Equipartition theorem
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In
The equipartition theorem makes quantitative predictions. Like the
Although the equipartition theorem makes accurate predictions in certain conditions, it is inaccurate when
Basic concept and simple examples
The name "equipartition" means "equal division," as derived from the Latin equi from the antecedent, æquus ("equal or even"), and partition from the noun, partitio ("division, portion").[2][3] The original concept of equipartition was that the total kinetic energy of a system is shared equally among all of its independent parts, on the average, once the system has reached thermal equilibrium. Equipartition also makes quantitative predictions for these energies. For example, it predicts that every atom of an inert noble gas, in thermal equilibrium at temperature T, has an average translational kinetic energy of 3/2kBT, where kB is the Boltzmann constant. As a consequence, since kinetic energy is equal to 1⁄2(mass)(velocity)2, the heavier atoms of xenon have a lower average speed than do the lighter atoms of helium at the same temperature. Figure 2 shows the Maxwell–Boltzmann distribution for the speeds of the atoms in four noble gases.
In this example, the key point is that the kinetic energy is quadratic in the velocity. The equipartition theorem shows that in thermal equilibrium, any degree of freedom (such as a component of the position or velocity of a particle) which appears only quadratically in the energy has an average energy of 1⁄2kBT and therefore contributes 1⁄2kB to the system's heat capacity. This has many applications.
Translational energy and ideal gases
The (Newtonian) kinetic energy of a particle of mass m, velocity v is given by
where vx, vy and vz are the Cartesian components of the velocity v. Here, H is short for Hamiltonian, and used henceforth as a symbol for energy because the Hamiltonian formalism 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 1⁄2kBT to the average kinetic energy in thermal equilibrium. Thus the average kinetic energy of the particle is 3/2kBT, 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 N particles is 3/2 N kB T.
It follows that the heat capacity of the gas is 3/2 N kB and hence, in particular, the heat capacity of a mole of such gas particles is 3/2NAkB = 3/2R, where NA is the Avogadro constant and R is the gas constant. Since R ≈ 2 cal/(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.[4]
The mean kinetic energy also allows the
where M = NAm is the mass of a mole of gas particles. This result is useful for many applications such as Graham's law of effusion, which provides a method for enriching uranium.[5]
Rotational energy and molecular tumbling in solution
A similar example is provided by a rotating molecule with
where ω1, ω2, and ω3 are the principal components of the angular velocity. By exactly the same reasoning as in the translational case, equipartition implies that in thermal equilibrium the average rotational energy of each particle is 3/2kBT. Similarly, the equipartition theorem allows the average (more precisely, the root mean square) angular speed of the molecules to be calculated.[6]
The tumbling of rigid molecules—that is, the random rotations of molecules in solution—plays a key role in the
Potential energy and harmonic oscillators
Equipartition applies to potential energies as well as kinetic energies: important examples include harmonic oscillators such as a spring, which has a quadratic potential energy
where the constant a describes the stiffness of the spring and q is the deviation from equilibrium. If such a one-dimensional system has mass m, then its kinetic energy Hkin is
where v and p = mv denote the velocity and momentum of the oscillator. Combining these terms yields the total energy[9]
Equipartition therefore implies that in thermal equilibrium, the oscillator has average energy
where the angular brackets denote the average of the enclosed quantity,[10]
This result is valid for any type of harmonic oscillator, such as a pendulum, a vibrating molecule or a passive electronic oscillator. Systems of such oscillators arise in many situations; by equipartition, each such oscillator receives an average total energy kBT and hence contributes kB to the system's heat capacity. This can be used to derive the formula for Johnson–Nyquist noise[11] 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 3N independent
By taking N to be the
However, this law is inaccurate at lower temperatures, due to quantum effects; it is also inconsistent with the experimentally derived third law of thermodynamics, according to which the molar heat capacity of any substance must go to zero as the temperature goes to absolute zero.[11] A more accurate theory, incorporating quantum effects, was developed by Albert Einstein (1907) and Peter Debye (1911).[12]
Many other physical systems can be modeled as sets of
Sedimentation of particles
Potential energies are not always quadratic in the position. However, the equipartition theorem also shows that if a degree of freedom x contributes only a multiple of xs (for a fixed real number s) to the energy, then in thermal equilibrium the average energy of that part is kBT/s.
