trying to reword 3/2 example in lead |
more on quadratics, need to revamp |
||
Line 7: | Line 7: | ||
==Simple formulation== |
==Simple formulation== |
||
[[Image:MaxwellBoltzmann.gif|right|thumb|420px|Figure 1. Probability density functions of the molecular speed for a few [[noble gas]]es at a [[temperature]] of 298.15 [[Kelvin|K]] (25 [[Celsius|°C]]). |
[[Image:MaxwellBoltzmann.gif|right|thumb|420px|Figure 1. Probability density functions of the molecular speed for a few [[noble gas]]es at a [[temperature]] of 298.15 [[Kelvin|K]] (25 [[Celsius|°C]]). The Maxwell-Boltzmann distribution applies to all atomic systems, liquids and solids as well as gases; all atoms, regardless of their type, receive the same average kinetic energy at a given temperature, by the equipartition theorem. Therefore, heavier atoms such as [[xenon]] (Xe-132) have a lower average speed than do lighter atoms such as [[helium]] (He-4). This difference in average speed may be used to separate molecules according to their mass by [[Graham's law|effusion]].]] |
||
The simplest statement of the equipartition theorem is that every particle in a physical system has the same [[kinetic energy]] in [[thermal equilibrium]], namely, ''(3/2)k<sub>B</sub>T''. The simplest way to show this is through the [[Maxwell-Boltzmann distribution]] of [[speed]]s ''v'' |
The simplest statement of the equipartition theorem is that every particle in a physical system has the same [[kinetic energy]] in [[thermal equilibrium]], namely, ''(3/2)k<sub>B</sub>T''. The simplest way to show this is through the [[Maxwell-Boltzmann distribution]] of [[speed]]s ''v'' |
||
Line 19: | Line 19: | ||
</math> |
</math> |
||
where the speed is the magnitude of the [[velocity]] [[vector (spatial)|vector]] <math>v = \sqrt{v_x^2 + v_y^2 + v_z^2}</math>. Since ''f(v)'' is a [[probability density function]], it has units of probability per speed or, equivalently, reciprocal speed as shown in Figure 1. This distribution |
where the speed is the magnitude of the [[velocity]] [[vector (spatial)|vector]] <math>v = \sqrt{v_x^2 + v_y^2 + v_z^2}</math>. Since ''f(v)'' is a [[probability density function]], it has units of probability per speed or, equivalently, reciprocal speed as shown in Figure 1. This distribution pertains to any system composed of atoms, and assumes only a [[canonical ensemble]], i.e., that the kinetic energies are distributed according to their [[Boltzmann factor]] at a temperature ''T''. The average kinetic energy equals |
||
:<math> |
:<math> |
||
Line 33: | Line 33: | ||
</math> |
</math> |
||
where ''N<sub>A</sub>'' is the [[Avogadro constant]], ''R = N<sub>A</sub>k<sub>B</sub>'' is the [[gas constant]], and ''M = N<sub>A</sub>m'' is the mass of one [[mole]] of |
where ''N<sub>A</sub>'' is the [[Avogadro constant]], ''R = N<sub>A</sub>k<sub>B</sub>'' is the [[gas constant]], and ''M = N<sub>A</sub>m'' is the mass of one [[mole]] of molecules. This result is useful for many applications such as [[Graham's law]] of [[effusion]], which is used in [[enriched uranium|purifying radioactive isotopes]] of [[uranium]]. |
||
The kinetic energy ''H<sup>kin</sup>'' is quadratic in the particle momentum ''p'' |
|||
:<math> |
|||
H^{kin} = \frac{1}{2} m v^{2} = \frac{p^{2}}{2m} |
|||
</math> |
|||
and a similar equipartition result applies to all quadratic terms of the energy. For example, the rotational energy of a molecule is quadratic in the [[angular momentum]] ''L'' |
|||
:<math> |
|||
H^{rot} = \frac{1}{2} I \omega^{2} = \frac{L^{2}}{2I} |
|||
</math> |
|||
where ''I'' and ω are the [[moment of inertia]] and the [[angular velocity]], respectively. Given the three degrees of rotational freedom, the average rotational energy in thermal equilibrium is likewise ''(3/2)k<sub>B</sub>T''. A [[spring (device)|spring]] also has a quadratic energy |
|||
:<math> |
|||
H^{spring} = \frac{1}{2} k q^{2} |
|||
</math> |
|||
where ''q'' is the spring extension. There being but one degree of freedom, the spring will have an average potential energy of ''(1/2)k<sub>B</sub>T''. |
|||
However, a more general version of the equipartition theorem can be derived. |
|||
==General formulation== |
==General formulation== |
Revision as of 19:02, 17 April 2007
In classical statistical mechanics, the equipartition theorem is a general formula that allows average energies of many physical systems to be calculated as a function of temperature. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. In its simplest form, equipartition states that equivalent types of energy have the same average value in thermal equilibrium. For example, according to the equipartition theorem, every molecule in an ideal gas has an average kinetic energy of (3/2)kBT and, contributes (3/2)kB to the system's heat capacity, where kB is the Boltzmann constant and T is the temperature in Kelvin.
