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Inclassicalstatistical mechanics, theequipartition theorem relates thetemperature of a system to its averageenergies. The equipartition theorem is also known as thelaw of equipartition,equipartition of energy, or simplyequipartition. The original idea of equipartition was that, inthermal equilibrium, energy is shared equally among all of its various forms; for example, the averagekinetic energy perdegree of freedom intranslational motion of a molecule should equal that inrotational motion.
The equipartition theorem makes quantitative predictions. Like thevirial theorem, it gives the total average kinetic and potential energies for a system at a given temperature, from which the system'sheat capacity can be computed. However, equipartition also gives the average values of individual components of the energy, such as the kinetic energy of a particular particle or the potential energy of a singlespring. For example, it predicts that every atom in amonatomicideal gas has an average kinetic energy of3/2kBT in thermal equilibrium, wherekB is theBoltzmann constant andT is the(thermodynamic) temperature. More generally, equipartition can be applied to anyclassical system inthermal equilibrium, no matter how complicated. It can be used to derive theideal gas law, and theDulong–Petit law for thespecific heat capacities of solids.[1] The equipartition theorem can also be used to predict the properties ofstars, evenwhite dwarfs andneutron stars, since it holds even whenrelativistic effects are considered.
Although the equipartition theorem makes accurate predictions in certain conditions, it is inaccurate whenquantum effects are significant, such as at low temperatures. When thethermal energykBT is smaller than the quantum energy spacing in a particulardegree 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 modelblack-body radiation—also known as theultraviolet catastrophe—ledMax Planck to suggest that energy in the oscillators in an object, which emit light, were quantized, a revolutionary hypothesis that spurred the development ofquantum mechanics andquantum field theory.
The name "equipartition" means "equal division," as derived from theLatinequi 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 totalkinetic 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 inertnoble gas, in thermal equilibrium at temperatureT, has an average translational kinetic energy of3/2kBT, wherekB is theBoltzmann constant. As a consequence, since kinetic energy is equal to1⁄2(mass)(velocity)2, the heavier atoms ofxenon have a lower average speed than do the lighter atoms ofhelium at the same temperature. Figure 2 shows theMaxwell–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, anydegree 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 of1⁄2kBT and therefore contributes1⁄2kB to the system'sheat capacity. This has many applications.
The (Newtonian) kinetic energy of a particle of massm, velocityv is given bywherevx,vy andvz are the Cartesian components of the velocityv. Here,H is short forHamiltonian, and used henceforth as a symbol for energy because theHamiltonian formalism plays a central role in the mostgeneral form of the equipartition theorem.
Since the kinetic energy is quadratic in the components of the velocity, by equipartition these three components each contribute1⁄2kBT to the average kinetic energy in thermal equilibrium. Thus the average kinetic energy of the particle is3/2kBT, 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 ofN particles is3/2N kB T.
It follows that theheat capacity of the gas is3/2N kB and hence, in particular, the heat capacity of amole of such gas particles is3/2NAkB =3/2R, whereNA is theAvogadro constant andR is thegas constant. SinceR ≈ 2cal/(mol·K), equipartition predicts that themolar 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 theroot mean square speedvrms of the gas particles to be calculated:whereM =NAm is the mass of a mole of gas particles. This result is useful for many applications such asGraham's law ofeffusion, which provides a method forenrichinguranium.[5]
A similar example is provided by a rotating molecule withprincipal moments of inertiaI1,I2 andI3. According to classical mechanics, therotational energy of such a molecule is given bywhereω1,ω2, andω3 are the principal components of theangular velocity. By exactly the same reasoning as in the translational case, equipartition implies that in thermal equilibrium the average rotational energy of each particle is3/2kBT. 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 therelaxations observed bynuclear magnetic resonance, particularlyprotein NMR andresidual dipolar couplings.[7] Rotational diffusion can also be observed by other biophysical probes such asfluorescence anisotropy,flow birefringence anddielectric spectroscopy.[8]
Equipartition applies topotential energies as well as kinetic energies: important examples includeharmonic oscillators such as aspring, which has a quadratic potential energywhere the constanta describes the stiffness of the spring andq is the deviation from equilibrium. If such a one-dimensional system has massm, then its kinetic energyHkin iswherev andp =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 energywhere the angular brackets denote the average of the enclosed quantity,[10]
This result is valid for any type of harmonic oscillator, such as apendulum, a vibrating molecule or a passiveelectronic oscillator. Systems of such oscillators arise in many situations; by equipartition, each such oscillator receives an average total energykBT and hence contributeskB to the system'sheat capacity. This can be used to derive the formula forJohnson–Nyquist noise[11] and theDulong–Petit law of solid heat capacities. The latter application was particularly significant in the history of equipartition.
