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 $\displaystyle N$ particles is defined as follows:

$\displaystyle Z_N := \int\limits_V \exp \left[ - \beta U (x_1 , \cdots , x_N) \right] d^3 x_1 \cdots d^3 x_N$

(1)

where $\displaystyle V$ is the volume enclosing the $\displaystyle N$ particles, $\displaystyle \beta$ a constant defined as

$\displaystyle \beta := \frac {1} {k_B T}$

(2)

with $\displaystyle k_B$ being the Boltzmann constant, $\displaystyle T$ the thermodynamic temperature^[2] $\displaystyle U$ the potential energy of interparticle forces, $\displaystyle \{ x_1 , \cdots , x_N \}$ the positions in the 3-D space $\displaystyle \mathbb R ^3$ of the $\displaystyle N$ particles, with $\displaystyle x_i = (x_i^1 , x_i^2 , x_i^3)$ and $\displaystyle x_i^j$ the $\displaystyle jth$ coordinate of the $\displaystyle ith$ particle, and $\displaystyle d^3 x_i = d x_i^1 d x_i^2 d x_i^3$ an infinitesimal volume. An example for the potential energy $\displaystyle U$ is the Lennard-Jones potential.

By setting $\displaystyle U=0$ , we have $\displaystyle Z_N = V^N$ . Since both $\displaystyle U$ and $\displaystyle k_B T$ have the dimension of energy, the integrand in Eq.(1) is dimensionless, and thus the configuration integral $\displaystyle Z_N$ has the dimension of $\displaystyle V^N$ . For this reason, some authors use the non-dimensionalized configuration integral obtained by dividing Eq.(1) by $\displaystyle V^N$ ; 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 self-contained manner that does not require too many prerequisites^[5].

1 Motivation
2 Systems with independent particles
- 2.1 Equilibrium, independent particles
- 2.2 Independent particles
  - 2.2.1 Distinguishable and identical particles
  - 2.2.2 Indistinguishable particles
3 Ideal monoatomic gases
- 3.1 Particle in a box
- 3.2 N particles in a box, partition function
4 Classical statistical mechanics, continuous energy
5 Further reading
6 Notes and references
7 Copyright

Note on equation numbering: To allow for the flexibility of inserting sections and equations, equation numbers are chosen to be always increasing and sequential within each section, but not necessarily continuous from one section to the next, i.e., there may be gaps in the equation numbers when going from one section to the next. On the other hand, when an equation is assigned a number, this number is unique to this equation, so there can not be any confusion when referring to the equation numbers.

To student readers:

The links to Wikipedia articles in the present article are intended to be good starting points; many of these articles are highly informative. While any document can contain errors, Wikipedia articles are particularly vulnerable to having errors since anyone, regardless of credentials, can anonymously edit these articles; see, e.g., Criticism of Wikipedia. In many instances, errors (pranks, or simply vandalism) appeared to be immediately discussed and corrected; sometimes errors may go for years without being corrected as experts don't usually read Wikipedia articles in their own field to correct these errors. I found a Wikipedia article whose original text was based on the book titled The Complete Idiot's Guide to Chemistry; this reference was later deleted, but the erroneous text has remained in the article ever since. It is therefore a good idea to verify with standard textbooks often provided as references, or with other articles that can be found online. Many articles simply list books as references without giving the page numbers, making it sometimes more time consuming to locate the source for verification (if at all possible). Sklogwiki is a wiki specialized in statistical mechanics and thermodynamics; there are often up-to-date references (papers), even though many articles there are being (or to be) completed; Citizendium is more like Wikipedia in breadth, but edited by experts with known identity and credentials, even though the coverage is not as broad and many articles are also being (or to be) completed. Many Wikipedia articles appeared to be written and edited by devoted students (and other enthusiasts).

Motivation

Disease research and drug design

Fig.1. Ligand-receptor-ligand binding. HIV 1 glycoprotein (ligand, red), CD4-glycoprotein (receptor, yellow), b12 antibody protein (ligand, green).