Similar to the virial theorem, equipartition gives the total average kinetic and potential energies for a given temperature, from which the specific heat of a system can be computed. However, equipartition goes further, giving the average values of individual types of energies. The equipartition theorem is also very general, pertaining to any classical system in thermal equilibrium, no matter how complicated. The equipartition theorem has broad applications, such as deriving the classical ideal gas law and the Dulong-Petit law of specific heats. The equipartition theorem holds even when relativistic effects are considered; for example, it may be used to derive the Chandrasekhar limit on the mass of a white dwarf star. However, the equipartition theorem holds only for ergodic systems in thermal equilibrium, which implies that all states with the same energy must be equally likely to be populated.
The equipartition theorem pertains to all classical systems, but is not strictly true when quantum physics is considered; equipartition is valid only at high temperatures, where the thermal energy kBT is much higher than the spacing between the quantum energy levels. At low temperatures, where the thermal energy is much lower than the energy spacing, the degrees of freedom are "frozen out"; their average energies and specific heats are much smaller than the value predicted by equipartition. For example, equipartition fails to accurately predict the specific heats of solids and diatomic gases; rather than being constant as predicted by equipartition, their specific heats decrease at low temperatures as various types of motion become "frozen out". This decrease was the first sign to physicists of the 19th century that classical physics was incorrect and that a new physics was needed. Equipartition's failure for electromagnetic radiation — the infamous ultraviolet catastrophe — led Albert Einstein to suggest that light itself was quantized into photons, a revolutionary hypothesis that spurred the development of quantum mechanics and quantum field theory. Einstein also developed a quantum mechanical theory to explain deviations in the specific heats of solids at low temperatures; as refined by Peter Debye, this theory hypothesizes that sound is quantized into phonons, in analogy with light.
Simple formulation
The simplest statement of the equipartition theorem is that every particle in a physical system has the same kinetic energy in thermal equilibrium, namely, (3/2)kBT. The simplest way to show this is through the Maxwell-Boltzmann distribution of speeds v
where the speed is the magnitude of the velocity vector . Since f(v) is a probability density function, it has units of probability per speed or, equivalently, reciprocal speed as shown in Figure 1. This distribution pertains to any system composed of atoms, and assumes only a canonical ensemble, i.e., that the kinetic energies are distributed according to their Boltzmann factor at a temperature T. The average kinetic energy equals
as stated by the equipartition theorem, where m is the mass of one molecule. The angular brackets symbolize taking the average of the enclosed quantity. Therefore, the root mean square speed vrms equals
where NA is the Avogadro constant, R = NAkB is the gas constant, and M = NAm is the mass of one mole of molecules. This result is useful for many applications such as Graham's law of effusion, which is used in purifying radioactive isotopes of uranium.
The kinetic energy Hkin is quadratic in the particle momentum p
and a similar equipartition result applies to all quadratic terms of the energy. For example, the rotational energy of a molecule is quadratic in the angular momentum L
where I and ω are the moment of inertia and the angular velocity, respectively. Given the three degrees of rotational freedom, the average rotational energy in thermal equilibrium is likewise (3/2)kBT. A spring also has a quadratic energy
where q is the spring extension. There being but one degree of freedom, the spring will have an average potential energy of (1/2)kBT. However, a more general version of the equipartition theorem can be derived.
