Configuration integral (statistical mechanics)
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The classical configuration integral, sometimes referred to as the configurational partition function^{[1]}, for a system with particles is defined as follows:
(1)
where is the volume enclosing the particles, a constant defined as
(2)
with being the Boltzmann constant, the thermodynamic temperature^{[2]} the potential energy of interparticle forces, the positions in the 3D space of the particles, with and the coordinate of the particle, and an infinitesimal volume. An example for the potential energy is the LennardJones potential.
By setting , we have . Since both and have the dimension of energy, the integrand in Eq.(1) is dimensionless, and thus the configuration integral has the dimension of . For this reason, some authors use the nondimensionalized configuration integral obtained by dividing Eq.(1) by ; see also Allen & Tildesley (1989)^{[3]}, p.41; McComb (2004)^{[4]}, p.95.
We begin to motivate by providing important applications of the configuration integral, then proceed to give a detailed derivation of Eq.(1) in a selfcontained manner that does not require too many prerequisites^{[5]}.


Motivation
Disease research and drug design
By way of motivation for learning about the configuration integral, we consider an important application of the configuration integral in the development of computational models for the ligandreceptor binding affinities, the study of which constitutes a most important problem in computational biochemistry; Swanson et al. (2004)^{[6]}, see also Receptor (biochemistry). Research in the prediction of binding affinities has been a continuing effort for more than half a century.
The Human Immunodeficiency Virus (HIV) that could induce AIDS (Acquired Immune Deficiency Syndrome) has wreaked havoc in several human communities in the world. An HIV virus, such as HIV1, destroys a human cell by first entering the cell through the cell membrane. To this end, the HIV1 virus would have its gp120 envelope glycoprotein bind first to the CR4 glycoprotein receptor in the cell membrane, then second to a chemokine receptor family (CXCR4 or CCR5) to initiate its entry into the cell; see the highly informative articles HIV, HIV structure and genome, and the references ^{[7]} ^{[8]}. A research program has been underway at NIH to develop HIV vaccine (particularly the socalled gp120 vaccines) by trying to understand, through atomistic simulations, a mechanism of how HIV virus evade antibody proteins that would block its binding to the chemokine receptor, thus preventing it from entering the cell^{[9]}. Fig.1 shows an atomistic model of a ligandreceptorligand binding involving the HIV1 virus gp120 envelope glycoprotein (ligand), the CR4 glycoprotein (receptor), and the b12 antibody protein (ligand).
In a ligandreceptor binding, a ligand is in general any molecule that binds to another molecule; the receiving molecule is called a receptor, which is a protein on the cell membrane or within the cell cytoplasm. Such binding can be represented by the chemical reaction describing noncovalent molecular association:
(3)
where represents the protein (receptor), B the ligand molecule, and the bound ligandreceptor.
A goal is to compute the change in the Gibbs energy^{[10]} for the above reaction^{[11]}, which is given in terms of the configuration integrals as follows ^{[12]}
(4)
where the quantities are the configuration integrals. For example, the configuration integral for protein is
(5)
The details on the other quantities are irrelevant for the present article, whose aim is to explain the origin of the configuration integral in statistical mechanics. Readers interested in understanding Eq.(4) are referred to Gilson et al. (1997)^{[12]} for a detailed derivation. A recent review of the ligandreceptor affinity calculation is given by Gilson & Zhou (2007)^{[13]}.
Classical partition function (sumoverstates)
In classical (no quantum effect) statistical mechanics, the configuration integral and the partition function are fundamental in the study of monoatomic, imperfect, classical gases and liquids. Once these functions are known, the thermodynamic properties can be calculated. For such a system with identical particles, the classical partition function^{[14]} is obtained by multiplying the configuration integral with a "momentum integral", i.e., an integral over the momentum space, whereas the configuration integral is an integral over the configuration space, with the product of the configuration space by the momentum space being the phase space^{[15]}. In other words, in parallel with the Hamiltonian being the sum of the kinetic energy and the potential energy, the partition function can be decomposed into a product of the "momentum integral" (related to the kinetic energy) and the configuration integral (related to the potential energy). On the other hand, unlike the configuration integral, which is in general difficult to integrate (because of the complex nature of the potential energy), this "momentum integral" can be easily integrated exactly (because of the simple nature of the kinetic energy) to yield a function of the number of particles and the thermodynamic temperature^{[2]} :
(6)
where the coefficient is defined as
Case Distinguishable particles Indistinguishable particles
(7)
and the thermal de Broglie wavelength defined as
(8)
where is the Planck constant. See further reading on .
It is sometimes mistaken to think of the configuration integral as the same as the partition function, modulo a multiplicative "constant". First, as seen in Eq.(6), the multiplicative factor of to obtain is not a constant, but a function of and , and is the result of integrating exactly the "momentum integral". Second, there are applications in which the configuration integral plays an important role, with no direct role for the "momentum integral", and therefore the partition function, such as the example of assessing the ligandreceptor binding affinity mentioned above.
The abstract terminology "partition function" is also known more concretely and unassumingly as the "sumoverstates", which conveys a clear and direct meaning of this function, as shown in Eq.(33); see, e.g., Kirkwood (1933)^{[16]}. Even though the name "partition function" (as used in statistical mechanics) is likely to first appear in 1922^{[17]}, in his classic book, Tolman (1938, 1979)^{[18]}, p.532, chose to use the name "sumoverstates", in agreement with what Planck (1932) used in German as "Zustandsumme" ^{[19]} ^{[20]} ^{[21]}. Other authors used the compromising, hybrid name "partition sum", e.g., Callen (1985), p.351^{[22]}. Khinchin (1949)^{[23]}, p.76, used the name "generating function" for the partition function^{[24]}.
For the particular case where there is no potential, i.e., , such as in the case of independent particles, the partition function takes a simple form.
As mentioned in the introduction, the configuration integral can be nondimensionalized by dividing by , with the nondimensionalized version denoted by . Then, the partition function can be written as the product of the partition function of ideal gas (i.e., when the potential is zero) by the nondimensionalized configuration integral . Thus, the configuration integral can be thought of as a correction factor for to obtain the partition function for the case where the potential is nonzero.
The power and elegance of statistical mechanics reside in its application to predict accurately the thermodynamics properties compared to experiments. The knowledge of the partition function (and the corresponding configuration integral) of a system is important since it allows for the calculation of thermodynamic properties. For instance, a most important relation—sometimes referred to as a fundamental relation; see Callen (1985), p.352^{[22]}—connecting statistical mechanics to thermodynamics is the relationship between the Helmholtz energy^{[10]} and the partition function ; McQuarrie (2000)^{[25]}, p.45:
(9)
with as defined in Eq.(2). Some authors such as Callen (1985) use the notation , instead of the more historical and customary notation , for the Helmholtz (free) energy^{[10]}^{[26]}. Eq.(9)_{2} plays an important role in the Jarzynski equality, also known as a nonequilibrium work relation; see the seminal paper by Jarzynski (1997)^{[27]}, where the notation was used for the Helmholtz energy^{[10]}^{[26]}.