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 of3N independentsimple harmonic oscillators, whereN denotes the number of atoms in the lattice. Since each harmonic oscillator has average energykBT, the average total energy of the solid is3N kBT, and its heat capacity is3N kB.
By takingN to be theAvogadro constantNA, and using the relationR =NAkB between thegas constantR and the Boltzmann constantkB, this provides an explanation for theDulong–Petit law ofspecific heat capacities of solids, which stated that the specific heat capacity (per unit mass) of a solid element is inversely proportional to itsatomic weight. A modern version is that the molar heat capacity of a solid is3R ≈ 6 cal/(mol·K).
However, this law is inaccurate at lower temperatures, due to quantum effects; it is also inconsistent with the experimentally derivedthird 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 byAlbert Einstein (1907) andPeter Debye (1911).[12]
Many other physical systems can be modeled as sets ofcoupled oscillators. The motions of such oscillators can be decomposed intonormal modes, like the vibrational modes of apiano string or theresonances of anorgan pipe. 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 calledergodicity, is important for the law of equipartition to hold.
Potential energies are not always quadratic in the position. However, the equipartition theorem also shows that if a degree of freedomx contributes only a multiple ofxs (for a fixed real numbers) to the energy, then in thermal equilibrium the average energy of that part iskBT/s.
There is a simple application of this extension to thesedimentation of particles undergravity.[13] For example, the haze sometimes seen inbeer can be caused by clumps ofproteins thatscatter light.[14] 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 alsodiffuse 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 ofbuoyant massmb. For an infinitely tall bottle of beer, thegravitational potential energy is given bywherez is the height of the protein clump in the bottle andg is theacceleration due to gravity. Sinces = 1, the average potential energy of a protein clump equalskBT. Hence, a protein clump with a buoyant mass of 10 MDa (roughly the size of avirus) would produce a haze with an average height of about 2 cm at equilibrium. The process of such sedimentation to equilibrium is described by theMason–Weaver equation.[15]
The equipartition of kinetic energy was proposed initially in 1843, and more correctly in 1845, byJohn James Waterston.[16] In 1859,James Clerk Maxwell argued that the kinetic heat energy of a gas is equally divided between linear and rotational energy.[17] 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.[18][19] Boltzmann applied the equipartition theorem to provide a theoretical explanation of theDulong–Petit law for thespecific heat capacities of solids.
The history of the equipartition theorem is intertwined with that ofspecific heat capacity, both of which were studied in the 19th century. In 1819, the French physicistsPierre Louis Dulong andAlexis 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 byJames Dewar andHeinrich Friedrich Weber showed that thisDulong–Petit law holds only at hightemperatures;[22] at lower temperatures, or for exceptionally hard solids such asdiamond, 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 classicalDrude model, metallic electrons act as a nearly ideal gas, and so they should contribute3/2NekB to the heat capacity by the equipartition theorem, whereNe 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.Boltzmann defended the derivation of his equipartition theorem as correct, but suggested that gases might not be inthermal equilibrium because of their interactions with theaether.[28]Lord Kelvin suggested that the derivation of the equipartition theorem must be incorrect, since it disagreed with experiment, but was unable to show how.[29] In 1900Lord Rayleigh instead put forward a more radical view that the equipartition theorem and the experimental assumption of thermal equilibrium wereboth 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.[30]Albert Einstein 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.[31] Einstein used the failure of equipartition to argue for the need of a new quantum theory of matter.[12]Nernst's 1910 measurements of specific heats at low temperatures[32] supported Einstein's theory, and led to the widespread acceptance ofquantum theory among physicists.[33]
The most general form of the equipartition theorem states that under suitable assumptions (discussedbelow), for a physical system withHamiltonian energy functionH and degrees of freedomxn, the following equipartition formula holds in thermal equilibrium for all indicesm andn:[6][10][13]Hereδmn is theKronecker delta, which is equal to one ifm =n and is zero otherwise. The averaging brackets is assumed to be anensemble average over phase space or, under an assumption ofergodicity, a time average of a single system.