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 ligand-receptor 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 HIV-1, destroys a human cell by first entering the cell through the cell membrane. To this end, the HIV-1 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 so-called 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 ligand-receptor-ligand binding involving the HIV-1 virus gp120 envelope glycoprotein (ligand), the CR4 glycoprotein (receptor), and the b12 antibody protein (ligand).

In a ligand-receptor 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:

$\displaystyle A + B \leftrightharpoons AB$

(3)

where $\displaystyle A$ represents the protein (receptor), B the ligand molecule, and $\displaystyle AB$ the bound ligand-receptor.

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]

$\displaystyle \Delta G^\circ_{AB} = - R T \log \left( \frac {C^\circ} {8 \pi^2} \right) \left( \frac {Z_{N,AB} Z_{N,O}} {Z_{N,A} Z_{N,B}} \right) + P^\circ \langle \Delta V_{AB} \rangle$

(4)

where the $\displaystyle Z$ quantities are the configuration integrals. For example, the configuration integral for protein $\displaystyle A$ is

$\displaystyle Z_{N,A} = \int \exp \left[ - \beta U(r_A , r_S) \right] d r_A d r_S$

(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 ligand-receptor affinity calculation is given by Gilson & Zhou (2007)^[13].

Classical partition function (sum-over-states)

In classical (no quantum effect) statistical mechanics, the configuration integral $\displaystyle Z_N$ and the partition function $\displaystyle Q_{class}$ 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 $\displaystyle N$ particles, the classical partition function^[14] $\displaystyle Q_{class}$ is obtained by multiplying the configuration integral $\displaystyle Z_N$ 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 $\displaystyle N$ of particles and the thermodynamic temperature^[2] $\displaystyle T$ :

$\displaystyle Q_{class} := \frac {Z_N} {C_N \Lambda ^{3N}} = \left( \frac {V^N} {C_N \Lambda ^{3N}} \right) \left( \frac {Z_N} {V^N} \right) = Q_{class}^0 Z_N^\star \ {\rm with} \ Q_{class}^0 := \frac {V^N} {C_N \Lambda ^{3N}} \ {\rm and} \ Z_N^\star := \frac {Z_N} {V^N}$

(6)

where the coefficient $\displaystyle C_N$ is defined as

$\, C_N$	Case
$\, 1$	Distinguishable particles
$\, N!$	Indistinguishable particles

(7)

and the thermal de Broglie wavelength $\displaystyle \Lambda$ defined as

$\displaystyle \Lambda (T) := \frac {h} {(2 \pi m k_B T) ^{1/2}}$

(8)

where $\displaystyle h$ is the Planck constant. See further reading on $\displaystyle \Lambda$ .

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 $\displaystyle Z_N$ to obtain $\displaystyle Q_{class}$ is not a constant, but a function of $\displaystyle N$ and $\displaystyle T$ , 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 ligand-receptor binding affinity mentioned above.

The abstract terminology "partition function" is also known more concretely and unassumingly as the "sum-over-states", 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 "sum-over-states", 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., $\displaystyle U=0$ , such as in the case of independent particles, the partition function $\displaystyle Q_{class}$ takes a simple form.

As mentioned in the introduction, the configuration integral $\displaystyle Z_N$ can be non-dimensionalized by dividing by $\displaystyle V^N$ , with the non-dimensionalized version denoted by $\displaystyle Z_N^\star$ . Then, the partition function $\displaystyle Q_{class}$ can be written as the product of the partition function $\displaystyle Q_{class}^0$ of ideal gas (i.e., when the potential $\displaystyle U$ is zero) by the non-dimensionalized configuration integral $\displaystyle Z_N^\star$ . Thus, the configuration integral $\displaystyle Z_N^\star$ can be thought of as a correction factor for $\displaystyle Q_{class}^0$ to obtain the partition function for the case where the potential is non-zero.