General formulation
In its most general form, the equipartition theorem states that
where H is the Hamiltonian energy function of a physical system, qk and pk are its kth generalized coordinate and generalized momentum, respectively, and the latter two equations follow from Hamiltonian mechanics.[1] The averaging brackets may refer either to the ensemble average over phase space (the usual meaning) or the time average of a single system over a long time. The generalized equipartition theorem holds in both the microcanonical ensemble,[2] when the total energy of the system is constant, and also in the canonical ensemble,[3][4] when the system is coupled to a heat bath with which it can exchange energy.
In many cases, H varies quadratically in xk, e.g., as a kinetic energy term hk = pk2/2m or as a potential energy hk = C qk2. In such cases, the average energy in that degree of freedom is ½kBT. Expressed another way, each quadratic degree of freedom contributes ½kB to the system's heat capacity. The motion of the particle has three degrees of freedom, namely, the x, y and z components of the velocity; therefore, the mean kinetic energy of a particle is 3 × ½kBT, as derived in the previous section. If the molecule is not a point particle and can rotate, its average rotational energy will likewise be (3/2)kBT, corresponding to the three degrees of freedom of rotation.
The equipartition theorem also states that the corresponding averages for different variables are always zero
when m≠n, as is the average for any generalized coordinate qk and its conjugate momentum pk
These various results can be summarized in the formula
where δmn is the Kronecker delta and xm and xn are any two variables in phase space, either generalized coordinates or generalized momenta. Derivations of this general formula are given at the end of the article; in the meantime, the formula is compared with other results and applied to various physical systems.
Requirement of ergodicity
The equipartition holds only for ergodic systems in thermal equilibrium. This implies that all states with the same energy must be equally likely to be populated. Consequently, it must be possible to exchange energy among its various forms in the system, or with an external heat bath in the canonical ensemble.
A commonly cited counter-example is that of a harmonic system with multiple normal modes. If the oscillator is isolated from the rest of the world, the energy in each mode is fixed and cannot be exchanged; hence, the equipartition theorem does not hold for such a system. These energies usually begin exchanging if sufficiently strong nonlinear terms are present in the energy function. However, the Kolmogorov–Arnold–Moser theorem states that energy will not be exchanged if the nonlinear perturbations are too small. Studies of the requirements for isolated systems to ensure ergodicity — and, thus, that the equipartition theorem holds — led to the modern chaos theory.
Relation to the virial theorem
The equipartition theorem is an extension of the virial theorem (proposed in 1870[5]), which relates the averages of the sums
where t represent time and the summation is over all the degrees of freedom k.[1] Two key differences are that the virial theorem relates the summed averages to each other, rather than individual averages, and also 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. However, these two types of averages should yield the same result, assuming ergodicity, and both have been used to estimate the total internal energy of complex physical systems.
History
The equipartition of kinetic energy was proposed initially in 1843, and more correctly in 1845, by John James Waterston.[6] A few years later, James Clerk Maxwell argued that the kinetic heat energy of a gas is equally divided between linear and rotational energy.[7] Ludwig Boltzmann expanded on this principle by showing that the average energy was divided equally among all the independent components of motion in the system (1876).[8][9] Boltzmann applied the equipartition theorem to providing a theoretical explanation of the Dulong-Petit law for the specific heats of solids.
The history of the equipartition theorem is intertwined with that of specific heats, 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 heats of solids at room temperature were almost all identical, roughly 6 cal/(mole·K).[10] Their law was used for many years as a technique of measuring atomic weights.[11] However, subsequent studies by James Dewar and Heinrich Friedrich Weber shows that this Dulong-Petit law holds only at high temperatures;[12] at lower temperatures, or for exceptionally stiff solids such as diamond, the specific heat was lower.[13]
Experimental observations of the specific heat of gases also raised concerns about the validity of the equipartition theorem. As described below, the theorem predicts that the specific heat of simple monatomic gases should be roughly 3 cal/(mole·K), whereas that of diatomic gases should be roughly 7 cal/(mole·K). Experiments confirmed the former prediction,[14] but found that the latter was instead 5 cal/(mole·K),[15] which falls to 3 cal/(mole·K) at very low temperatures.[16] More generally, molecules were believed to be composed of parts (atoms) already in the 19th century, and should have much higher specific heats than observed, as noted first by Maxwell in 1875.[17]
The failure of the equipartition theorem to account for the specific heats of solids and gases was addressed in several ways. Boltzmann defended the derivation of his equipartition theorem as correct, but suggested that gases might not be in thermal equilibrium because of their interactions with the aether.[18] Lord Kelvin suggested that the derivation of the equipartition theorem must be incorrect, since it disagreed with experiment, but was unable to show how.[19] Finally, Lord Rayleigh defended both the derivation and the experimental assumption of thermal equilibrium, and noted the need for a new principle that would provide an "escape from the destructive simplicity" of the equipartition theorem.[20] Albert Einstein provided that escape, by showing in 1907 that these anomalies in the specific heat were due to quantum effects, specifically the quantization of energy in the elastic modes of the solid.[21] Einstein used the failure of the equipartition theorem to argue for the need of a new quantum theory of matter, years ahead of other physicists.[11] A refinement of Einstein's theory by Peter Debye[22] is still the principal theory of specific heats in solids at all temperatures, and led to the prediction of phonons.