The general equipartition theorem holds in both themicrocanonical ensemble,[10] when the total energy of the system is constant, and also in thecanonical ensemble,[6][34] when the system is coupled to aheat bath with which it can exchange energy. Derivations of the general formula are givenlater in the article.
The general formula is equivalent to the following two:
If a degree of freedomxn appears only as a quadratic termanxn2 in the HamiltonianH, then the first of these formulae implies thatwhich 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 bys, applies to energies of the formanxns.
The degrees of freedomxn are coordinates on thephase space of the system and are therefore commonly subdivided intogeneralized position coordinatesqk andgeneralized momentum coordinatespk, wherepk is theconjugate momentum toqk. In this situation, formula 1 means that for allk,
Using the equations ofHamiltonian mechanics,[9] these formulae may also be written
Similarly, one can show using formula 2 thatand
The general equipartition theorem is an extension of thevirial theorem (proposed in 1870[35]), which states thatwheret denotestime.[9] Two key differences are that the virial theorem relatessummed rather thanindividual averages to each other, and it does not connect them to thetemperatureT. Another difference is that traditional derivations of the virial theorem use averages over time, whereas those of the equipartition theorem use averages overphase space.
Ideal gases provide an important application of the equipartition theorem. As well as providing the formulafor the average kinetic energy per particle, the equipartition theorem can be used to derive theideal gas law from classical mechanics.[6] Ifq = (qx,qy,qz) andp = (px,py,pz) denote the position vector and momentum of a particle in the gas, andF is the net force on that particle, thenwhere the first equality isNewton's second law, and the second line usesHamilton's equations and the equipartition formula. Summing over a system ofN particles yields
ByNewton'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 pressureP of the gas. HencewheredS is the infinitesimal area element along the walls of the container. Since thedivergence of the position vectorq isthedivergence theorem implies thatwheredV is an infinitesimal volume within the container andV is the total volume of the container.
Putting these equalities together yieldswhich immediately implies theideal gas law forN particles:wheren =N/NA is the number of moles of gas andR =NAkB is thegas 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 thepartition function.[36]
A diatomic gas can be modelled as two masses,m1 andm2, joined by aspring ofstiffnessa, which is called therigid rotor-harmonic oscillator approximation.[20] The classical energy of this system iswherep1 andp2 are the momenta of the two atoms, andq is the deviation of the inter-atomic separation from its equilibrium value. Every degree of freedom in the energy is quadratic and, thus, should contribute1⁄2kBT to the total average energy, and1⁄2kB to the heat capacity. Therefore, the heat capacity of a gas ofN diatomic molecules is predicted to be7N·1⁄2kB: the momentap1 andp2 contribute three degrees of freedom each, and the extensionq contributes the seventh. It follows that the heat capacity of a mole of diatomic molecules with no other degrees of freedom should be7/2NAkB =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 onlyincrease the predicted specific heat, not decrease it.[26] This discrepancy was a key piece of evidence showing the need for aquantum theory of matter.