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] $\displaystyle A$ and the partition function $\displaystyle Q$ ; McQuarrie (2000)^[25], p.45:

$\displaystyle A (T,V,N) = - k_B T \log Q (T,V,N) \Longleftrightarrow Q = \exp [- A / (k_B T)] = \exp (-\beta A)$

(9)

with $\displaystyle \beta = 1 / (k_B T)$ as defined in Eq.(2). Some authors such as Callen (1985) use the notation $\displaystyle F$ , instead of the more historical and customary notation $\displaystyle A$ , for the Helmholtz (free) energy^[10]^[26]. Eq.(9)₂ plays an important role in the Jarzynski equality, also known as a non-equilibrium work relation; see the seminal paper by Jarzynski (1997)^[27], where the notation $\displaystyle F$ 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 $\displaystyle Z_N$ 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 $\displaystyle [\exp(-\beta U) - 1]$ provides a systematic method to approximately calculate the configuration integral $\displaystyle Z_N$ . This method is known as cluster expansion, which Maria Göppert-Mayer 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 $\displaystyle Q_{class}$ in Eq.(6) and the classical configuration integral $\displaystyle Z_N$ 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 $\displaystyle P$ , volume $\displaystyle V$ , temperature $\displaystyle T$ , magnetic field $\displaystyle M$ , 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 time-independent 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 micro-constituents (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.44-48, kinetic theory, ensemble approach and ergodic theory; Section 2.III, pp.48-59, Gibbs' statistical mechanics; Section 2.IV, pp.59-71, 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, coarse-graining (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 hand-waving. 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 quantum-mechanically 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 quantum-mechanical 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 quantum-mechanically identical, which means the same as being indistinguishable. For a philosophical discussion, see French (2006) ^[39].

Distinguishable and identical particles

The classical Hamiltonian $\displaystyle H$ of a particle is given by

$\displaystyle H (x,p) = K + U = \frac {1} {2} m v^2 + U (x) = \frac {p^2} {2 m} + U (x)$

(21)

where $\displaystyle x$ represents the position vector of the particle, $\displaystyle p = mv$ its linear momentum, $\displaystyle K$ its kinetic energy, $\displaystyle U$ its potential energy, $\displaystyle m$ its mass, and $\displaystyle v$ its velocity.

For a system of $\displaystyle N$ independent particles, the Hamiltonian is

$\displaystyle H = \sum_{i=1}^{N} H_i = \sum_{i=1}^{N} \left[ \frac {p_i^2} {2 m_i} + U (x_i) \right]$

(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 1-D, the system is quantized (see canonical quantization ) by replacing the classical Hamiltonian $\displaystyle H$ by the Hamiltonian operator $\displaystyle \mathcal H$ in which the momentum is replace by the momentum operator

$\displaystyle p = - \imath \hbar \frac{\partial}{\partial x} \ {\rm where} \ \hbar = \frac{h}{2 \pi}$

(23)

with $\displaystyle \imath$ being the unit imaginary number, so that

$\displaystyle \mathcal H = - \frac {\hbar ^2} {2 m} \frac {\partial ^2} {\partial x^2} + U (x)$

(24)

In 3-D, the Hamiltonian operator $\displaystyle \mathcal H$ takes the form:

$\displaystyle \mathcal H = - \frac {\hbar ^2} {2 m} {\rm div} + U(x)$

(25)

where $\displaystyle {\rm div}$ is the divergence operator

$\displaystyle {\rm div} = \frac {\partial} {\partial x^i} \frac {\partial} {\partial x^i} = \frac {\partial^2} {(\partial x^1)^2} + \frac {\partial^2} {(\partial x^2)^2} + \frac {\partial^2} {(\partial x^3)^2} = \frac {\partial^2} {(\partial x)^2} + \frac {\partial^2} {(\partial y)^2} + \frac {\partial^2} {(\partial z)^2}$