Basic applications
Simple harmonic oscillators
The simplest application of the equipartition theorem is to a simple harmonic oscillator.[2] A typical oscillator is a particle of mass m attached to a spring of stiffness k, for which the energy is the sum of the kinetic and potential energies
where q is the extension of the spring from equilibrium, and p is the momentum of the particle.[1] The energy function H is quadratic in both q and p, indicating that they each contribute ½kBT to the total average energy. Since
and
the total average energy in this system is kBT
This result is valid for any type of harmonic oscillator, such as a pendulum, a vibrating molecule or a passive electronic oscillator; for instance, the equipartition theorem can be used to derive the formula for Johnson-Nyquist noise.
The equipartition theorem also holds for sets of independent oscillators, such as the normal modes of coupled oscillators, e.g., the modes of a piano string, the resonances of an organ pipe, or the lattice vibrations in a solid.[4] If there are N independent oscillators, then their average total energy is NkBT, and their specific heat at constant volume is NkB. In particular, a mole of such oscillators would have a specific heat of NAkB=R where NA is Avogadro's number and R is the universal gas constant, roughly 2 cal/(mole·K). This is the explanation for the Dulong-Petit law of molar specific heats of solids; each mole of atoms in the lattice contributes 3R≈6 cal/(mole·K) to the specific heat, since each atom can oscillate in three independent directions.
Non-relativistic ideal gases
Ideal gases provide another good illustration of the equipartition theorem.[4] The classical kinetic energy of a single particle of mass m is given by
where (px, py, pz) are the Cartesian components of the momentum p of the particle. For the px component, the equipartition theorem equals
and similarly for the py and pz components. Adding them together and dividing by two gives the average kinetic energy of a particle in three dimensions
In an ideal gas, there is no potential energy; by assumption, the particles have no internal degrees of freedom and move independently of one another. Therefore, the total energy consists only of their kinetic energies; quantitatively, the average total energy U of an ideal gas of N particles is 3/2 N kBT and its specific heat is 3/2 N kB. Thus, a mole of a monatomic gas should have a specific heat at constant volume of (3/2)NAkB=(3/2)R where again NA is Avogadro's number and R is the universal gas constant, roughly 2 cal/(mole·K). Therefore, the predicted molar specific heat should be roughly 3 cal/(mole·K), which was confirmed by experiment.[14] From the mean kinetic energy, the root mean square speed of the molecules can be calculated.
The ideal gas law
The ideal gas law can be derived from the equipartition theorem. The equation
and its counterparts for the qy and qz components lead to the formula
where q is the position vector of the particle and F is the net force on that particle. Summing over the N particles yields
Since the particles don't interact, the only force on them is the inwards pressure P applied by the walls of their container. Therefore, quantity to be averaged is
where dS is an infinitesimal area on the walls of the container. Using the divergence theorem, this integral can be shown to be constant
where dV is an infinitesimal volume within the container and V is the total volume of the container. This follows because the divergence of the position vector q in three dimensions is three
Putting this together
yields the ideal gas law for N particles
Diatomic gases
A diatomic gas can be modelled as two masses, m1 and m2, joined by a spring of stiffness k. The classical energy of this system is
where q is the deviation of the inter-atomic separation from its equilibrium value. Each degree of freedom in the Hamiltonian energy function is quadratic and, thus, should contribute ½kBT to the total average energy, and ½kB. Therefore, the specific heat of a gas of N diatomic molecules is predicted to be 7N · ½kB; for each molecule, p1 and p2 each contribute three degrees of freedom, whereas q contributes the seventh. Therefore, the specific heat of a mole of diatomic molecules with no other degrees of freedom should be (7/2)NAkB=(7/2)R where again NA is Avogadro's number and R is the universal gas constant, roughly 2 cal/(mole·K). Therefore, the predicted molar specific heat should be roughly 7 cal/(mole·K); however, the measured value is roughly 5 cal/(mole·K)[15] and falls to 3 cal/(mole·K) at very low temperatures.[16] This disagreement cannot be explained by using a more complex model of the molecule, since adding more degrees of freedom can only increase the classical specific heat, not decrease it. This discrepancy was a key piece of evidence showing the need for a quantum theory of matter.
