma = \Delta E \frac \int_ \ldots d\Gamma, where the ellipses represent the integrand. From this, it follows that is proportional to :\Gamma = \Delta E \ \frac = \Delta E \ \rho(E), where is the density of states. By the usual definitions of
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
, the entropy equals , and the
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
is defined by :\frac = \frac = k_\text \frac = k_\text \frac\,\frac .


The canonical ensemble

In the
canonical ensemble In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the heat b ...
, the system is in
thermal equilibrium Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in ...
with an infinite heat bath at
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
(in kelvins). The probability of each state in
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
is given by its Boltzmann factor times a normalization factor \mathcal, which is chosen so that the probabilities sum to one :\mathcal \int e^ d\Gamma = 1, where . Using Integration by parts for a phase-space variable the above can be written as : \mathcal \int e^ d\Gamma = \mathcal \int d[x_k e^] d\Gamma_k - \mathcal \int x_k \frac d\Gamma, where , i.e., the first integration is not carried out over . Performing the first integral between two limits and and simplifying the second integral yields the equation : \mathcal \int \left[ e^ x_ \right]_^ d\Gamma_+ \mathcal \int e^ x_ \beta \frac d\Gamma = 1, The first term is usually zero, either because is zero at the limits, or because the energy goes to infinity at those limits. In that case, the equipartition theorem for the canonical ensemble follows immediately : \mathcal \int e^ x_k \frac \,d\Gamma = \left\langle x_k \frac \right\rangle = \frac = k_\text T. Here, the averaging symbolized by \langle \ldots \rangle is the
ensemble average In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents ...
taken over the
canonical ensemble In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the heat b ...
.


The microcanonical ensemble

In the microcanonical ensemble, the system is isolated from the rest of the world, or at least very weakly coupled to it. Hence, its total energy is effectively constant; to be definite, we say that the total energy is confined between and . For a given energy and spread , there is a region of
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
in which the system has that energy, and the probability of each state in that region of
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
is equal, by the definition of the microcanonical ensemble. Given these definitions, the equipartition average of phase-space variables (which could be either or ) and is given by :\begin \left\langle x_ \frac \right \rangle &= \frac \, \int_ x_ \frac \,d\Gamma\\ &=\frac\, \frac \int_ x_ \frac \,d\Gamma\\ &= \frac \,\frac \int_ x_ \frac \,d\Gamma, \end where the last equality follows because is a constant that does not depend on . Integration by parts, Integrating by parts yields the relation :\begin \int_ x_ \frac \,d\Gamma &= \int_ \frac \bigl( x_m ( H - E ) \bigr) \,d\Gamma - \int_ \delta_ ( H - E ) d\Gamma \\ &= \delta_ \int_ ( E - H ) \,d\Gamma, \end since the first term on the right hand side of the first line is zero (it can be rewritten as an integral of ''H'' − ''E'' on the hypersurface where ). Substitution of this result into the previous equation yields : \left\langle x_m \frac \right\rangle = \delta_ \frac \, \frac \int_ \left( E - H \right)\,d\Gamma = \delta_ \frac \, \int_ \,d\Gamma = \delta_ \frac. Since \rho = \frac the equipartition theorem follows: : \left\langle x_ \frac \right\rangle = \delta_ \left(\frac \frac\right)^ = \delta_ \left(\frac \right)^ = \delta_ k_\text T. Thus, we have derived the #General formulation for all energies, general formulation of the equipartition theorem : \left\langle x_ \frac \right\rangle = \delta_ k_\text T, which was so useful in the #Applications, applications described above.