(26)

The Schrödinger equation then takes the form

$\displaystyle \mathcal H \psi = \varepsilon \psi$

(27a)

where $\displaystyle \psi$ is a wave function (an eigenfunction), and $\displaystyle \varepsilon$ the corresponding energy (eigenvalue). The energy and the corresponding wave function constitute an eigenpair, called a quantum state; there are infinitely many such eigenpairs

$\displaystyle \{ (\varepsilon_i , \psi_i), i = 1, \cdots , \infty \}$

(27b)

There will be multiple eigenvalues; the multiplicity of an eigenvalue is called a degeneracy number denoted by $\displaystyle \omega$ .

The eigenpairs can be grouped by energy level $\displaystyle \ell$ , i.e., by the numerical values of the energy (eigenvalue) $\displaystyle \varepsilon$ ; each energy level $\displaystyle \ell$ thus has $\displaystyle \omega_\ell$ quantum states (eigenpairs) with the same energy value $\displaystyle \varepsilon_\ell$ , but with different wave functions $\displaystyle \psi_{\ell k}$ , $\displaystyle k = 1 , \cdots , \omega_\ell$ . The set of quantum states at energy level $\displaystyle \varepsilon_\ell$ is

$\displaystyle \{ (\varepsilon_\ell , \psi_{\ell k}), k = 1 , \cdots , \omega_\ell \}$

For a system of $\displaystyle N$ independent particles, similar to Eq.(22), the system Hamiltonian operator is the sum of the Hamiltonian operator of individual particle:

$\displaystyle \mathcal H = \sum_{a=1}^{N} {\mathcal H}_a \ {\rm with} \ \mathcal H_a \psi_a = \varepsilon_a \psi_a$

(28)

We reserve the index $\displaystyle a$ to designate the particle number, the index $\displaystyle i$ for the quantum state, i.e., the eigenpair number, and the index $\displaystyle \ell$ for the energy level.

Consider the system wave function of the form

$\displaystyle \psi = \prod_{a=1}^{N} \psi_a$

(29)

Then (cf. Hill (1986), p.60),

$\displaystyle \mathcal H \psi = \sum_{a=1}^{N} {\mathcal H}_a \left( \prod_{b=1}^{N} \psi_b \right) = \sum_{a=1}^{N} \left( \prod_{\stackrel{b=1}{b \ne a}}^{N} \psi_b \right) \underbrace{ {\mathcal H}_a \psi_a }_{\varepsilon_a \psi_a} = \underbrace{ \left( \sum_{a=1}^{N} \varepsilon_a \right) }_{\varepsilon} \psi = \varepsilon \psi$

(30)

Thus, the energy of the system is the sum of the energies of individual particles:

$\displaystyle \varepsilon = \sum_{a=1}^{N} \varepsilon_a$

(31)

Hidden in Eq.(31) is the sum over all possible quantum states for each particle. Let $\displaystyle j_a$ represent the state index for particle $\displaystyle a$ , and $\displaystyle \varepsilon_{a , j_a}$ the energy corresponding to state $\displaystyle j_a$ of particle $\displaystyle a$ . Then

$\displaystyle \varepsilon_{j_1, \cdots , j_N} = \sum_{a=1}^{N} \varepsilon_{a, j_a}$

(32)

is the energy corresponding to the system state identified by the n-tuple $\displaystyle (j_1 , \cdots , j_N)$ .