Failures due to quantum effects
The equipartition theorem breaks down when the thermal energy kBT is significantly smaller than the spacing between energy levels. The theorem 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 below. 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 and quantum field theory.[11] Other, more subtle quantum effects can lead to corrections to equipartition, such as identical particles and continuous symmetries.
To illustrate this breakdown, consider the average energy in a single (quantum) harmonic oscillator. Its energy levels are given by En = nħω, where ħ = h/2π is Planck's constant , ω is the fundamental frequency of the oscillator, and n is an integer. The probability of a given energy level being populated in the canonical ensemble is given by its Boltzmann factor
where β=1/kBT and the denominator Z is the partition function, here a geometric series
Its average energy is given by
Substitution of the formula for Z gives the final result[2]
At high temperatures, when the thermal energy kBT is much greater than the spacing ħω between energy levels, the exponential argument βħω is much less than one and the average energy becomes kBT, in agreement with the equipartition theorem. However, at low temperatures, when βħω>>1, the average energy goes to zero — the higher excited energy levels are "frozen out". For 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. For example, this reasoning was used by Albert Einstein to resolve the ultraviolet catastrophe of blackbody radiation.[23] 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.[23][24] However, by the reasoning above, the average energy in the higher-ω modes goes to zero as ω goes to infinity; moreover, Planck's law of black body radiation, which describes the experimental distribution of energy in the modes, follows from the same reasoning.[23]
Symmetry in a quantum system introduces another, more subtle correction to the equipartition theorem. Excited rotational states are impossible for systems with a continuous symmetry, such as the rotation of a diatomic gas about the axis connecting the centers of the atoms. Since such states do not exist, they are effectively frozen out; hence, the system has fewer effective degrees of freedom. Thus, the diatomic molecule has a molar specific heat of (5/2)R, instead of (7/2)R; the vibrational degree of freedom is frozen out at room temperature, and one rotational degree of freedom is frozen out because of symmetry. At even lower temperatures, the two remaining rotational modes are frozen out, giving a molar specific heat of (3/2)R, corresponding to the three degrees of translational freedom. Other symmetries, such as those involving identical particles, can also produce deviations from the specific heats predicted by the equipartition theorem.
More advanced applications
Anharmonic oscillators
The anharmonic oscillator — one in which the potential energy is not quadratic in the extension q — provides a complementary view of the equipartition theorem. For simplicity, let the energy function have the form
where C and m are arbitrary constants, the equipartition theorem predicts that
Thus, the average potential energy equals kBT/m, not kBT/2 as for the quadratic harmonic oscillator.
More generally, if the energy function of a one-dimensional system is expanded in a Taylor series in its generalized coordinate q
the equipartition theorem predicts that
Contrary to the other examples cited here, this latter equipartition result does not allow the average potential energy to be written in simple form
Extreme relativistic ideal gases
An instructive counterpoint to the ideal gas law is an extreme relativistic ideal gas,[4] such as occur in white dwarf and neutron stars.[2] In such cases, the kinetic energy of a single particle is given by a different formula
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
Thus, the average energy in the extreme relativistic case is twice that of the non-relativistic case; the average total energy U of an extreme relativistic ideal gas of N particles is 3 N kBT.
Non-ideal gases
The equipartition theorem may also be used to derive the energy and pressure of non-ideal gases,[4] which are gases in which the particles interact with one another. As a simple illustration, the gas particles may be assumed to be spherically symmetric (isotropic) and interacting only by a pair potential V(r), where r is the distance between the two particles. It is customary to introduce a radial distribution function g(r) such that the probability density of finding another particle at a distance r is given by 4πr2ρ g(r), where ρ=N/V is the mean density of the gas.[25] In this case, the mean potential energy for one particle equals the integral
where the factor of two is introduced to avoid double counting the interactions; in other words, each particle gets half of its interaction with another particle. Therefore, the total energy for N particles equals
By an analogous argument, it is possible to derive the non-ideal pressure equation
Molecular tumbling in solution
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 couplings. Rotational diffusion can also be observed by other biophysical probes such as fluorescence anisotropy, flow birefringence and dielectric spectroscopy.