There is a simple application of this extension to the
where z is the height of the protein clump in the bottle and
History
The equipartition of kinetic energy was proposed initially in 1843, and more correctly in 1845, by
The history of the equipartition theorem is intertwined with that of specific heat capacity, 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.[21] Their law was used for many years as a technique for measuring atomic weights.[12] However, subsequent studies by James Dewar and Heinrich Friedrich Weber showed that this Dulong–Petit law holds only at high temperatures;[22] at lower temperatures, or for exceptionally hard solids such as diamond, the specific heat capacity was lower.[23]
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,[4] but found that molar heat capacities of diatomic gases were typically about 5 cal/(mol·K),[24] and fell to about 3 cal/(mol·K) at very low temperatures.[25] Maxwell noted in 1875 that the disagreement between experiment and the equipartition theorem was much worse than even these numbers suggest;[26] 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.[27] According to the classical Drude model, metallic electrons act as a nearly ideal gas, and so they should contribute 3/2 NekB to the heat capacity by the equipartition theorem, where Ne 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.[27]
Several explanations of equipartition's failure to account for molar heat capacities were proposed.
General formulation of the equipartition theorem
The most general form of the equipartition theorem states that under suitable assumptions (discussed
Here δmn is the Kronecker delta, which is equal to one if m = n and is zero otherwise. The averaging brackets is assumed to be an
The general equipartition theorem holds in both the
The general formula is equivalent to the following two:
If a degree of freedom xn appears only as a quadratic term anxn2 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 anxns.
The degrees of freedom xn are coordinates on the
Using the equations of Hamiltonian mechanics,[9] 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 (proposed in 1870[35]), which states that
where t denotes time.[9] 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 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.
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 from classical mechanics.[6] 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
where the first equality is
By
where dS is the infinitesimal area element along the walls of the container. Since the divergence of the position vector q is
the divergence theorem implies that
where dV is an infinitesimal volume within the container and V is the total volume of the container.
Putting these equalities together yields
which immediately implies the ideal gas law for N particles:
where n = N/NA is the number of moles of gas and R = NAkB 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.[36]
Diatomic gases
A diatomic gas can be modelled as two masses, m1 and m2, joined by a spring of stiffness a, which is called the rigid rotor-harmonic oscillator approximation.[20] The classical energy of this system is
where p1 and p2 are the momenta of the two atoms, and q 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 1⁄2kBT to the total average energy, and 1⁄2kB to the heat capacity. Therefore, the heat capacity of a gas of N diatomic molecules is predicted to be 7N·1⁄2kB: the momenta p1 and p2 contribute three degrees of freedom each, and the extension q contributes the seventh. It follows that the heat capacity of a mole of diatomic molecules with no other degrees of freedom should be 7/2NAkB = 7/2R 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)[24] and fall to 3 cal/(mol·K) at very low temperatures.[25] 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.[26] 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
Taking the derivative of H with respect to the px momentum component gives the formula
and similarly for the py and pz 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 N particles, it is 3 NkBT.
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
The total mean potential energy of the gas is therefore , where N is the number of particles in the gas, and the factor 1⁄2 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,[6] 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 q (the
where C and s are arbitrary real constants. In these cases, the law of equipartition predicts that
Thus, the average potential energy equals kBT/s, not kBT/2 as for the quadratic harmonic oscillator (where s = 2).
More generally, a typical energy function of a one-dimensional system has a
for non-negative integers n. There is no n = 1 term, because at the equilibrium point, there is no net force and so the first derivative of the energy is zero. The n = 0 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[38]
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 of a particle from the Langevin equation.[6] According to that equation, the motion of a particle of mass m with velocity v is governed by Newton's second law
where Frnd 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 that opposes the particle's motion through the solution. The drag force is often written Fdrag = −γv; therefore, the time constant τ equals m/γ.