The partition function (or sum-over-states) of particle $\displaystyle a$ is of the form

$\displaystyle q_a = \sum_{\stackrel{j_a}{{\rm (states)}}} \exp (- \beta \varepsilon_{a , j_a})$

(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)

$\begin{align} \displaystyle Q & = \sum_{\stackrel{(j_1 , \cdots , j_N)}{{\rm (states)}}} \exp (- \beta \varepsilon_{j_1, \cdots , j_N}) = \sum_{\stackrel{(j_1 , \cdots , j_N)}{{\rm (states)}}} \exp ( - \beta \sum_{a=1}^{N} \varepsilon_{a, j_a} ) = \sum_{\stackrel{(j_1 , \cdots , j_N)}{{\rm (states)}}} \prod_{a=1}^{N} \exp (- \beta \varepsilon_{a , j_a}) \\ & = \prod_{a=1}^{N} \sum_{\stackrel{j_a}{{\rm (states)}}} \exp (- \beta \varepsilon_{a , j_a}) = \prod_{a=1}^{N} q_a \end{align}$

(34)

In addition to being independent and distinguishable, if the particles are also identical, then

$\displaystyle q_1 = \cdots = q_N = q$

(35)

and

$\displaystyle Q = q^N$

(36)

Indistinguishable particles

Quantum-mechanically identical particles are indistinguishable. Roughly speaking, each n-tuple $\displaystyle (j_1 , \cdots , j_N)$ has $\displaystyle N!$ identical permutations, and thus the partition function $\displaystyle Q$ in Eq.(36) should be divided by $\displaystyle N!$ , i.e.,

$\displaystyle Q = \frac {q^N} {N!}$

(51)

The justification for Eq.(51) is actually more sophisticated. Consider identical, indistinguishable particles, labeled $\displaystyle \{ a, b, c, \cdots \}$ for the convenience of making the argument. Consider different quantum states labeled $\displaystyle \{ j_a , j_b , j_c , \cdots\}$ with $\displaystyle j_a \ne j_b \ne j_c \ne \cdots$ . By permutations, in the partition function $\displaystyle q^N$ in Eq.(36), there are $\displaystyle N!$ identical terms of the form

$\displaystyle \exp \left[ - \beta ( \varepsilon_{a, j_a} + \varepsilon_{b, j_b} + \varepsilon_{c, j_c} + \cdots ) \right]$

(52)

Only one term among the $\displaystyle N!$ should be counted in the partition function.

But there also terms such as

$\displaystyle \exp \left[ - \beta ( \varepsilon_{a, {\color{Red}\underset{=}{j_a}}} + \varepsilon_{b, {\color{Red}\underset{=}{j_a}}} + \varepsilon_{c, j_c} + \cdots ) \right]$

(53)

where the energy level of the first two particles are the same; by permutations of the last $\displaystyle (N-1)$ particles, there are $\displaystyle (N-1)!$ 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 Bose-Einstein statistics for bosons, but not allowed in the Fermi-Dirac 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, $\displaystyle 10 k_B T$ , much larger than the number of particles $\displaystyle N$ , 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

$\displaystyle Q \approx \frac {q^N} {N!}$

(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 Bose-Einstein statistics and the Fermi-Dirac statistics as temperature increases $\displaystyle T \rightarrow \infty$ .

Ideal monoatomic gases

There are $\displaystyle N$ independent and indistinguishable (quantum-mechanically identical) particles in a cubic box of side length $\displaystyle L$ . To compute the partition function $\displaystyle Q_{class} = \frac{q^N}{N!}$ 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 3-D time-independent Schrödinger equation for a particle of mass $\displaystyle m$ , in a box, as given

$\displaystyle - \frac {\hbar^2} {2m} {\rm div} \psi + U(x) \psi = \varepsilon \psi$

(61)

with zero potential inside the box, i.e., $\displaystyle U(x) = 0$ , we obtain the following energy levels (eigenvalues)

$\displaystyle \varepsilon_{l_x , l_y , l_z} = \frac {h^2 (l_x^2 + l_y^2 + l_z^2)} {8 m L^2}$

(62)

where $\displaystyle \{ l_x , l_y , l_z \} \in \mathbb N^3$ are the quantum numbers, which take natural values in $\displaystyle \mathbb N$ , i.e., $\displaystyle l_x, l_y , l_z = 1, 2, 3, \cdots$ .