Physics of fluid turbulence
The configuration integral in particular, and statistical mechanics in general, have been used in the modelling of fluid turbulence; see, e.g., McComb (1992)^{[28]}, p.193.
Calculating the configuration integral
The dependence of the interparticle forces on the distance between the particles makes the evaluation of the configuration integral in Eq.(1) "extremely difficult"; such evaluation has been driving much of the research in classical statistical mechanics; McQuarrie (2000)^{[25]}, p.116.
There are two methods to calculate a configuration integral: (i) approximate methods, and (ii) direct numerical integration.
For dilute gases in which the potential is of the usual type, an expansion of the integrand of Eq.(1) in the powers of provides a systematic method to approximately calculate the configuration integral . This method is known as cluster expansion, which Maria GöppertMayer contributed to develop; See Assael et al. (1996)^{[29]}, p.49; Huang (1987), p.213^{[30]}.
The direct numerical evaluation of the configuration integral is discussed in Tafipolsky et al. (2005)^{[31]}.
As mentioned, we will build up below the fundamental concepts that lead to the expression for the classical partition function in Eq.(6) and the classical configuration integral in Eq.(1), so that all terms in these expressions are explained, without requiring too many prerequisites.
Systems with independent particles
Equilibrium, independent particles
According to the orthodox thermodynamic theory, a thermodynamic system is in equilibrium if its thermodynamic state, which is a set of values for its thermodynamic parameters (i.e., macroscopic parameters such as pressure , volume , temperature , magnetic field , etc.), remains constant in time; see, e.g., Sklar (1993)^{[32]}, p.22, Huang (1987)^{[30]}, p.3. A quantitative condition of equilibrium can be described as the partial time derivative of the distribution density being zero, i.e., the distribution density is timeindependent at any fixed point in the phase space; this condition is related to the Liouville theorem; see, e.g., Tolman (1979), p.55.
A more advanced and abstract concept of equilibrium came from the development of the kinetic theory, the criticisms of this theory, and the response of the proponents of the kinetic theory to these criticisms. In this concept, equilibrium does not characterize any single macroscopic state of the system, but rather a class of macrostates with each macrostate having its own probability; this approach is known as a reduction of thermodynamics to statistical mechanics; Sklar (1993)^{[32]}, p.23. This theory in turn had been a subject of criticism as to how the probabilistic assumptions, thought to be derived from the microconstituents (atomic structure), were introduced into the theory.
Another line of reduction of thermodynamics to statistical mechanics allowed for the modeling of fluctuations around an equilibrium state. The work on this theory can be traced back to Einstein. Here, unlike the orthodox thermodynamic theory, an isolated system in contact with a heat bath at a constant temperature would have a range of internal temperatures and internal energy contents, centered on the temperature and internal energy of macroscopic equilibrium as predicted in the orthodox thermodynamic theory. This work was described by Sklar (1993) as of "surpassing elegance".
Deeper foundational issues on the definition of equilibrium will become more abstract and complex. The interested reader is referred to Sklar (1993)^{[32]}, Chapter 2, Section 2.II.6, pp.4448, kinetic theory, ensemble approach and ergodic theory; Section 2.III, pp.4859, Gibbs' statistical mechanics; Section 2.IV, pp.5971, criticism of Gibbs' approach by the Ehrenfests. Chapter 5, p.156, detail discussion of equilibrium theory. A shorter version can be found in Sklar (2004)^{[33]}.
Basically, the historical development of statistical mechanics is rather entangled, with many branches, foundational issues, and problems to explore and resolve. Such confusion manifests itself through the existence of a dozen or so schools of thoughts in statistical mechanics with conflicting approaches: Ergodic theory, coarsegraining (Markovianism), interventionism, BBGKY hierarchy ^{[34]} ^{[32]}, Jaynes, the Brussels school with De Donder^{[35]}, Prigogine, etc.; see Uffink (2004)^{[36]}, ^{[37]}. Disagreements among some of these schools can be found in further reading. The confusion came from the works of Boltzmann himself, who pursued different lines of thoughts; he would often abandon a line of thoughts, only to come back to it several years later^{[36]}.
Here, we only need to use the orthodox definition of equilibrium for the purpose of explaining the origin of the configuration integral, and follow Hill (1960, 1986)^{[38]} and McQuarrie (2000)^{[25]}.
The particles in a system are considered as independent when there is a weak interaction among them that only involves collision between the particles or between a particle and the surrounding wall. There is no (or negligeable) interparticle forces. Without interaction among the particles, the system cannot reach an equilibrium, Hill (1986), p.59.
One way to think of this problem is by considering an unconfined space, with no walls, in which the particles can move without colliding against each other. Assuming no external forces, such as gravitational forces, the particles continue to move on a straight line with constant velocity. Macroscopically, there cannot be an equilibrium state. In other words, to achieve a macroscopic equilibrium, it is necessary that there be at least the kind of "weak interaction" mentioned above.
Admittedly, the above concept of equilibrium and independence among the particles is somewhat intuitive and handwaving. It would be desirable to put the above concept on a more solid theoretical footing.
Independent particles
For systems with independent particles, there are two cases to consider: (1) Identical and distinguishable particles, and (2) Indistinguishable (or quantummechanically identical) particles.
It should be noted that in case (1), even though the particles are distinguishable, e.g., by their positions, they are identical in all other properties. An example would be the model of a monoatomic crystal, in which each atom is attached to a particular lattice site, and cannot jump to another lattice site. While these atoms are identical to each other, they are distinguishable by their locations in the crystal lattice; see Hill (1986), p.61.
Case (2) is related to a quantummechanical system in which identical particles are indistinguishable.
There may be a confusion in the use of the adjective "identical" in both cases. To distinguish the above two different types of "identical" particles, in case (1), we say the particles are identical (but distinguishable), whereas in case (2), we say that the particles are quantummechanically identical, which means the same as being indistinguishable. For a philosophical discussion, see French (2006) ^{[39]}.
Distinguishable and identical particles
The classical Hamiltonian of a particle is given by
(21)
where represents the position vector of the particle, its linear momentum, its kinetic energy, its potential energy, its mass, and its velocity.