Limitations


Requirement of ergodicity

The law of equipartition holds only for ergodic hypothesis, ergodic systems in
thermal equilibrium Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in ...
, which implies that all states with the same energy must be equally likely to be populated. Consequently, it must be possible to exchange energy among all its various forms within the system, or with an external
heat bath In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is al ...
in the
canonical ensemble In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the heat b ...
. The number of physical systems that have been rigorously proven to be ergodic is small; a famous example is the dynamical billiards, hard-sphere system of Yakov G. Sinai, Yakov Sinai. The requirements for isolated systems to ensure ergodic theory, 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 is usually a good assumption. A commonly cited counter-example where energy is ''not'' shared among its various forms and where equipartition does ''not'' hold in the microcanonical ensemble is a system of coupled harmonic oscillators. If the system is isolated from the rest of the world, the energy in each
normal mode A normal mode of a dynamical system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencies. ...
is constant; energy is not transferred from one mode to another. Hence, equipartition does not 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 will not be exchanged unless the nonlinear perturbations are strong enough; if they are too small, the energy will remain trapped in at least some of the modes. Another way ergodicity can be broken is by the existence of nonlinear soliton symmetries. In 1953, Enrico Fermi, Fermi, John Pasta, Pasta, Stanislaw Ulam, Ulam and Mary Tsingou, Tsingou conducted Fermi–Pasta–Ulam–Tsingou problem, computer simulations of a vibrating string that included a non-linear term (quadratic in one test, cubic in another, and a piecewise linear approximation to a cubic in a third). They found that the behavior of the system was quite different from what intuition based on equipartition would have led them to expect. Instead of the energies in the modes becoming equally shared, the system exhibited a very complicated quasi-periodic behavior. This puzzling result was eventually explained by Kruskal and Zabusky in 1965 in a paper which, by connecting the simulated system to the Korteweg–de Vries equation led to the development of soliton mathematics.


Failure due to quantum effects

The law of equipartition breaks down when the thermal energy is significantly smaller than the spacing between energy levels. Equipartition no longer holds because it is a poor approximation to assume that the energy levels form a smooth continuum (theory), continuum, which is required in the #Derivations, derivations of the equipartition theorem above. Historically, the failures of the classical equipartition theorem to explain specific heats and blackbody radiation were critical in showing the need for a new theory of matter and radiation, namely,
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
and
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
. To illustrate the breakdown of equipartition, consider the average energy in a single (quantum) harmonic oscillator, which was discussed above for the classical case. Neglecting the irrelevant zero-point energy term, its quantum energy levels are given by , where is the Planck constant, is the fundamental frequency of the oscillator, and is an integer. The probability of a given energy level being populated in the
canonical ensemble In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the heat b ...
is given by its Boltzmann factor :P(E_n) = \frac, where and the denominator is the partition function, here a geometric series :Z = \sum_^ e^ = \frac. Its average energy is given by : \langle H \rangle = \sum_^ E_ P(E_) = \frac \sum_^ nh\nu \ e^ = -\frac \frac = -\frac. Substituting the formula for gives the final result :\langle H \rangle = h\nu \frac. At high temperatures, when the thermal energy is much greater than the spacing between energy levels, the exponential argument is much less than one and the average energy becomes , in agreement with the equipartition theorem (Figure 10). However, at low temperatures, when , the average energy goes to zero—the higher-frequency energy levels are "frozen out" (Figure 10). As another example, the internal excited electronic states of a hydrogen atom do not contribute to its specific heat as a gas at room temperature, since the thermal energy (roughly 0.025 electronvolt, eV) is much smaller than the spacing between the lowest and next higher electronic energy levels (roughly 10 eV). Similar considerations apply whenever the energy level spacing is much larger than the thermal energy. This reasoning was used by
Max Planck Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918. Planck made many substantial contributions to theoretical p ...
and
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
, among others, to resolve the
ultraviolet catastrophe The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was the prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium would emit an unbounded quantity of energy ...
of blackbody radiation.. An s:A Heuristic Model of the Creation and Transformation of Light, English translation is available from Wikisource. 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 , there would be an infinite amount of energy in the container. However, by the reasoning above, the average energy in the higher-frequency modes goes to zero as ''ν'' 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 symmetry, 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 matter, degenerate fermion gas. Such gases are important for the structure of
white dwarf A white dwarf is a stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very dense: its mass is comparable to the Sun's, while its volume is comparable to the Earth's. A white dwarf's faint luminosity comes fro ...
and
neutron star A neutron star is the collapsed core of a massive supergiant star, which had a total mass of between 10 and 25 solar masses, possibly more if the star was especially metal-rich. Except for black holes and some hypothetical objects (e.g. w ...
s. At low temperatures, a fermionic condensate, 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.


See also

* Kinetic theory of gases, Kinetic theory * Quantum statistical mechanics


Notes and references


Further reading

* * * * * * * * ASIN B00085D6OO *


External links


Applet demonstrating equipartition in real time for a mixture of monatomic and diatomic gases
{{Webarchive, url=https://web.archive.org/web/20200806154151/https://webphysics.davidson.edu/physlet_resources/thermo_paper/thermo/examples/ex20_4.html , date=2020-08-06
The equipartition theorem in stellar physics
written by Nir J. Shaviv, an associate professor at the Racah Institute of Physics in the Hebrew University of Jerusalem. Physics theorems Laws of thermodynamics Statistical mechanics theorems