Condition for approximation of partition function

As mentioned above, the number of quantum states $\displaystyle \Phi$ available between the ground state and the ground state plus $\displaystyle 10 k_B T$ should be much larger than the number of particles $\displaystyle N$ for the approximation in Eq.(54) to be valid, i.e.,

$\displaystyle \Phi \gg N$

(63)

Thus if we can connect the number $\displaystyle \Phi$ of quantum states to a given maximum energy level $\displaystyle \varepsilon_0$ , then we can establish an energetic condition for which the approximation in Eq.(54) is valid.

Consider Eq.(62) and the 3-D space of quantum numbers $\displaystyle \{ l_x , l_y , l_z \} \in \mathbb N^3$ . Each point of natural-number 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 $\displaystyle (l_x^2 + l_y^2 + l_z^2)$ in Eq.(62), let's consider a sphere in the space of quantum numbers, centered at the origin, and having a radius $\displaystyle R$ such that

$\displaystyle R^2 = (l_x^2 + l_y^2 + l_z^2) = \frac {8 m L^2 \varepsilon_{l_x, l_y, l_z}} {h^2}$

(64)

Setting $\displaystyle \varepsilon_{l_x, l_y, l_z} = \varepsilon_0$ , we would have the expression of the radius $\displaystyle R_0$ such that the quantum states on the surface of that sphere would have the energy level $\displaystyle \varepsilon_0$

$\displaystyle R_0 = \sqrt{ \frac {8 m L^2 \varepsilon_0} {h^2} }$

(65)

Fig.2. Particle in a box, quantum states, degeneracy

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 $\displaystyle R_0$ contains all quantum states with energy levels less than $\displaystyle \varepsilon_0$ , i.e., this volume is equal to the number of quantum states with energy less than $\displaystyle \varepsilon_0$ :

$\displaystyle \Phi (\varepsilon_0) = \frac{1}{8} \frac{4 \pi R_0^3}{3} = \frac{\pi R_0^3}{6} = \frac{\pi}{6} \left( \frac {8 m L^2 \varepsilon_0} {h^2} \right)^{3/2} = \frac{\pi}{6} \left( \frac {8 m \varepsilon_0} {h^2} \right)^{3/2} V$

(66)

with $\displaystyle V=L^3$ being the volume of the cube of length $\displaystyle L$ . Fig.2 illustrates the quantum states in the space of quantum numbers $\displaystyle (l_x , l_y)$ : The circle with radius $\displaystyle R_0$ corresponds to the energy level $\displaystyle \varepsilon_0$ ; the quantum states outside the band corresponding to the energy levels $\displaystyle \varepsilon_0$ and $\displaystyle \varepsilon_0 + d \varepsilon$ 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 $\displaystyle \varepsilon_0$ ; cf. McQuarrie (2000), p.11, Hill (1986), p.75.

Now, take $\displaystyle \varepsilon_0$ of the order of $\displaystyle k_B T$ , i.e.,

$\displaystyle \varepsilon_0 = \mathcal O (k_B T)$

(67)

then the condition in Eq.(63) becomes

$\displaystyle \frac{\pi}{6} \left( \frac {8 m k_B T} {h^2} \right)^{3/2} V \gg N \ {\rm or} \ \frac {V} {N} \gg \frac {6} {\pi} \left( \frac {2 \pi} {8} \right)^{3/2} \left( \frac {h} {\sqrt{2 \pi m k_B T}} \right)^3 = 1.33 \ \Lambda^3$

(68)

where $\displaystyle \Lambda$ is the thermal de Broglie wavelength in Eq.(8). The factor 1.33 can be dismissed from the inequality in Eq.(68), which becomes

$\displaystyle \left( \frac {V} {N} \right)^{1/3} \gg \Lambda$

(69)

i.e., for the partition function $\displaystyle Q$ in Eq.(51) to be a good approximation, the average distance between the particles should be much greater than the thermal de Broglie wavelength $\displaystyle \Lambda$ ; 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 $\displaystyle Q$ in Eq.(51).^[40]

Discrete energy, continuous energy

Here, there is a potential confusion due to the use of the notation $\displaystyle \omega$ to designate the degeneracy^[41] in both the discrete (quantum) energy case and in the continuous energy case.