For a system of independent particles, the Hamiltonian is
(22)
Even though it is possible to explain the configuration integral strictly within the framework of classical statistical mechanics, it is more general and simpler to develop the formulation within the framework of quantum statistical mechanics, which includes the classical statistical mechanics as a particular case; Hill (1986), p.2.
For a single particle in 1D, the system is quantized (see canonical quantization ) by replacing the classical Hamiltonian by the Hamiltonian operator in which the momentum is replace by the momentum operator
(23)
with being the unit imaginary number, so that
(24)
In 3D, the Hamiltonian operator takes the form:
(25)
where is the divergence operator
(26)
The Schrödinger equation then takes the form
(27a)
where is a wave function (an eigenfunction), and the corresponding energy (eigenvalue). The energy and the corresponding wave function constitute an eigenpair, called a quantum state; there are infinitely many such eigenpairs
(27b)
There will be multiple eigenvalues; the multiplicity of an eigenvalue is called a degeneracy number denoted by .
The eigenpairs can be grouped by energy level , i.e., by the numerical values of the energy (eigenvalue) ; each energy level thus has quantum states (eigenpairs) with the same energy value , but with different wave functions , . The set of quantum states at energy level is
For a system of independent particles, similar to Eq.(22), the system Hamiltonian operator is the sum of the Hamiltonian operator of individual particle:
(28)
We reserve the index to designate the particle number, the index for the quantum state, i.e., the eigenpair number, and the index for the energy level.
Consider the system wave function of the form
(29)
Then (cf. Hill (1986), p.60),
(30)
Thus, the energy of the system is the sum of the energies of individual particles:
(31)
Hidden in Eq.(31) is the sum over all possible quantum states for each particle. Let represent the state index for particle , and the energy corresponding to state of particle . Then
(32)
is the energy corresponding to the system state identified by the ntuple .
The partition function (or sumoverstates) of particle is of the form
(33)
where the sum is over all quantum states.
Likewise, the partition function for a system of independent and distinguishable particles is (cf. Hill (1986), p.60)
(34)
In addition to being independent and distinguishable, if the particles are also identical, then
(35)
and
(36)
Indistinguishable particles
Quantummechanically identical particles are indistinguishable. Roughly speaking, each ntuple has identical permutations, and thus the partition function in Eq.(36) should be divided by , i.e.,
(51)
The justification for Eq.(51) is actually more sophisticated. Consider identical, indistinguishable particles, labeled for the convenience of making the argument. Consider different quantum states labeled with . By permutations, in the partition function in Eq.(36), there are identical terms of the form
(52)
Only one term among the should be counted in the partition function.
But there also terms such as
(53)
where the energy level of the first two particles are the same; by permutations of the last particles, there are such terms. There are many other similar terms in which a subset of two or more particles have an identical energy level.
Terms like those in Eq.(53), with repeated energy levels, are allowed in the BoseEinstein statistics for bosons, but not allowed in the FermiDirac statistics for fermions.
But in the limiting case in which each particle has a number of quantum states between the molecular ground state and the molecular ground state plus, say, , much larger than the number of particles , then the number of terms such as those in Eq.(52) is much larger than the number of terms such as those in Eq.(53), since there are many different quantum states to choose from; see Hill (1986), p.63. Hence, the partition function can be approximated by
(54)
Thus Eq.(51) should actually be thought of as an approximation, rather than exact equality.
The above limiting case for which Eq.(51) is valid is called the classical statistics or Boltzmann statistics , which is the limit of the BoseEinstein statistics and the FermiDirac statistics as temperature increases .
Ideal monoatomic gases
There are independent and indistinguishable (quantummechanically identical) particles in a cubic box of side length . To compute the partition function of this system, we need to know the energy levels of a single particle in a box, which is a classic problem.
Particle in a box
Energy levels
By solving the 3D timeindependent Schrödinger equation for a particle of mass , in a box, as given
(61)
with zero potential inside the box, i.e., , we obtain the following energy levels (eigenvalues)
(62)
where are the quantum numbers, which take natural values in , i.e., .
Condition for approximation of partition function
As mentioned above, the number of quantum states available between the ground state and the ground state plus should be much larger than the number of particles for the approximation in Eq.(54) to be valid, i.e.,
(63)
Thus if we can connect the number of quantum states to a given maximum energy level , then we can establish an energetic condition for which the approximation in Eq.(54) is valid.
Consider Eq.(62) and the 3D space of quantum numbers . Each point of naturalnumber coordinates in this space corresponds to a quantum state, which can be thought of as occupying a unit cube with a unit volume in this space. Because of the factor in Eq.(62), let's consider a sphere in the space of quantum numbers, centered at the origin, and having a radius such that
(64)
Setting , we would have the expression of the radius such that the quantum states on the surface of that sphere would have the energy level
(65)
Since the quantum numbers are natural numbers (strictly positive integers), the quantum states lie in an octant (1/8th of the sphere). Thus, the volume of an octant with radius contains all quantum states with energy levels less than , i.e., this volume is equal to the number of quantum states with energy less than :
(66)
with being the volume of the cube of length . Fig.2 illustrates the quantum states in the space of quantum numbers : The circle with radius corresponds to the energy level ; the quantum states outside the band corresponding to the energy levels and are represented small open circles; the quantum states inside that "energy band" are the small solid circles; the number of small solid circles is the degeneracy at energy level ; cf. McQuarrie (2000), p.11, Hill (1986), p.75.
Now, take of the order of , i.e.,
(67)
then the condition in Eq.(63) becomes
(68)
where is the thermal de Broglie wavelength in Eq.(8). The factor 1.33 can be dismissed from the inequality in Eq.(68), which becomes
(69)
i.e., for the partition function in Eq.(51) to be a good approximation, the average distance between the particles should be much greater than the thermal de Broglie wavelength ; otherwise, quantum effects will not be negligible.
It is seen that the condition in Eq.(69) led to the approximation in Eq.(54), and is therefore the sufficient condition for the validity of the application of the classical or Boltzmann statistics as expressed in the partition function in Eq.(51).^{[40]}
Discrete energy, continuous energy
Here, there is a potential confusion due to the use of the notation to designate the degeneracy^{[41]} in both the discrete (quantum) energy case and in the continuous energy case.