Equipartition was used above to derive the classicalideal gas law fromNewtonian mechanics. However,relativistic effects become dominant in some systems, such aswhite dwarfs andneutron stars,[10] and the ideal gas equations must be modified. The equipartition theorem provides a convenient way to derive the corresponding laws for an extreme relativisticideal gas.[6] In such cases, the kinetic energy of asingle particle is given by the formulaTaking the derivative ofH with respect to thepx momentum component gives the formulaand similarly for thepy andpz components. Adding the three components together giveswhere 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: forN particles, it is3NkBT.
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 throughconservative forces whose potentialU(r) depends only on the distancer between the particles.[6] This situation can be described by first restricting attention to a single gas particle, and approximating the rest of the gas by aspherically symmetric distribution. It is then customary to introduce aradial distribution functiong(r) such that theprobability density of finding another particle at a distancer from the given particle is equal to4πr2ρg(r), whereρ =N/V is the meandensity of the gas.[37] It follows that the mean potential energy associated to the interaction of the given particle with the rest of the gas isThe total mean potential energy of the gas is therefore, whereN is the number of particles in the gas, and the factor1⁄2 is needed because summation over all the particles counts each interaction twice.Adding kinetic and potential energies, then applying equipartition, yields theenergy equationA similar argument,[6] can be used to derive thepressure equation
An anharmonic oscillator (in contrast to a simple harmonic oscillator) is one in which the potential energy is not quadratic in the extensionq (thegeneralized position which measures the deviation of the system from equilibrium). Such oscillators provide a complementary point of view on the equipartition theorem.[38][39] Simple examples are provided by potential energy functions of the formwhereC ands are arbitraryreal constants. In these cases, the law of equipartition predicts thatThus, the average potential energy equalskBT/s, notkBT/2 as for the quadratic harmonic oscillator (wheres = 2).
More generally, a typical energy function of a one-dimensional system has aTaylor expansion in the extensionq:for non-negativeintegersn. There is non = 1 term, because at theequilibrium point, there is no net force and so the first derivative of the energy is zero. Then = 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 formuladoesnot allow the average potential energy to be written in terms of known constants.
The equipartition theorem can be used to derive theBrownian motion of a particle from theLangevin equation.[6] According to that equation, the motion of a particle of massm with velocityv is governed byNewton's second lawwhereFrnd is a random force representing the random collisions of the particle and the surrounding molecules, and where thetime constant τ reflects thedrag force that opposes the particle's motion through the solution. The drag force is often writtenFdrag = −γv; therefore, the time constantτ equalsm/γ.
The dot product of this equation with the position vectorr, after averaging, yields the equationfor Brownian motion (since the random forceFrnd is uncorrelated with the positionr). Using the mathematical identitiesandthe basic equation for Brownian motion can be transformed intowhere the last equality follows from the equipartition theorem for translational kinetic energy:The abovedifferential equation for (with suitable initial conditions) may be solved exactly:On small time scales, witht ≪τ, the particle acts as a freely moving particle: by theTaylor series of theexponential function, the squared distance grows approximatelyquadratically:However, on long time scales, witht ≫τ, the exponential and constant terms are negligible, and the squared distance grows onlylinearly:This describes thediffusion of the particle over time. An analogous equation for the rotational diffusion of a rigid molecule can be derived in a similar way.