For the discrete-energy 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 $\displaystyle \mathbb N$ , e.g., $\displaystyle k = 1 , 2 , 3, \cdots$ . (2b) Without a summation index, but using the notation for energy $\displaystyle \varepsilon$ to designate a discrete variable that takes values in the set of distinct energy levels, i.e., $\displaystyle \varepsilon \in \{ \varepsilon_1 , \varepsilon_2 , \cdots \}$ , such that $\displaystyle \varepsilon_1 \ne \varepsilon_2 \ne \cdots$ , i.e., each energy level $\displaystyle \varepsilon_k$ has a different value of energy; there is no value that is repeated. The 3 ways of writing the summation in the partition function $\displaystyle q$ (i.e., 1, 2a, 2b above) for a single particle are presented below

$\displaystyle q = \sum_{\stackrel{j}{{\rm (states)}}} \exp (- \beta \varepsilon_j) = \sum_{\stackrel{k}{{\rm (levels)}}} \omega_k \exp (- \beta \varepsilon_k) = \sum_{\stackrel{\varepsilon}{{\rm (levels)}}} \omega(\varepsilon) \exp (- \beta \varepsilon)$

(70)

For the continuous-energy case, some authors wrote the partition function $\displaystyle q$ as [Hill (1986), p.77; McQuarrie (2000), p.82]

$\displaystyle q = \int\limits_0^\infty \omega(\varepsilon) \exp (- \beta \varepsilon) d \varepsilon$

(*)

Here is the confusion: The quantity $\displaystyle \omega (\varepsilon)$ in Eq.(70) is the degeneracy at the energy level $\displaystyle \varepsilon$ , i.e., the number of quantum states at the same energy level $\displaystyle \varepsilon$ , whereas the quantity $\displaystyle \omega (\varepsilon)$ in Eq.(*) is not the degeneracy, but the number of quantum states per unit energy at the energy level $\displaystyle \varepsilon$ . McQuarrie (2000), p.82, called the factor $\displaystyle \omega (\varepsilon) d \varepsilon$ in Eq.(*) the "effective degeneracy", which is not immediately clear at first encounter. The dimensions of these two $\displaystyle \omega$ '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 $\displaystyle \overline \omega$ , for the number of quantum states per unit energy:

$\displaystyle q = \int\limits_0^\infty \overline \omega(\varepsilon) \exp (- \beta \varepsilon) d \varepsilon$

(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

$\displaystyle q = \sum_{\stackrel{l_x, l_y, l_z}{{\rm (states)}}} \exp (- \beta \varepsilon_{l_x, l_y, l_z})$

(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 $\displaystyle (l_x, l_y, l_z)$ . Such is the case if

$\displaystyle \beta \Delta \varepsilon = \frac {\Delta \varepsilon} {k_B T} \ll 1$

(73)

with $\displaystyle \beta$ defined in Eq.(2), and $\displaystyle \Delta \varepsilon$ the increment in energy level due to an increment of the indices $\displaystyle (l_x, l_y, l_z)$ . Based on the expression of the energy level in Eq.(62), consider a unit increment of the quantum numbers from $\displaystyle ( \hat l_x, \hat l_y, \hat l_z )$ to $\displaystyle ( \hat l_x + 1, \hat l_y, \hat l_z )$ , the increment in the energy level is of order

$\displaystyle \Delta \varepsilon = \mathcal O \left( \frac {h^2} {8 m L^2} \right) = \mathcal O \left( \frac {h^2} {8 m V^{2/3}} \right)$

Thus, using the expression for the thermal de Broglie wavelength $\displaystyle \Lambda$ in Eq.(2), we have