The equipartition theorem allows the mean angular speeds of the molecule to be calculated. The rotational energy of the molecule is given by
where I1-3 and ω1-3 are the moments of inertia and angular velocities about the molecule's three principal axes, respectively. This energy can also be written in terms of the corresponding angular momenta L1-3
where Lk = Ikωk. Application of the equipartition theorem yields the relation
and similarly for L2 and L3. Thus, each rotational degree of freedom possesses a mean rotational kinetic energy of ½ kBT and contributes ½ kB to the specific heat, from which the mean rotational speeds about each principal axis can be calculated. As another consequence, the mean rotational energy of the molecule equals the mean translational energy of the molecule
Brownian motion
The equipartition theorem can be used to derive the Brownian motion of a particle from the Langevin equation. According to that equation, the motion of a particle of mass m 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.
Taking the dot product of this equation with the position vector r and averaging yields the basic 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
This equation may be solved exactly
For small times, t << τ, the particle acts as a freely moving particle; the squared distance grows quadratically
However, for long times, t >> τ, the squared distance grows only linearly with the time
which 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 has long been used as a tool in astrophysics. It may be used to estimate stellar temperatures or the Chandrasekhar limit on the mass of white dwarf stars. It is also sometimes used in modeling complex stellar events, such as the collapse of a star into a black hole.
The average temperature of a star can be estimated from the equipartition theorem. Assuming spherical symmetry, the total gravitational potential energy can be estimated by integration over the radius
where M(r) is the mass within a radius r and ρ is the stellar density; G represents the gravitational constant. Assuming for simplicity a constant density throughout the star, this integration yields the formula
where M and R are the star's total mass and radius. Hence, the star's temperature is given by the equipartition theorem
where N is the number of particles in the star. This number equals roughly (M/mp) — mp is the mass of one proton — since most stars are composed mainly of hydrogen. Substitution of the mass and radius of the Sun yields an estimated solar temperature of T = 14 million Kelvin, very close to the actual temperature of 15 million Kelvin.
Derivations
Derivations of the equipartition theorem can be found in many statistical mechanics textbooks, both for the microcanonical ensemble[2][4] and for the canonical ensemble.[3][4] To help explain those derivations, the following notation is introduced.
Derivations of the equipartition theorem involve averages over phase space, which is the set of generalized coordinates q and their conjugate momenta p needed to completely specify the state of the system. The symbol dΓ represents an infinitesimal volume of phase space
We use the symbol Γ to represent the volume of the phase space where the energy H lies between two limits, E and E+ΔE
It is generally assumed that ΔE is very small, ΔE<<E. Similarly, Σ represents 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 Γ 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 Σ(E), and the temperature T is defined by
Derivation for the canonical ensemble
In the canonical ensemble, the system is in thermal equilibrium with an infinite heat bath at temperature T (in Kelvin). The probability of each state in phase space is given by its Boltzmann factor times a normalization factor , which is chosen so that the probabilities sum to one
where β = 1/kBT. Integration by parts for a phase-space variable xk (which could be either qk or pk) between two limits a and b yields the equation
where dΓk=dΓ/dxk, i.e., the first integration is not carried out over xk. 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 ensemble average taken over the canonical ensemble.
Derivation for 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 H is confined between E and E+ΔE. For a given energy E and spread ΔE, there is a region of phase space Γ 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 qkor 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
The first term on the right-hand side is zero, since it can be re-cast an integral of (H - E) on the hypersurface where H = E.
Substitution of this result in the above equation
yields the equipartition theorem
See also
- Degrees of freedom (physics and chemistry)
- Virial theorem
- Kinetic theory
- Heat capacity
- Statistical mechanics
- Quantum statistical mechanics
- Ultraviolet catastrophe
References
- ^ a b c Goldstein, H (1980). Classical Mechanics (2nd. ed ed.). Addison-Wesley. ISBN 0-201-02918-9.