For the discreteenergy case, the summation in the partition function for a single particle can be written in two ways (cf. Eq.(33)): (1) In terms of the quantum states, and (2) in terms of the energy levels. The summation in terms of the energy levels itself has been written in two ways: (2a) With a summation index for the energy level; this summation index (a discrete variable) takes values in the set of natural numbers , e.g., . (2b) Without a summation index, but using the notation for energy to designate a discrete variable that takes values in the set of distinct energy levels, i.e., , such that , i.e., each energy level has a different value of energy; there is no value that is repeated. The 3 ways of writing the summation in the partition function (i.e., 1, 2a, 2b above) for a single particle are presented below
(70)
For the continuousenergy case, some authors wrote the partition function as [Hill (1986), p.77; McQuarrie (2000), p.82]
(*)
Here is the confusion: The quantity in Eq.(70) is the degeneracy at the energy level , i.e., the number of quantum states at the same energy level , whereas the quantity in Eq.(*) is not the degeneracy, but the number of quantum states per unit energy at the energy level . McQuarrie (2000), p.82, called the factor in Eq.(*) the "effective degeneracy", which is not immediately clear at first encounter. The dimensions of these two 's are different from each other (one is number of quantum states, the other is number of quantum states per unit energy). Thus it is better to write Eq.(*) with a different notation, say , for the number of quantum states per unit energy:
(71)
The case of continuous energy has two useful applications: (1) approximate a densely populated spectrum of discrete quantum energy levels in the evaluation of the partition function (see the next few subsections), (2) use in classical statistical mechanics where the energy varies continously, as opposed to the discrete energy in quantum mechanics.
Approximate summation by integration
With the expression in Eq.(62) for the energy levels for a particle in a box, the partition function expression in Eq.(70) becomes
(72)
The summation in Eq.(72) can be approximated by an integration of the type shown in Eq.(71) if the summand in Eq.(72) changes essentially continuously with increments of the indices . Such is the case if
(73)
with defined in Eq.(2), and the increment in energy level due to an increment of the indices . Based on the expression of the energy level in Eq.(62), consider a unit increment of the quantum numbers from to , the increment in the energy level is of order
Thus, using the expression for the thermal de Broglie wavelength in Eq.(2), we have
(74)
With the restriction that the average distance between the particles much larger than the thermal de Broglie wavelength, as expressed in Eq.(69), so that the approximated partition function in Eq.(54) become accurate, we have
(75)
If is of the order of the Avogadro number, then
(76)
which largely satisfies the condition in Eq.(73)^{[42]}, so that the summation in the partition function in Eq.(72) can be approximated by the integration as expressed in Eq.(71).
Effective degeneracy, order of magnitude
Now that we have introduced the different notation for the number of quantum states per unit energy as shown in Eq.(71), the "effective degeneracy" can be written as
(81)
We note immediately that the effective degeneracy is not the same as the degeneracy , hence the difference in notation.
As illustrated in Fig.1, the effective degeneracy is the number of quantum states lying inside the band formed by the circle with radius corresponding to the energy level and a (slightly) larger circle corresponding to the energy level ; Hill (1986), p.77; McQuarrie (2000), p.11. We have
(82)
To give an idea about the magnitude of the effective degeneracy, consider the following numerical data with (in SI units)
 Temperature
 Mass
 Box length
 Increment of energy
If we just look at the order of magnitude, then
and thus
which is a large number for a simple system like a particle in a box at room temperature; cf. McQuarrie (2000), p.11. The order of magnitude of , i.e., the number of quantum states per unit energy, is then
which is much larger than the effective degeneracy .
Partition function, thermal de Broglie wavelength
Using Eq.(82), the partition function in Eq.(71) for a particle in a box can now be evaluated as follows (Hill (1986), p.77)
(91)
where , defined in Eq.(8), is called the thermal de Broglie wavelength. In Eq.(91), we made use of the following integration result of the Gamma function:
(92)
Integrating by parts the Gamma function in Eq.(92)_{1}, we obtain:
(93)
Next, by changing the variable , we obtain
(94)
using the integration result in Eq.(162), noting that the domain of integration here is half that in Eq.(162). For more details on the Gamma function, the readers are referred to Sebah & Gourdon (2002)^{[43]}.
The thermal de Broglie wavelength has the dimension of length (of course): In the numerator of , the Planck constant has the dimension of energy times time, i.e., force times length times time . In the denominator of , the term has the dimension of energy, i.e., force times length , or equivalently mass times velocity squared . Thus, the denominator has the dimension of mass times velocity , or momentum. The dimension of is then
(95)
See further reading on .
N particles in a box, partition function
The partition function of a system with independent, identical, indistinguishable particles can now be written as
(101)
It can be verified that in the absence of a potential energy, i.e., (since the particles are independent; there is no interparticle forces), the configuration integral , and the partition function in Eq.(6) is reduced to Eq.(101).
Helmholtz energy
From Eq.(9), the Helmholtz energy^{[10]}^{[26]} of this system can now be written as
by using the Stirling approximation for , i.e.,
and . Thus,
(102)
which can be used to calculate the thermodynamic properties of the system; see Hill (1986), p.77.
Thermodynamic properties
"the average physicist is made a little uncomfortable by thermodynamics. He is suspicious of its ostensible generality, and he doesn't quite see how anybody has a right to expect to achieve that kind of generality. He finds much more congenial the approach of statistical mechanics, with its analysis reaching into the details of those microscopic processes which in their large aggregates constitute the subject matter of thermodynamics. He feels, rightly or wrongly, that by the methods of statistical mechanics and kinetic theory he has achieved a deeper insight." P.W. Bridgman, The nature of thermodynamics, 1941, p.3.
Once the expression for the Helmholtz energy is available, one can then obtain the expressions for the thermodynamic properties of the canonical ensemble, i.e., entropy , pressure , chemical potentials . In addition, since , we can also obtain the expression for the total internal energy of the system.
Recall that the independent variables of the internal energy for the canonical ensemble are entropy , volume , and the particle numbers for different components (or species); we write . We have (McQuarrie (2000), p.17)
(111)
being the chemical potential for species .
With the Legendre transformation
(112)
we have
(113)
making the independent variables for the Helmholtz energy . From Eq.(113) and using Eq.(9), we have (cf. Hill (1986), p.19)
(114)
(115)
(116)
Finally, using Eq.(114) in Eq.(9), we obtain the expression for the total internal energy
(117)
with defined in Eq.(2).
Now using the expression for the Helmholtz energy in Eq.(102) and Eqs.(114)(117), we obtain the following expressions for the thermodynamic properties for an ideal monoatomic gas (i.e., a system of distinguishable, identical, and independent particles):
where and are constants with respect to ; see Eq.(102). Thus, the entropy for ideal monoatomic gas takes the form (cf. Hill (1986), p.79)
(118)
Similarly, the pressure takes the familiar form of the ideal gas law:
(119)
Chemical potential for the case with one species:
(120)
where Eq.(119) had been used in the last equation. Thus, (cf. Hill (1986), p.80)
(121)
The internal energy of an ideal monoatomic gas follows from the expression for in Eq.(102) and Eq.(112), and the expression for in Eq.(118)
(122)
which is purely the kinetic energy of the system, without the potential energy, since there were no interparticle forces. The amount of kinetic energy per momentum degree of freedom (dof) of the system is (there are momentum dofs)^{[44]}. Of course, the same result shown in Eq.(122) can be obtained by differentiating the partition function using Eq.(117)_{2} or Eq.(117)_{3}. Eq.(119) for pressure and Eq.(122) for internal energy are called the "classical results" since they can be derived from using only the kinetic theory, without a need for a quantummechanical setting. Here, we recover the classical results starting from a quantummechanical setting, which is reassuring, and therein lies the beauty and power of statistical mechanics.