The dot product of this equation with the position vector r, after averaging, yields the equation
for Brownian motion (since the random force Frnd is uncorrelated with the position r). 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 for (with suitable initial conditions) may be solved exactly:
On small time scales, with t ≪ τ, the particle acts as a freely moving particle: by the Taylor series of the exponential function, the squared distance grows approximately quadratically:
However, on long time scales, with t ≫ τ, the exponential and constant terms are negligible, and the squared distance grows only linearly:
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 have long been used as a tool in astrophysics.[40] As examples, the virial theorem may be used to estimate stellar temperatures or the Chandrasekhar limit on the mass of white dwarf stars.[41][42]
The average temperature of a star can be estimated from the equipartition theorem.[43] Since most stars are spherically symmetric, the total gravitational potential energy can be estimated by integration
where M(r) is the mass within a radius r and ρ(r) is the stellar density at radius r; G represents the gravitational constant and R the total radius of the star. Assuming a constant density throughout the star, this integration yields the formula
where M is the star's total mass. Hence, the average potential energy of a single particle is
where N is the number of particles in the star. Since most stars are composed mainly of ionized hydrogen, N equals roughly M/mp, where mp 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) is partly fortuitous.[44]
Star formation
The same formulae may be applied to determining the conditions for star formation in giant molecular clouds.[45] 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—is no longer valid, i.e., when the gravitational potential energy exceeds twice the kinetic energy
Assuming a constant density ρ for the cloud
yields a minimum mass for stellar contraction, the Jeans mass MJ
Substituting the values typically observed in such clouds (T = 150 K, ρ = 2×10−16 g/cm3) gives an estimated minimum mass of 17 solar masses, which is consistent with observed star formation. This effect is also known as the
Derivations
Kinetic energies and the Maxwell–Boltzmann distribution
The original formulation of the equipartition theorem states that, in any physical system in thermal equilibrium, every particle has exactly the same average translational kinetic energy, 3/2kBT.[47] However, this is true only for ideal gas, and the same result can be derived from the Maxwell–Boltzmann distribution. 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 z-component
Since different components of velocity are independent of each other, the average translational kinetic energy is given by
Notice, the Maxwell–Boltzmann distribution 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.[36]
Quadratic energies and the partition function
More generally, the equipartition theorem states that any
in the formula for Z. 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 textbooks, both for the microcanonical ensemble[6][10] and for the canonical ensemble.[6][34] They involve taking averages over the phase space 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
Secondly, the infinitesimal volume
of the phase space is introduced and used to define the volume Σ(E, ΔE) of the portion of phase space where the energy H of the system lies between two limits, E and E + ΔE:
In this expression, ΔE is assumed to be very small, ΔE ≪ E. Similarly, Ω(E) is defined to be the total volume of phase space where the energy is less than E:
Since ΔE is very small, the following integrations are equivalent
where the ellipses represent the integrand. From this, it follows that Σ is proportional to ΔE
where ρ(E) is the density of states. By the usual definitions of statistical mechanics, the entropy S equals kB log Ω(E), and the temperature T is defined by
The canonical ensemble
In the
where β = 1/(kBT). Using Integration by parts for a phase-space variable xk the above can be written as
where dΓk = dΓ/dxk, i.e., the first integration is not carried out over xk. Performing the first integral between two limits a and b and simplifying the second integral yields the equation
The first term is usually zero, either because xk 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
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.[10] Hence, its total energy is effectively constant; to be definite, we say that the total energy H is confined between E and E+dE. For a given energy E and spread dE, there is a region of phase space Σ in which the system has that energy, and the probability of each state in that region of phase space is equal, by the definition of the microcanonical ensemble. Given these definitions, the equipartition average of phase-space variables xm (which could be either qk or pk) and xn is given by
where the last equality follows because E is a constant that does not depend on xn. 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 H = E).
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
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.[50] If the system is isolated from the rest of the world, the energy in each normal mode 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 simple example is an ideal gas of a finite number of colliding particles in a round vessel. Due to the vessel's symmetry, the angular momentum of such a gas is conserved. Therefore, not all states with the same energy are populated. This results in the mean particle energy being dependent on the mass of this particle, and also on the masses of all the other particles.[51]
Another way ergodicity can be broken is by the existence of nonlinear
Failure due to quantum effects
The law of equipartition breaks down when the thermal energy kBT 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
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
where β = 1/kBT and the denominator Z is the partition function, here a geometric series
Its average energy is given by
Substituting the formula for Z gives the final result[10]
At high temperatures, when the thermal energy kBT is much greater than the spacing hν between energy levels, the exponential argument βhν is much less than one and the average energy becomes kBT, in agreement with the equipartition theorem (Figure 10). However, at low temperatures, when hν ≫ kBT, 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 kBT (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, among others, to resolve the ultraviolet catastrophe of black-body radiation.[52] 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 kBT, there would be an infinite amount of energy in the container.[52][53] 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.[52]
Other, more subtle quantum effects can lead to corrections to equipartition, such as
See also
Notes and references
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- ^ "partition". Online Etymology Dictionary. Retrieved 2008-12-20..