{{cite book}}
:|edition=
has extra text (help) - ^ a b c d e Huang, K (1987). Statistical Mechanics (2nd ed. ed.). John Wiley and Sons. pp. pp. 136–138. ISBN 0-471-81518-7.
{{cite book}}
:|edition=
has extra text (help);|pages=
has extra text (help) - ^ a b Tolman, RC (1938). The Principles of Statistical Mechanics. New York: Dover Publications. pp. pp. 93–98. ISBN 0-486-63896-0.
{{cite book}}
:|pages=
has extra text (help) - ^ a b c d e f g Pathria, RK (1972). Statistical Mechanics. Pergamon Press. pp. pp. 43–48, 73–74. ISBN 0-08-016747-0.
{{cite book}}
:|pages=
has extra text (help) - ^ Clausius, R (1870). "Ueber einen auf die Wärme anwendbaren mechanischen Satz". Annalen der Physik. 141: 124–130.
Clausius, RJE (1870). "On a Mechanical Theorem Applicable to Heat". Philosophical Magazine, Ser. 4. 40: 122–127. - ^ Brush, SG (1976). The Kind of Motion We Call Heat, Volume 1. Amsterdam: North Holland. pp. 134–159. ISBN 978-0444870094.
Brush, SG (1976). The Kind of Motion We Call Heat, Volume 2. Amsterdam: North Holland. pp. 336–339. ISBN 978-0444870094.
Waterston, JJ (1851). "On the physics of media that are composed of free and elastic molecules in a state of motion". British Association Reports. 21: 6. (abstract). Not published in full until Philos. Trans. R. Soc. London. A183: 1–79. 1893.{{cite journal}}
: Missing or empty|title=
(help) Reprinted J.S. Haldane, ed. (1928). The collected scientific papers of John James Waterston. Edinburgh: Oliver & Boyd. - ^ Maxwell, JC. "Illustrations of the Dynamical Theory of Gases". In WD Niven (ed.). The Scientific Papers of James Clerk Maxwell. New York: Dover. pp. Vol.1, pp. 377–409. ISBN 978-0486495606. 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". Wiener Berichte. 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". Wiener Berichte. 74: 553–560.
- ^ Petit, AT (1819). "Recherches sur quelques points importants de la théorie de la chaleur". Annales de Chimie et de Physique. 10: 395–413.
{{cite journal}}
: Unknown parameter|coauthors=
ignored (|author=
suggested) (help) - ^ a b c Pais, A (1982). Subtle is the Lord. Oxford University Press. ISBN 0-19-853907-X.
- ^ Dewar, J (1872). "The Specific Heat of Carbon at High Temperatures". Philosophical Magazine. 44: 461–.
Weber, HF (1872). "Die specifische Wärme des Kohlenstoffs". Annalen der Physik. 147: 311–319.
Weber, HF (1875). "Die specifische Wärmen der Elemente Kohlenstoff, Bor und Silicium". Annalen der Physik. 154: 367–423, 553–582. - ^ de la Rive, A (1840). "Unknown". Annales de Chimie et de Physique. 75: 113–.
{{cite journal}}
: Unknown parameter|coauthors=
ignored (|author=
suggested) (help)
Regnault, HV (1841). "Recherches sur la chaleur spécifique des corps simples et des corps composés (deuxième Mémoire)". Annales de Chimie et de Physique. 1 (3me Série): 129–207. Read at l'Académie des Sciences on 11 January 1841.
Wigand, A (1907). "Unknown". Annalen der Physik. 22: 99–. - ^ a b Kundt, A (1876). "Ueber die specifische Wärme des Quecksilbergases". Annalen der Physik. 157: 353–369.
{{cite journal}}
: Unknown parameter|coauthors=
ignored (|author=
suggested) (help) - ^ a b Wüller, A (1896). Lehrbuch der Experimentalphysik. Leipzig: Teubner. pp. Vol. 2, 507ff.
- ^ a b Eucken, A (1912). "The molecular heat of hydrogen at low temperatures (in German)". Sitzungsberichte der königlichen Preussischen Akademie der Wissenschaften. 1912: 141–151.
- ^ Maxwell, JC. "On the Dynamical Evidence of the Molecular Constitution of Bodies". In WD Niven (ed.). The Scientific Papers of James Clerk Maxwell. New York: Dover. pp. Vol.2, pp.418–438. ISBN 978-0486495606. A lecture delivered by Prof. Maxwell at the Chemical Society on 18 February 1875.