In this section, we have connected statistical mechanics and thermodynamics; such connection, having a starting point in statistical mechanics, is often known in philosophy as a reduction of thermodynamics to statistical mechanics, leading to an alternative name statistical thermodynamics for the field; see Sklar (2004)^{[33]}.
Average continuous energy, equipartition
The internal energy in Eq.(122) can be obtained by computing the average energy as follows. From the partition function Eq.(70)_{1} for discrete energy of a single particle in a box, the probability that the particle is found in state is^{[24]}
(123)
and the average energy of a single particle is (cf. Hill (1986), p.12)
(124)
For the continous energy case, we obtain from Eq.(71) the probability for a single particle to be within the energy band as (Hill (1986), p.77)
(125)
and the average energy of a single particle as
(126)
Next, using Eq.(91), i.e., the expression for , in Eq.(126), we have the average energy of a single particle as (after cancelling out the common factor)
(127)
where we have made use of Eq.(92)_{3} and Eq.(93)_{2}. For a system with particles, we then obtain the same result as in Eq.(122), based on Eq.(31) (system energy is the sum of particle energy).
The above result for the ideal gas law in Eq.(119) and for the internal energy in Eqs.(122) and (127) can also be obtained using the equipartition theorem (Tolman (1979), p.93).
Classical statistical mechanics, continuous energy
In the limit of large quantum numbers, quantum statistics would asymptotically approach classical statistics. As temperature increases, terms corresponding to larger quantum numbers provide more important contribution to the overall sum in the canonical ensemble partition function. All results obtained from classical statistics, which is often easier to use, would be some limits of quantum statistics. Classical statistics is a special case of quantum statistics. We follow Hill (1986), p.112, to develop classical statistics inductively from the quantum statistics results obtained above.
Onedimensional harmonic oscillator
Energy
Classical case (continuous energy)
The potential energy in a classical harmonic oscillator (springmass system) with linear forcedisplacement relationship is a quadratic form in the coordinate
(131)
where is the spring stiffness coefficient ^{[45]}, and the displacement.
From Eq.(21), the classical Hamiltonian is now written as
(132)
The classical frequency of this simple oscillator takes the form ^{[46]}
(133)
Thus, the Hamiltonian can be written in terms of the frequency as follows (cf. Hill (1986), p.113)
(134)
The Hamiltonian in Eq.(134) is the total internal energy of the system, and is continuous in terms of the phasespace variables .
Quantum case (discrete energy)
The energy levels of a quantum harmonic oscillator, obtained by solving Eq.(27a), are discrete and nondegenerate
(141)
where is the Planck constant, the classical frequency in Eq.(133), , and the circular frequency.
Partition function
We begin by considering the partition function for the quantum harmonic oscillator, with discrete energy levels and its limiting case, then "generalize" to the partition function for the classical harmonic oscillator with continuous energy. The limiting case of the quantum partition function will be use to determine the constant for the classical partition function.
Quantum case
Using Eq.(70)_{1}, the quantum partition function can be written as
(151)
As temperature rises, i.e., , thus ; a development in Taylor series leads to an asymptotic function in term of for the partition function (cf. Hill (1986), p.89)
(152)
The above asymptotic function will be used to fix the proportionality constant in the classical partition function for the case with continuous energy.
Another way of obtaining the asymptotic function in Eq.(152) is to follow the same line of argument made further above to approximate summation by integration. Since is very small, is nearly equal to 1, and thus would be essentially continuous with , i.e., the increment
is very small. In this case, the summation in Eq.(151) can be approximated by an integration, i.e.,
(153)
since
with
Average quantum energy, failure of equipartition
First, using Eq.(70)_{1}, we can rewrite Eq.(124) as follows
(156)
Compare Eq.(156)_{3} to Eq.(117)_{3}. Next, with the partition function for a quantum harmonic oscillator given in Eq.(151)_{4}, using Eq.(156)_{3}, we obtain the average energy for the quantum harmonic oscillator as (cf. Tolman (1979), p.379; McQuarrie (2000), p.121, p.132)
(157)
with the limit of as being the result of using the same method as in Eq.(152). Thus we have equipartition of energy for high temperature , half from the kinetic energy and half from the potential energy (cf. Eq.(127) for a particle in a 3D box where there was only kinetic energy, with zero potential energy). At low temperature , there was no equipartition of energy, i.e., equipartition does not work at low temperature due to quantum effects. Fig.3 shows the quantum correction to equipartition at low temperature for the quantum harmonic oscillator and for the electromagnetic oscillator (which is used to explain blackbody radiation)^{[47]}.
Classical case, phase integral
A logical "generalization" of the quantum partition function for discrete energy levels in Eq.(151)_{1}, i.e.,
is to replace the quantum summation with an integral for continuous energy to obtain the classical partition function . But it is not that simple; to make sure that the classical partition function agrees with the quantum partition function in the limit when , i.e.,
in Eqs.(152)(153), we need to add a proportionality constant in the expression for :
(161)
where we made use of the integration results
(162)
and
(163)
Thus (cf. Hill (1986), p.113)
(164)
The integral in Eq.(164) is carried over the whole phase space is called the phase integral.
Particle in a box
Recall from Eq.(72) that the quantum partition function for a particle in a box takes the form
At sufficiently high temperature, , the condition in Eq.(73) is satisfied, i.e.,
and the quantum partition function can be approximated by an integration, leading to the expression in Eq.(91), i.e.,
Consider the following "generalization" of Eq.(72) to the case with continuous energy
(171)
with the Hamiltonian (or energy) being simply the kinetic energy (the potential energy is zero)
(172)
and
(173)
Using the integration result in Eq.(163), and the fact that the classical partition function is the limiting case of the quantum partition function as , we have
(174)
Thus,
(175)
Hence, for a particle in a box with 3 momentum dofs and 3 position dofs^{[44]} (see Hill (1986), p.115):
(176)
System of indistinguishable particles
Zero potential energy (without interparticle forces)
In general, for a particle system in a box with momentum dofs and with zero potential energy, the Hamiltonian is purely kinetic energy
(181)
and the classical partition function takes the form
(182)
For a system of particles in 3D space , we have , which represents both momentum dofs and position dofs^{[44]}. Without a zero potential energy, the integration can be carried out to yield
(183)
The above partition function is for distinguishable particles. For indistinguishabe particles, a division by yields the classical partition function
(184)
which is the same as in Eq.(101).
