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 b ...
, 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. 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 acce ...
per
degree of freedom in
translational motion of a molecule should equal that in
rotational motion.
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 cap ...
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), a season of the year
* Spring (device), a mechanical device that stores energy
* Spring (hydrology), a natural source of water
* Spring (mathematics), a geometric surface in the shape of a h ...
. For example, it predicts that every atom in a
monatomic ideal gas 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 constan ...
and ''T'' is the
(thermodynamic) temperature. More generally, equipartition can be applied to any
classical system
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classica ...
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 s ...
, and the
Dulong–Petit law 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 ...
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 are significant, such as at low temperatures. When the
thermal energy 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 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 spe ...
—also known as the
ultraviolet catastrophe—led
Max Planck 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 languages, 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 ...
''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 acce ...
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 ch ...
, 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 constan ...
. 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 ta ...
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 (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 cap ...
. This has many applications.
Translational energy and ideal gases
The (Newtonian) kinetic energy of a particle of mass , velocity is given by
:
where , and are the Cartesian components of the velocity . Here, is short for
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) ''moment ...
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 cap ...
of the gas is and hence, in particular, the heat capacity of a
mole 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 ...
and ''R'' is the
gas constant. Since ''R'' ≈ 2
cal Cal or CAL may refer to:
Arts and entertainment
* ''Cal'' (novel), a 1983 novel by Bernard MacLaverty
* "Cal" (short story), a science fiction short story by Isaac Asimov
* ''Cal'' (1984 film), an Irish drama starring John Lynch and Helen Mir ...
/(
mol·
K), equipartition predicts that the
molar heat capacity 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 of the gas particles to be calculated:
:
where is the mass of a mole of gas particles. This result is useful for many applications such as
Graham's law
Graham's law of effusion (also called Graham's law of diffusion) was formulated by Scottish physical chemist Thomas Graham in 1848.Keith J. Laidler and John M. Meiser, ''Physical Chemistry'' (Benjamin/Cummings 1982), pp. 18–19 Graham found ...
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 escap ...
, 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 , and . The rotational energy of such a molecule is given by
:
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 object ...
. 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, particularly
protein NMR 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 Weber ...
,
flow birefringence and
dielectric spectroscopy.
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 equilibrium position, experiences a restoring force ''F'' proportional to the displacement ''x'':
\vec F = -k \vec x,
where ''k'' is a positive const ...
s such as a
spring
Spring(s) may refer to:
Common uses
* Spring (season), a season of the year
* Spring (device), a mechanical device that stores energy
* Spring (hydrology), a natural source of water
* Spring (mathematics), a geometric surface in the shape of a h ...
, which has a quadratic potential energy
:
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
:
where and denote the velocity and momentum of the oscillator. Combining these terms yields the total energy
:
Equipartition therefore implies that in thermal equilibrium, the oscillator has average energy
:
where the angular brackets
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 th ...
, a vibrating molecule or a passive
electronic oscillator
An electronic oscillator is an electronic circuit that produces a periodic, oscillating electronic signal, often a sine wave or a square wave or a triangle wave. Oscillators convert direct current (DC) from a power supply to an alternating ...
. 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 cap ...
. 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 reg ...
and the
Dulong–Petit law 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 th ...
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 ...
, and using the relation between the
gas constant and the Boltzmann constant , this provides an explanation for the
Dulong–Petit law 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 theor ...
(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, Netherland ...
(1911).
Many other physical systems can be modeled as sets of
coupled oscillators. 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 s ...
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 oscil ...
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, res ...
s that
scatter
Scatter may refer to:
* Scattering, in physics, the study of collisions
* Statistical dispersion or scatter
* Scatter (modeling), a substance used in the building of dioramas and model railways
* Scatter, in computer programming, a parameter in ...
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 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 . For an infinitely tall bottle of beer, the
gravitational potential energy is given by
:
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 ...
due to gravity. Since , the average potential energy of a protein clump equals . Hence, a protein clump with a buoyant mass of 10
MDa (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 Ivanovsk ...
) 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.
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 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 the Sande ...
. 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 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 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 and
Alexis Thérèse Petit 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 decis ...
showed that this
Dulong–Petit law 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 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 chemically stable form of carbon at room temperature and pressure, b ...
, 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
Maxwell may refer to:
People
* Maxwell (surname), including a list of people and fictional characters with the name
** James Clerk Maxwell, mathematician and physicist
* Justice Maxwell (disambiguation)
* Maxwell baronets, in the Baronetage of ...
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
The Drude model of electrical conduction was proposed in 1900 by Paul Drude to explain the transport properties of electrons in materials (especially metals). Basically, Ohm's law was well established and stated that the current ''J'' and voltag ...
, 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 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 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. A ...
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 theor ...
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 among physicists.
General formulation of the equipartition theorem
The most general form of the equipartition theorem states that under suitable assumptions (discussed
below
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred Below ...
), for a physical system with
Hamiltonian energy function and degrees of freedom , the following equipartition formula holds in thermal equilibrium for all indices and :
:
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
is assumed to be an
ensemble average 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 hea ...