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- ^ ISBN 0-471-56658-6.
- ^ ISBN 0-19-853907-X.
- ^ .
- PMID 16366710.
- .
- doi:10.1098/rsta.1892.0001. Reprinted J.S. Haldane, ed. (1928). The collected scientific papers of John James Waterston. Edinburgh: Oliver & Boyd.) Waterston's key paper was written and submitted in 1845 to the
Waterston, JJ (1843). Thoughts on the Mental Functions. (reprinted in his Papers, 3, 167, 183.)
Waterston, JJ (1851). British Association Reports. 21: 6.{{cite journal}}
: Missing or empty|title=
(helpLord Rayleigh, who criticized the original reviewer for failing to recognize the significance of Waterston's work. Waterston managed to publish his ideas in 1851, and therefore has priority over Maxwell for enunciating the first version of the equipartition theorem. - ISBN 978-0-486-49560-6. Read by Prof. Maxwell at a Meeting of the British Association at Aberdeen on 21 September 1859.
- ^ Boltzmann, L (1871). "Einige allgemeine Sätze über Wärmegleichgewicht (Some general statements on thermal equilibrium)". Wiener Berichte (in German). 63: 679–711. In this preliminary work, Boltzmann showed that the average total kinetic energy equals the average total potential energy when a system is acted upon by external harmonic forces.
- ^ Boltzmann, L (1876). "Über die Natur der Gasmoleküle (On the nature of gas molecules)". Wiener Berichte (in German). 74: 553–560.
- ^ ISBN 978-1-891389-15-3.
- Annales de Chimie et de Physique(in French). 10: 395–413.
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- ^ a b Wüller, A (1896). Lehrbuch der Experimentalphysik (Textbook of Experimental Physics) (in German). Leipzig: Teubner. Vol. 2, 507ff.
- ^ a b Eucken, A (1912). "Die Molekularwärme des Wasserstoffs bei tiefen Temperaturen (The molecular specific heat of hydrogen at low temperatures)". Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften (in German). 1912: 141–151.
- ^ ISBN 0-486-61534-0. ASIN B000GW7DXY. A lecture delivered by Prof. Maxwell at the Chemical Society on 18 February 1875.
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- ^ Nernst, W (1910). "Untersuchungen über die spezifische Wärme bei tiefen Temperaturen. II. (Investigations into the specific heat at low temperatures)". Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften (in German). 1910: 262–282.
- LCCN 73151106.
- ^ ISBN 0-486-63896-0.
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Clausius, RJE (1870). "On a Mechanical Theorem Applicable to Heat". Philosophical Magazine - ^ a b Vu-Quoc, L., Configuration integral (statistical mechanics), 2008. this wiki site is down; see this article in the web archive on 2012 April 28.
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- ^ a b Tolman, RC (1927). Statistical Mechanics, with Applications to Physics and Chemistry. Chemical Catalog Company. pp. 76–77.
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- ^ Collins, GW (1978). The Virial Theorem in Stellar Astrophysics. Pachart Press.
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- ^ Kourganoff, V (1980). Introduction to Advanced Astrophysics. Dordrecht, Holland: D. Reidel. pp. 59–60, 134–140, 181–184.
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- ^ Arnold, VI; Avez A (1957). Théorie ergodique des systèms dynamiques (in French). Gauthier-Villars, Paris. (English edition: Benjamin-Cummings, Reading, Mass. 1968).
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Further reading
- ISBN 0-471-81518-7.
- ISBN 0-486-63896-0.
- ISBN 0-08-023039-3.
- Mandl, F (1971). Statistical Physics. John Wiley and Sons. pp. 213–219. ISBN 0-471-56658-6.
- Mohling, F (1982). Statistical Mechanics: Methods and Applications. John Wiley and Sons. pp. 137–139, 270–273, 280, 285–292. ISBN 0-470-27340-2.
- ISBN 0-08-016747-0.
- ISBN 0-262-16049-8.
- Tolman, RC (1927). Statistical Mechanics, with Applications to Physics and Chemistry. Chemical Catalog Company. pp. 72–81. ASIN B00085D6OO
- ISBN 0-486-63896-0.
External links
- Applet demonstrating equipartition in real time for a mixture of monatomic and diatomic gases Archived 2020-08-06 at the Wayback Machine
- 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.