An interpretation for the division by in Eq.(184) can be given as follows. Consider the case with two particles moving in a 1D space, with a 2D phase space. The phase integral in Eq.(184) can be approximated by a quadruple summation
(185a)
where
(185b)
is an infinitesimal area in the phase space, as represented by a green square in Fig.4. The sets and are sets of discrete values of the continuous variables and , for :
(186)
The phasespace coordinates of particle are denoted by . In the summation in Eq.(185a), there are two terms that correspond respectively to (i) particle 1 (red dot in Fig.4) at position in phase space, i.e., , and particle 2 (blue dot in Fig.4) at position in phase space, i.e., , and (ii) particle 2 (blue dot in Fig.4) at position in phase space, i.e., , and particle 1 (red dot in Fig.4) at position in phase space, i.e., . In the case where the particles are indistinguishable, the above two cases represent only a single quantum state, i.e., the summation in Eq.(185a) thus double counted the number of states, and thus should be divided by the number of particles, i.e., .
For a system with indistinguishable particles, each state is counted times, and thus a division by in Eq.(184) (cf. Hill (1986), p.117). The above argument to justify for the factor was attributed to Gibbs, and resolved the Gibbs paradox when mixing identical gases; see, e.g., Huang (1987), p.141. Kirkwood (1934)^{[16]}, considered this Gibbs argument "somewhat arbitrary", provided a quantummechanical approach to obtain the factor ; see also Hill (1986), p.462. On the other hand, Buchdahl (1974)^{[48]} provided a justification based on purely classical statistical mechanics.
A classical partition function (e.g., Eq.(182)) with a correction factor such as shown in Eq.(184) is sometimes referred to as a semiclassical partition function; see, e.g., Kirkwood (1934), Reichl (1998), p.359. In the literature, there was some confusion in the terminologies used: The phase integral—which Kirkwood (1933)^{[16]} referred to as the Gibbs phase integral, and which does not have the factor —was sometimes called the "classical partition function" by some authors, see, e.g., Zwanzig (1957)^{[49]}. On the other hand, some other authors, such as Hill (1986), referred to the phase integral with the factor , with or without the correction factor , as the "classical partition function".
Nonzero potential energy (with interparticle forces)
When the potential energy is nonzero, i.e., there are interparticle forces, which is the more interesting and general case, the Hamiltonian is written as
(201)
where is the momentum of particle along the th coordinate direction, for and , the potential energy, and the position of particle in 3D space. In this case, the classical partition function in Eq.(184)_{1} becomes Eq.(6), after integrating out the kinetic energy part (first term in Eq.(201)), i.e.,
(6)
When there the potential function , we obtain , thus recovering Eq.(184)_{2}; see Hill (1986), p.118.
Further reading
 Thermal de Broglie wavelength
 Boltzmann's Work in Statistical Physics, The Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.), First published Wed 17 Nov, 2004.
 Brussels or BrusselsAustin school in statistical mechanics
 Google search with keywords "brussels school statistical mechanics"
 Some Philosophical Influences on Ilya Prigogine's Statistical Mechanics
 Bishop, R.C., BrusselsAustin Nonequilibrium Statistical Mechanics in the Early Years: Similarity Transformations between Deterministic Probabilistic Descriptions, arXiv:physics/0304018v3, [physics.classph], 7 Apr 2003.
 Shalizi, C., Ilya Prigogine, Notebooks, 20 Aug 2007.
 Entropy in statistical mechanics, Chap.11, dissertation, University of Utretch, Netherlands.
 Disagreements with the BrusselsAustin school
 Bricmont, J., Science of Chaos or Chaos in Science?, arXiv:chaodyn/9603009v1, Mar 1996.
 Dr. Prigogine's dissipative structure theory and some critical comments on this theory
Notes and references
 ↑ See, e.g., G.H. Wannier, Statistical Physics, Dover Publications, New York, 1987 (p.244, Read from Google Book); K. Lucas, Molecular Models for Fluids, Cambridge University Press, 2007 (p.270, Read from Google Book).
 ↑ ^{2.0} ^{2.1} Some wiki pages for temperature: Wikipedia Sklogwiki (where the article has not been completed, but the reference to journal articles are quite uptodate). Jones, E.R., Fahrenheit and Celsius, a history, The Physics Teacher, Nov. 1980, Vol.18, No.8, pp.594595. PBS NOVA program Absolute Zero, aired in Jan 2008, or watch this program online. C. Shiller, Motion Mountain, (encyclopedic, unusual, free, online physics textbook), 21st edition, Dec 2007, Part I, Section on Temperature, pp.259280.
 ↑ Allen, M.P., and Tildesley, D.J., Computer Simulation of Liquids, Oxford University Press, 1989. Read from Google Book.
 ↑ McComb, W.D., Renormalization Methods: A Guide for Beginners, Oxford University Press, 2004. Read from Google Book.
 ↑ The intention here is to make the article more useful for learning and research, rather than simply providing a list of formulas with superficial explanation. For this matter, see several useful articles in mathematics, e.g., Calculus of variations, etc. or even the classical mechanics article classical harmonic oscillator, which is closer to the present article.
 ↑ J. M. J. Swanson, R. H. Henchman, and J. A. McCammon, Revisiting Free Energy Calculations: A Theoretical Connection to MM/PBSA and Direct Calculation of the Association Free Energy, Biophys J. 2004 January; 86(1): 67–74.
 ↑ Kwong, P., et al. HIV1 evades antibodymediated neutralization through conformational masking of receptorbinding sites, Nature 420, 678682 (12 December 2002).
 ↑ Kwong, P., et al. Structure of an HIV gp120 envelope glycoprotein in complex with the CD4 receptor and a neutralizing human antibody, Nature 393, 648659 (18 June 1998)
 ↑ Peter D. Kwong, Ph.D., Vaccine Research Center, Structural Biology Laboratory, National Institute of Allergy and Infectious Diseases, National Institute of Health, Description of Research Program, 2007.