The equipartition theorem and the relatedvirial theorem have long been used as a tool inastrophysics.[40] As examples, the virial theorem may be used to estimate stellar temperatures or theChandrasekhar limit on the mass ofwhite 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 totalgravitationalpotential energy can be estimated by integrationwhereM(r) is the mass within a radiusr andρ(r) is the stellar density at radiusr;G represents thegravitational constant andR the total radius of the star. Assuming a constant density throughout the star, this integration yields the formulawhereM is the star's total mass. Hence, the average potential energy of a single particle iswhereN is the number of particles in the star. Since moststars are composed mainly ofionizedhydrogen,N equals roughlyM/mp, wheremp is the mass of one proton. Application of the equipartition theorem gives an estimate of the star's temperatureSubstitution of the mass and radius of theSun yields an estimated solar temperature ofT = 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]
The same formulae may be applied to determining the conditions forstar formation in giantmolecular 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, thevirial theorem—is no longer valid, i.e., when the gravitational potential energy exceeds twice the kinetic energyAssuming a constant densityρ for the cloudyields a minimum mass for stellar contraction, the Jeans massMJSubstituting 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 theJeans instability, after the British physicistJames Hopwood Jeans who published it in 1902.[46]
The original formulation of the equipartition theorem states that, in any physical system inthermal equilibrium, every particle has exactly the same average translationalkinetic energy,3/2kBT.[47] However, this is true only forideal gas, and the same result can be derived from theMaxwell–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 thez-component
Since different components of velocity are independent of each other, the average translational kinetic energy is given by
Notice, theMaxwell–Boltzmann distribution should not be confused with theBoltzmann 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]
More generally, the equipartition theorem states that anydegree of freedomx which appears in the total energyH only as a simple quadratic termAx2, whereA is a constant, has an average energy of1⁄2kBT in thermal equilibrium. In this case the equipartition theorem may be derived from thepartition functionZ(β), whereβ = 1/(kBT) is the canonicalinverse temperature.[48] Integration over the variablex yields a factorin the formula forZ. The mean energy associated with this factor is given byas stated by the equipartition theorem.
General derivations of the equipartition theorem can be found in manystatistical mechanics textbooks, both for themicrocanonical ensemble[6][10] and for thecanonical ensemble.[6][34]They involve taking averages over thephase space of the system, which is asymplectic manifold.
To explain these derivations, the following notation is introduced. First, the phase space is described in terms ofgeneralized position coordinatesqj together with theirconjugate momentapj. The quantitiesqj completely describe theconfiguration of the system, while the quantities(qj,pj) together completely describe itsstate.
Secondly, the infinitesimal volumeof the phase space is introduced and used to define the volumeΣ(E, ΔE) of the portion of phase space where the energyH of the system lies between two limits,E andE + 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 thanE:
SinceΔE is very small, the following integrations are equivalentwhere the ellipses represent the integrand. From this, it follows thatΣ is proportional toΔEwhereρ(E) is thedensity of states. By the usual definitions ofstatistical mechanics, theentropyS equalskB log Ω(E), and thetemperatureT is defined by
In thecanonical ensemble, the system is inthermal equilibrium with an infinite heat bath attemperatureT (in kelvins).[6][34] The probability of each state inphase space is given by itsBoltzmann factor times anormalization factor, which is chosen so that the probabilities sum to onewhereβ = 1/(kBT). UsingIntegration by parts for a phase-space variablexk the above can be written aswheredΓk =dΓ/dxk, i.e., the first integration is not carried out overxk. Performing the first integral between two limitsa andb and simplifying the second integral yields the equation
The first term is usually zero, either becausexk 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 theensemble average taken over thecanonical 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 energyH is confined betweenE andE+dE. For a given energyE and spreaddE, there is a region ofphase spaceΣ in which the system has that energy, and the probability of each state in that region ofphase space is equal, by the definition of the microcanonical ensemble. Given these definitions, the equipartition average of phase-space variablesxm (which could be eitherqk orpk) andxn is given by
where the last equality follows becauseE is a constant that does not depend onxn.Integrating by parts yields the relationsince the first term on the right hand side of the first line is zero (it can be rewritten as an integral ofH −E on thehypersurface whereH =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 theoremwhich was so useful in theapplications described above.