,
when the system is coupled to a
heat bath 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:
#
#
If a degree of freedom ''x
n'' appears only as a quadratic term ''a
nx
n''
2 in the Hamiltonian ''H'', then the first of these formulae implies that
:
which is twice the contribution that this degree of freedom makes to the average energy
. 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 ''a
nx
ns''.
The degrees of freedom ''x
n'' 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 usuall ...
of the system and are therefore commonly subdivided into
generalized position coordinates ''q
k'' and
generalized momentum coordinates ''p
k'', where ''p
k'' 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 cl ...
to ''q
k''. In this situation, formula 1 means that for all ''k'',
:
Using the equations of
Hamiltonian mechanics,
these formulae may also be written
:
Similarly, one can show using formula 2 that
:
and
:
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
:
where ''t'' denotes
time
Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
.
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 usuall ...
.
Applications
Ideal gas law
Ideal gases provide an important application of the equipartition theorem. As well as providing the formula
:
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 s ...
from classical mechanics.
If q = (''q
x'', ''q
y'', ''q
z'') and p = (''p
x'', ''p
y'', ''p
z'') denote the position vector and momentum of a particle in the gas, and
F is the net force on that particle, then
:
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 mo ...
, and the second line uses
Hamilton's equations and the equipartition formula. Summing over a system of ''N'' particles yields
:
By
Newton's third law 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
:
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 ...
of the position vector is
:
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 ...
implies that
:
where is an infinitesimal volume within the container and is the total volume of the container.
Putting these equalities together yields
:
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 s ...
for ''N'' particles:
:
where is the number of moles of gas and is the
gas constant. 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), a season of the year
* Spring (device), a mechanical device that stores energy
* Spring (hydrology), a natural source of water
* Spring (mathematics), a geometric surface in the shape of a h ...
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 ...
, which is called the ''rigid rotor-harmonic oscillator approximation''.
The classical energy of this system is
:
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 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 s ...
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 motio ...
. 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 ...
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.
In such cases, the kinetic energy of a
single particle is given by the formula
:
Taking the derivative of with respect to the momentum component gives the formula
:
and similarly for the and components. Adding the three components together gives
:
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 forces 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 distribution. It is then customary to introduce a
radial distribution function such that the
probability density 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. Mathematicall ...
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
:
The total mean potential energy of the gas is therefore
, 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''
:
A similar argument,
can be used to derive the ''pressure equation''
:
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
:
where and are arbitrary
real constants. In these cases, the law of equipartition predicts that
:
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 the extension :
:
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 languag ...
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
:
In contrast to the other examples cited here, the equipartition formula
:
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
In physics, a Langevin equation (named after Paul Langevin) is a stochastic differential equation describing how a system evolves when subjected to a combination of deterministic and fluctuating ("random") forces. The dependent variables in a Lang ...
.
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 mo ...
:
where is a random force representing the random collisions of the particle and the surrounding molecules, and where the
time constant τ reflects the
drag force
In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding flu ...
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
:
for Brownian motion (since the random force is uncorrelated with the position ). Using the mathematical identities
:
and
:
the basic equation for Brownian motion can be transformed into
:
where the last equality follows from the equipartition theorem for translational kinetic energy:
:
The above
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
for
(with suitable initial conditions) may be solved exactly:
:
On small time scales, with , the particle acts as a freely moving particle: by the
Taylor series
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 ser ...
of the
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
, the squared distance grows approximately ''quadratically'':
:
However, on long time scales, with , the exponential and constant terms are negligible, and the squared distance grows only ''linearly'':
:
This describes the
diffusion
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 ...
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
Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline said, Astrophysics "seeks to ascertain the nature of the h ...
. As examples, the virial theorem may be used to estimate stellar temperatures or the
Chandrasekhar limit
The Chandrasekhar limit () is the maximum mass of a stable white dwarf star. The currently accepted value of the Chandrasekhar limit is about ().
White dwarfs resist gravitational collapse primarily through electron degeneracy pressure, compar ...
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 ...
stars.
The average temperature of a star can be estimated from the equipartition theorem. Since most stars are spherically symmetric, the total
gravitational
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 str ...
potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potenti ...
can be estimated by integration
:
where is the mass within a radius and is the stellar density at radius ; represents the
gravitational constant
The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...
and the total radius of the star. Assuming a constant density throughout the star, this integration yields the formula
:
where is the star's total mass. Hence, the average potential energy of a single particle is
:
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
Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-to ...
, equals roughly , where is the mass of one proton. Application of the equipartition theorem gives an estimate of the star's temperature
:
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%
relative error
The approximation error in a data value is the discrepancy between an exact value and some ''approximation'' to it. This error can be expressed as an absolute error (the numerical amount of the discrepancy) or as a relative error (the absolute er ...
) is partly fortuitous.
Star formation
The same formulae may be applied to determining the conditions for
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
:
Assuming a constant density for the cloud
:
yields a minimum mass for stellar contraction, the Jeans mass
:
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
Sir James Hopwood Jeans (11 September 187716 September 1946) was an English physicist, astronomer and mathematician.
Early life
Born in Ormskirk, Lancashire, the son of William Tulloch Jeans, a parliamentary correspondent and author. 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 acce ...