 ↑ ^{10.0} ^{10.1} ^{10.2} ^{10.3} ^{10.4} The Gibbs energy is customarily known as the "Gibbs free energy", and the Helmholtz energy the "Helmholtz free energy". The name "free energy" was originally used by Gibbs in 18761878, and by Lewis and Randall, Thermodynamics and the Free Energy of Chemical Substances, McGrawHill, 1923; see the Glossary of coined terms in J. Andraos' Named Things in Chemistry & Physics. The IUPAC recommends to use either energy (more precise) or function (vague), instead of "free energy". Some comments in the literature related to the use of "free energy": "A ... worst situation exists with the Gibbs and the Helmholtz functions [energy]. The misnomer free energy (which, by the way, is not energy but is freely misused) is employed by American chemists to designate the Gibbs function, but by Europeans to designate the Helmholtz function. To make matter worse, both groups insist on using the same symbol for the two different functions," p.350 in M.W. Zemansky, Fashions in Thermodynamics, Am. J. Phys., Vol.25, No.6, pp.349351, Sep 1957. Of course, the Gibbs energy and the Helmholtz energy are different kinds of energy. An example of an American physicist who associated "free energy" to the "Helmholtz free energy" is Huang (1987), p.22 and Index p.489. "IUPAC banished the ambiguous term “free energy” decades ago," p.754 in R. Battino, "Mysteries" of the First and Second Laws of Thermodynamics, J. Chem. Edu., Vol.84, No.5, pp.753755, May 2007. The ambiguity of the name "free energy" can be seen in the Interactive Link Maps that show the many connections of other concepts to the "free energy" (which in the past was used for both the Gibbs energy and the Helmholtz energy) in the Gold Book, the IUPAC Compendium of Chemical Terminology. The above two comments spanned exactly half a century, railing against the use of "free energy", but without a complete success as the terminology has been ingrained in many who grew up with it; see, e.g., Bevan Ott & BoerioGoates (2000), p.xvi. It is therefore important to know both the currently recommended and the past usage of different terminologies so to read the classic (or even modern) works. Here, we adhere to the IUPAC recommendations—and I went "free" hunting in this article.
 ↑ After the kinetic contributions of each species have cancelled.
 ↑ ^{12.0} ^{12.1} M. K. Gilson, J. A. Given, B. L. Bush and J. A. McCammon, The statisticalthermodynamic basis for computation of binding affinities: A critical review. Biophysical Journal 72: 10471069 (1997).
 ↑ M. K. Gilson and H.X. Zhou, Calculation of ProteinLigand Binding Affinities, Annual Review of Biophysics and Biomolecular Structure Vol. 36: 2142 (Volume publication date June 2007).
 ↑ Some wiki pages on partition function: Sklogwiki Wikipedia
 ↑ See, e.g., Tolman (1979), p.45.
 ↑ ^{16.0} ^{16.1} ^{16.2} J.G. Kirkwood, Quantum Statistics of Almost Classical Assemblies, Phys. Rev. 44, 31  37 (1933), 45, 116  117 (1934).
 ↑ Tolman (1938, 1979) pointed to R.H. Fowler's book in 1936 where the name "partition function" was used, whereas Tolman chose to stick with the name "sumoverstates". A search of the archive of the Physical Review turned up an earlier paper by Mac Gillavry (1930), which referred to a paper by Fowler (1922) in the Philosophical Magazine for a relationship between entropy and "partition function". Later on, I found a long obituary for Fowler (by his colleague E.A. Milne, Obituary Notices of Fellows of the Royal Society, 1945, pp.6078), where his work in 1922 on the partition of energy was mentioned on p.69; his 1922 papers were coauthored with C.G. Darwin, see p.75. The works of Fowler and Darwin appeared to be so well known that Huang (1987)'s chapter on the general properties of the partition function, p.193, referred to the "DarwinFowler method" without mentioning the original papers. The DarwinFowler method is an exact derivation of the MaxwellBoltzmann statistics without the need to appeal to the Stirling approximation of . Tolman (1979), footnote p.567, also referred to the treatments of partition function by "Darwin and Fowler" without citing the original papers. Since the DarwinFowler papers of 1922 are not accessible online, the closest possible way to read about their work would be H.N.V. Temperley, Statistical mechanics and the partition of numbers I. The transition of liquid helium, Proc. Roy. Soc. London, A, 1949, pp.361375. For a different, more recent tack, see also H.J. Schmidt and J. Schnack, Partition functions and symmetric polynomials, Am. J. Phys., Vol.70, No.1, pp.5357, 2002. For more historical details and the relationship between Fowler, LennardJones, McCrea, etc., see K. Gavroglu and A. Simoes, Preparing the ground for quantum chemistry in Great Britain : the work of the physicist R. H. Fowler and the chemist N. V. Sidgwick, Brit. J. Hist. Sci., Vol.35, pp.187212, 2002.
 ↑ Tolman, R.C., The Principles of Statistical Mechanics, Dover Publications, 1979. Orginally published by Oxford University Press in 1938. Book review by W.H. McCrea, The Mathematical Gazette, pp.415417, 1939. Many would disagree with McCrea since it is so convenient and consistent (notation and viewpoint) to have the quantum mechanics part written by the same author in this classic book; "almost a book within a book" (back cover).
 ↑ More often written as "Zustandssumme" in the German literature, with "Zustand(s)" being German for "state(s)", and "summe" for "sum"; see EnglishGerman dictionary, by W. Schneider. Some other translators are not good at translating the above word, such as Babel Fish, freeTranslation.com, Dictionary.com (translated "zustand" to "was entitled").
 ↑ Tolman (1979), footnote p.567, cited Fowler (1936) Statistical Mechanics, second edition, Cambridge, as an example of an author who used the name "partition function", instead of "sumoverstates".
 ↑ For this reason ("Zustandssumme"), some authors use the letter to denote the partition function, and the letter the configuration integral; see, e.g., Reichl (1998). Kirkwood (1933) used for "sumoverstates".
 ↑ ^{22.0} ^{22.1} Callen, H., Thermodynamics and an Introduction to Thermostatistics, 2nd edition, Wiley, New York, 1985. Book review by R.B. Griffith, Am. J. Phys., Sep. 1987, Vol.55, No.9, pp.860861. Excellent overview and review of four books on thermodynamics and thermal physics (Callen's was one) from a pedagogical viewpoint by H.L. Scott, Am. J. Phys., Feb 1998, Vol.66, No.2, pp. 164167. Review of 1960 edition of Callen's book by P.W. Bridgman, Am. J. Phys., Oct 1960, Vol.28, No.7, p.684.
 ↑ Khinchin, A.I., Mathematical Foundations of Statistical Mechanics, Dover Publications, New York, 1949.