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The law of equipartition holds only forergodic systems inthermal equilibrium, which implies that all states with the same energy must be equally likely to be populated.[10] Consequently, it must be possible to exchange energy among all its various forms within the system, or with an externalheat bath in thecanonical ensemble. The number of physical systems that have been rigorously proven to be ergodic is small; a famous example is thehard-sphere system ofYakov Sinai.[49] The requirements for isolated systems to ensureergodicity—and, thus equipartition—have been studied, and provided motivation for the modernchaos theory ofdynamical systems. A chaoticHamiltonian system need not be ergodic, although that is usually a good assumption.[50]
A commonly cited counter-example where energy isnot shared among its various forms and where equipartition doesnot 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 eachnormal 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 theenergy function, energy may be transferred between the normal modes, leading to ergodicity and rendering the law of equipartition valid. However, theKolmogorov–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 nonlinearsoliton symmetries. In 1953,Fermi,Pasta,Ulam andTsingou conductedcomputer simulations of a vibrating string that included a non-linear term (quadratic in one test, cubic in another, and a piecewise linear approximation to a cubic in a third). They found that the behavior of the system was quite different from what intuition based on equipartition would have led them to expect. Instead of the energies in the modes becoming equally shared, the system exhibited a very complicated quasi-periodic behavior. This puzzling result was eventually explained by Kruskal and Zabusky in 1965 in a paper which, by connecting the simulated system to theKorteweg–de Vries equation led to the development of soliton mathematics.
The law of equipartition breaks down when the thermal energykBT 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 smoothcontinuum, which is required in thederivations of the equipartition theorem above.[6][10] Historically, the failures of the classical equipartition theorem to explainspecific heats andblack-body radiation were critical in showing the need for a new theory of matter and radiation, namely,quantum mechanics andquantum field theory.[12]
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 irrelevantzero-point energy term since it can be factored out of the exponential functions involved in the probability distribution, the quantum harmonic oscillator energy levels are given byEn =nhν, whereh is thePlanck constant,ν is thefundamental frequency of the oscillator, andn is an integer. The probability of a given energy level being populated in thecanonical ensemble is given by itsBoltzmann factorwhereβ = 1/kBT and the denominatorZ is thepartition function, here ageometric series
Its average energy is given by
Substituting the formula forZ gives the final result[10]
At high temperatures, when the thermal energykBT is much greater than the spacinghν between energy levels, the exponential argumentβhν is much less than one and the average energy becomeskBT, in agreement with the equipartition theorem (Figure 10). However, at low temperatures, whenhν ≫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 energykBT (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 byMax Planck andAlbert Einstein, among others, to resolve theultraviolet catastrophe ofblack-body radiation.[52] The paradox arises because there are an infinite number of independent modes of theelectromagnetic field in a closed container, each of which may be treated as a harmonic oscillator. If each electromagnetic mode were to have an average energykBT, 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 asν 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 asidentical particles andcontinuous symmetries. The effects of identical particles can be dominant at very high densities and low temperatures. For example, thevalence electrons in a metal can have a mean kinetic energy of a fewelectronvolts, which would normally correspond to a temperature of tens of thousands of kelvins. Such a state, in which the density is high enough that thePauli exclusion principle invalidates the classical approach, is called adegenerate fermion gas. Such gases are important for the structure ofwhite dwarf andneutron stars.[citation needed] At low temperatures, afermionic analogue of theBose–Einstein condensate (in which a large number of identical particles occupy the lowest-energy state) can form; suchsuperfluid electrons are responsible[dubious –discuss] forsuperconductivity.
{{cite journal}}:Missing or empty|title= (help)Waterston's key paper was written and submitted in 1845 to theRoyal Society. After refusing to publish his work, the Society also refused to return his manuscript and stored it among its files. The manuscript was discovered in 1891 byLord 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.{{cite book}}:ISBN / Date incompatibility (help) A lecture delivered by Prof. Maxwell at the Chemical Society on 18 February 1875.{{cite book}}:ISBN / Date incompatibility (help) Re-issued in 1987 by MIT Press asKelvin's Baltimore Lectures and Modern Theoretical Physics: Historical and Philosophical Perspectives (Robert Kargon and Peter Achinstein, editors).ISBN 978-0-262-11117-1{{cite book}}:ISBN / Date incompatibility (help){{cite book}}:ISBN / Date incompatibility (help){{cite book}}:ISBN / Date incompatibility (help){{cite book}}:ISBN / Date incompatibility (help)