, .
However, this is true only for
ideal gas, 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
:
with this equation, we can calculate the mean square velocity of the -component
:
Since different components of velocity are independent of each other, the average translational kinetic energy is given by
:
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 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 , where is the canonical
inverse temperature. Integration over the variable yields a factor
:
in the formula for . The mean energy associated with this factor is given by
:
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 b ...
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 hea ...
.
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 usuall ...
of the system, which is a
symplectic manifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sym ...
.
To explain these derivations, the following notation is introduced. First, the phase space is described in terms of
generalized position coordinates together with their
conjugate momenta
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 cl ...
. The quantities completely describe the
configuration
Configuration or configurations may refer to:
Computing
* Computer configuration or system configuration
* Configuration file, a software file used to configure the initial settings for a computer program
* Configurator, also known as choice bo ...
of the system, while the quantities together completely describe its
state.
Secondly, the infinitesimal volume
:
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 :
:
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 :
:
Since is very small, the following integrations are equivalent
:
where the ellipses represent the integrand. From this, it follows that is proportional to
:
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 b ...
, the
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
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
:
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 hea ...
, 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 usuall ...
is given by its
Boltzmann factor times a
normalization factor , which is chosen so that the probabilities sum to one
:
where . Using
Integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivat ...
for a phase-space variable the above can be written as
:
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
:
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
:
Here, the averaging symbolized by
is the
ensemble average 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 hea ...
.
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 usuall ...
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 usuall ...
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
:
where the last equality follows because is a constant that does not depend on .
Integrating by parts yields the relation
:
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
:
Since
the equipartition theorem follows:
:
Thus, we have derived the
general formulation of the equipartition theorem
:
which was so useful in the
applications described above.
Limitations
Requirement of ergodicity
The law of equipartition holds only for
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 ...
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 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 hea ...
. The number of physical systems that have been rigorously proven to be ergodic is small; a famous example is the
hard-sphere system of
Yakov Sinai. The requirements for isolated systems to ensure
ergodicity—and, thus equipartition—have been studied, and provided motivation for the modern
chaos theory
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to hav ...
of
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
s. 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
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 ...
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
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the me ...
symmetries. In 1953,
Fermi
Enrico Fermi (; 29 September 1901 – 28 November 1954) was an Italian (later naturalized American) physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the nuclear age" and t ...
,
Pasta
Pasta (, ; ) is a type of food typically made from an unleavened dough of wheat flour mixed with water or eggs, and formed into sheets or other shapes, then cooked by boiling or baking. Rice flour, or legumes such as beans or lentils, ...
,
Ulam Ulam may refer to:
* ULAM, the ICAO airport code for Naryan-Mar Airport, Russia
* Ulam (surname)
* Ulam (salad), a type of Malay salad
* ''Ulam'', a Filipino term loosely translated to viand or side dish; see Tapa (Filipino cuisine)
* Ulam, the l ...
and
Tsingou conducted
computer simulations
Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be dete ...
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, which is required in the
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
Zero-point energy (ZPE) is the lowest possible energy that a quantum mechanical system may have. Unlike in classical mechanics, quantum systems constantly fluctuate in their lowest energy state as described by the Heisenberg uncertainty pri ...
term, its quantum energy levels are given by , where is the
Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
, is the
fundamental frequency
The fundamental frequency, often referred to simply as the ''fundamental'', is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. I ...
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 hea ...
is given by its
Boltzmann factor
:
where and the denominator is the
partition function, here a
geometric series
:
Its average energy is given by
:
Substituting the formula for gives the final result
:
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
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 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 theor ...
, among others, to resolve the
ultraviolet catastrophe of
blackbody radiation.
[. An English translation is available from ]Wikisource
Wikisource is an online digital library of free-content textual sources on a wiki, operated by the Wikimedia Foundation. Wikisource is the name of the project as a whole and the name for each instance of that project (each instance usually re ...
. The paradox arises because there are an infinite number of independent modes of the
electromagnetic field
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...
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
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
electronvolt
In physics, an electronvolt (symbol eV, also written electron-volt and electron volt) is the measure of an amount of kinetic energy gained by a single electron accelerating from rest through an electric potential difference of one volt in vacu ...
s, 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
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formula ...
invalidates the classical approach, is called a
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 ...
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 analogue of the
Bose–Einstein condensate
In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.6 ...
(in which a large number of identical particles occupy the lowest-energy state) can form; such
superfluid electrons are responsible for
superconductivity
Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
.
See also
*
Kinetic theory
*
Quantum statistical mechanics
Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems. In quantum mechanics a statistical ensemble (probability distribution over possible quantum states) is described by a density operator ''S'', which is ...
Notes and references
Further reading
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* ASIN B00085D6OO
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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
The Hebrew University of Jerusalem (HUJI; he, הַאוּנִיבֶרְסִיטָה הַעִבְרִית בִּירוּשָׁלַיִם) is a public research university based in Jerusalem, Israel. Co-founded by Albert Einstein and Dr. Chaim Weiz ...
.
Physics theorems
Laws of thermodynamics
Statistical mechanics theorems