- ^ Boltzmann, L (1895). "On certain Questions of the Theory of Gases". Nature. 51: 413–415.
- ^ Thomson, W (1904). Baltimore Lectures. Baltimore: Johns Hopkins University Press. pp. Sec. 27. Re-issued in 1987 by MIT Press as Kelvin's Baltimore Lectures and Modern Theoretical Physics: Historical and Philosophical Perspectives (Robert Kargon and Peter Achinstein, editors). Template:ISBN-13
- ^ Rayleigh, JWS (1900). "The Law of Partition of Kinetic Energy". Philosophical Magazine. 49: 98–118.
- ^ Einstein, A (1907). "Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme". Annalen der Physik. 22: 180–190.
Einstein, A (1907). "Berichtigung zu meiner Arbeit: 'Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme'". Annalen der Physik. 22: 800.
Einstein, A (1911). "Eine Beziehung zwischen dem elastischen Verhalten and der spezifischen Wärme bei festen Körpern mit einatomigem Molekül". Annalen der Physik. 34: 170–174.
Einstein, A (1911). "Bemerkung zu meiner Arbeit: 'Eine Beziehung zwischen dem elastischen Verhalten and der spezifischen Wärme bei festen Körpern mit einatomigem Molekül'". Annalen der Physik. 34: 590.
Einstein, A (1911). "Elementare Betrachtungen über die thermische Molekularbewegung in festen Körpern". Annalen der Physik. 35: 679–694. - ^ Debye, P (1912). "Zur Theorie der spezifischen Wärmen". Annalen der Physik. 39: 789–839.
- ^ a b c Einstein, A (1905). "Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt". Annalen der Physik. 17: 132–148. Template:De icon. An English translation is available from Wikisource.
- ^ Rayleigh, JWS (1900). "Remarks upon the Law of Complete Radiation". Philosophical Magazine. 49: 539–540.
- ^ McQuarrie, DA (2000). Statistical Mechanics (revised 2nd ed. ed.). University Science Books. ISBN 978-1891389153.
{{cite book}}
:|edition=
has extra text (help)
Further reading
- Pathria, RK (1972). Statistical Mechanics. Pergamon Press. pp. pp. 43–48, 73–74. ISBN 0-08-016747-0.
{{cite book}}
:|pages=
has extra text (help)
- Huang, K (1987). Statistical Mechanics (2nd ed. ed.). John Wiley and Sons. pp. pp. 136–138. ISBN 0-471-81518-7.
{{cite book}}
:|edition=
has extra text (help);|pages=
has extra text (help)
- Pauli, W (1973). Pauli Lectures on Physics: Volume 4. Statistical Mechanics. MIT Press. pp. pp. 27–40. ISBN 0-262-16049-8.
{{cite book}}
:|pages=
has extra text (help)
- Khinchin, AI (1949). Mathematical Foundations of Statistical Mechanics (G. Gamow, translator). New York: Dover Publications. pp. pp. 93–98. ISBN 0-486-63896-0.
{{cite book}}
:|pages=
has extra text (help)
- Mohling, F (1982). Statistical Mechanics: Methods and Applications. John Wiley and Sons. pp. pp. 137–139, 270–273, 280, 285–292. ISBN 0-470-27340-2.
{{cite book}}
:|pages=
has extra text (help)
- Landau, LD (1980). Statistical Physics, Part 1 (3rd ed. ed.). Pergamon Press. pp. pp. 129–132. ISBN 0-08-023039-3.
{{cite book}}
:|edition=
has extra text (help);|pages=
has extra text (help); Unknown parameter|coauthors=
ignored (|author=
suggested) (help)
- Mandl, F (1971). Statistical Physics. John Wiley and Sons. pp. pp. 213–219. ISBN 0-471-56658-6.
{{cite book}}
:|pages=
has extra text (help)
- Tolman, RC (1938). The Principles of Statistical Mechanics. New York: Dover Publications. pp. pp. 93–98. ISBN 0-486-63896-0.
{{cite book}}
:|pages=
has extra text (help)
- Tolman, RC (1927). Statistical Mechanics, with Applications to Physics and Chemistry. Chemical Catalog Company. pp. pp. 72–81.
{{cite book}}
:|pages=
has extra text (help)
External links
- 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.