 ↑ ^{24.0} ^{24.1} While the meaning of "sumoverstates" is clear, many books on statistical mechanics did not explain the meaning of the name "partition function", except for simply identifying formula such as that in Eq.(6) as the "partition function". It is likely that the original reason for using the names partition function and generating function is in number theory: A partition of an integer is a way to write an integer as a sum of integers, and a partition function of an integer is a the number of partitions of that integer. See, e.g., Partition function P, Partition function Q, Partition function q, Generating function, and Partition (number theory). In statistical mechanics, the partition function could be thought of as related to a partition of unity (total probability) into a sum (over states) in which each term represents the probability that the system occupies the corresponding state; see Eq.(123) and also meaning and significance of partition function. As mentioned in a footnote, it was Darwin and Fowler who introduced the name "partition function" in their papers in 1922 on the partition of energy.
 ↑ ^{25.0} ^{25.1} ^{25.2} McQuarrie, D.A., Statistical Mechanics , 2nd edition, University Science Books, Sausalito, CA, 2000.
 ↑ ^{26.0} ^{26.1} ^{26.2} It is not clear whether there is a different historical connotation in the choice of , which stands for "Free", instead of , which stands for the German word "Arbeit", meaning "work", to avoid conjuring up the painful memory of Nazi concentration camps, where the motto or slogan "Arbeit macht frei" was built into their gates. Even though "frei", meaning "free", was also part of this slogan, it is not the same as "Free" standing by itself. Callen (1985), p.146, wrote that was an "internationally adopted symbol for the Helmholtz potential" (energy), and that the "International Unions of Physics and Chemistry" (which cannot, or no longer, be found using a web search) recommended for enthalpy in "agreement with almost universal usage". The IUPAC (also from Wikipedia) recommends the symbol for the Helmholtz energy; see their list of symbols for thermodynamics and statistics. See also Section 7. Nomenclature in the Recommendations for nomenclature and tables in biochemical thermodynamics, 1994.
 ↑ Jarzynski, C., Nonequilibrium Equality for Free Energy Differences, Phys. Rev. Lett. 78, pp.26902693 (1997).
 ↑ McComb, W.D., The Physics of Fluid Turbulence, Oxford University Press, USA (March 11, 1992). Read from Google Book.
 ↑ M.J. Assael, J.P. Martin Trusler, T.F. Tsolakis, Thermophysical Properties of Fluids: An Introduction to Their Prediction, Imperial College Press (June 1996). Read from Google Book.
 ↑ ^{30.0} ^{30.1} Huang, K., Statistical Mechanics , second edition, Wiley, New York, 1987.
 ↑ M. Tafipolsky and R. Schmid, Calculation of rotational partition functions by an efficient Monte Carlo importance sampling technique, Journal of Computational Chemistry, Volume 26, Issue 15, Pages 1579  1591, 2005.
 ↑ ^{32.0} ^{32.1} ^{32.2} ^{32.3} Sklar, L., Physics and Chance: Philosophical issues in the foundations of statistical mechanics , Cambridge University Press, New York, 1993.
 ↑ ^{33.0} ^{33.1} Sklar, L., Philosophy of Statistical Mechanics (dynamic version), The Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.). Static Summer 2004 Edition with minor correction. Static Summer 2001 Edition, first appeared on 12 Apr 2001.
 ↑ Reichl, L., A Modern Course in Statistical Physics, 2nd edition, Wiley, 1998. Book review by J.H. Luscombe, Am. J. Phys., Dec 1999, Vol.67, No.12, pp.12851287.
 ↑ See also Some Philosophical Influences on Ilya Prigogine's Statistical Mechanics.
 ↑ ^{36.0} ^{36.1} Uffink, J., Boltzmann's Work in Statistical Physics, The Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.), 2004.
 ↑ Kovac, J., Review of Modern Thermodynamics: From Heat Engines to Dissipative Structures (by Dilip Kodepudi and Ilya Prigogine), J. of Chemical Education, Nov 1999.
 ↑ Hill, T.L., An Introduction to Statistical Thermodynamics , Dover Publications, 1986. Originally published by AddisonWesley in 1960.
 ↑ French, S., Identity and Individuality in Quantum Theory, The Stanford Encyclopedia of Philosophy, ed. by E. N. Zalta, 2006 version with subtantive content change.
 ↑ It is mentioned in Hill (1986), p.118, that Eq.(69) is the "necessary and sufficient" condition for the validity of the application of the partition function in Eq.(51). It is, however, not clear that Eq.(69) is the necessary condition for Eq.(51); if such was the case, one must be able to show that Eq.(51) implies Eq.(69). Thus far, it has been shown only that Eq.(69) implies Eq.(51).
 ↑ The degeneracy of an energy level is the multiplicity of the eigenvalue obtained when solving the eigenvalue problem in Eq.(61).
 ↑ In Hill (1986), p.77, the order in Eq.(76) was instead of ; it is not clear which value of Hill (1986) used. In any case, this difference does not change the conclusion.
 ↑ Sebah, P., and Gourdon, X., Introduction to the Gamma function (PostScript), 2002. The html version may not be readable, depending on how your browser was set up.
 ↑ ^{44.0} ^{44.1} ^{44.2} The phase space for a system with particles in 3D space has dofs, momentum dofs and position dofs.
 ↑ The spring stiffness coefficient is sometimes referred to in the physics/chemistry literature as the "force constant" (e.g., Hill (1986), p.113) denoted by .
 ↑ In classicalmechanics literature, the frequency is usually denoted by ; in quantummechanics literature, the frequency is denoted by .
 ↑ See also the Wikipedia feature article "Equipartition theorem" at 17:42, 6 February 2008, Section "Failure due to quantum effects", where the average energy had an expression different from that in Eq.(157); the reason was because the expression for the energy of the quantum harmonic oscillator was taken to be (missing ) instead of the correct expression given in Eq.(141). This error was first introduced almost a year ago, as of Mon, 11 Feb 2008, in the version at 19:20, 29 March 2007; even though the limit of the average energy as was the same as in Eq.(157), the accompanying plot in the version at 17:42, 6 February 2008 is clearly different from the correct plot for low ; see Fig.3. On the other hand, the problem can be fixed based on a légerdemain feat of Planck to explain blackbody radiation: Here, just arbitrarily remove the problematic term from the energy of the quantum harmonic oscillator (Tolman (1979), p.381); what is left, i.e., is called the energy of an electromagnetic oscillator; see, e.g., p.246 in Atkins, P. and de Paula, J., Physical Chemistry, 8th edition, W.H. Freeman, New York, 2006.
 ↑ Buchdahl, H.A. Remark on the Factor 1/N! in the Partition Function, Am. J. Phys., Vol.42, No.1, pp.5153, Jan 1974.
 ↑ R.W. Zwanzig, Transition from Quantum to "Classical" Partition Function, Phys. Rev. 106, 13  15 (1957).
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