$\displaystyle \frac {\Delta \varepsilon} {k_B T} = \mathcal O \left( \frac {h^2} {8 m k_B T V^{2/3}} \right) = \mathcal O \left( \frac {\Lambda^2} {V^{2/3}} \right)$

(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

$\displaystyle \frac {\Delta \varepsilon} {k_B T} = \mathcal O \left( \frac {\Lambda^2} {V^{2/3}} \right) \ll \mathcal O \left( \frac {1} {N^{2/3}} \right)$

(75)

If $\displaystyle N$ is of the order of the Avogadro number, then

$\displaystyle \frac {\Delta \varepsilon} {k_B T} \ll \mathcal O ((10^{23})^{-2/3}) = \mathcal O (10^{-15})$

(76)

which largely satisfies the condition in Eq.(73)^[42], so that the summation in the partition function $\displaystyle q$ 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 $\displaystyle \overline \omega$ for the number of quantum states per unit energy as shown in Eq.(71), the "effective degeneracy" can be written as

$\displaystyle \omega (\varepsilon, d \varepsilon) = \overline \omega (\varepsilon) d \varepsilon$

(81)

We note immediately that the effective degeneracy $\displaystyle \omega (\varepsilon, d \varepsilon)$ is not the same as the degeneracy $\displaystyle \omega(\varepsilon)$ , hence the difference in notation.

As illustrated in Fig.1, the effective degeneracy $\displaystyle \omega (\varepsilon_0, d \varepsilon)$ is the number of quantum states lying inside the band formed by the circle with radius $\displaystyle R_0$ corresponding to the energy level $\displaystyle \varepsilon_0$ and a (slightly) larger circle corresponding to the energy level $\displaystyle \varepsilon_0 + d \varepsilon$ ; Hill (1986), p.77; McQuarrie (2000), p.11. We have

$\displaystyle \omega (\varepsilon_0, d \varepsilon) = \overline \omega (\varepsilon_0) d \varepsilon = \frac {d \Phi (\varepsilon_0)} {d \varepsilon} d \varepsilon = \frac {\pi} {4} \left( \frac {8 m} {h^2} \right)^{3/2} V \sqrt{\varepsilon_0} \ d \varepsilon$

(82)

To give an idea about the magnitude of the effective degeneracy, consider the following numerical data with $\displaystyle \varepsilon_0 = k_B T$ (in SI units)

Boltzmann constant $\displaystyle k_B = 1.38 \times 10^{-23} \, J \cdot K^{-1}$

Temperature $\displaystyle T = 300^\circ K$

Mass $\displaystyle m = 10^{-22} g = 10^{-25} Kg$

Planck constant $\displaystyle h = 6.62 \times 10^{-34} \, J \cdot s$

Box length $\displaystyle L = 10 cm = 10^{-1} m$

Increment of energy $\displaystyle d \varepsilon = 0.01 \varepsilon_0$

If we just look at the order of magnitude, then

$\displaystyle k_B = O(10^{-23}) \, J \cdot K^{-1}$

$\displaystyle T = O(100)^\circ K$

$\displaystyle \varepsilon_0 = k_B T = O(10^{-21}) J$

$\displaystyle m = O(10^{-25}) Kg$

$\displaystyle h = O(10^{-34}) \, J \cdot s$

$\displaystyle V = L^3 = O(10^{-3}) m$

and thus

$\displaystyle \omega(\varepsilon_0, d \varepsilon) = O \left[ \left( \frac {10^{25}} {(10^{-34})^2} \right)^{3/2} \cdot 10^{-3} \cdot (10^{-21})^{1/2} \cdot 10^{-2} \cdot 10^{-21} \right] = O(10^{28})$

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 $\displaystyle \overline \omega (\varepsilon)$ , i.e., the number of quantum states per unit energy, is then

$\displaystyle \overline \omega(\varepsilon_0) = O(10^{50})$

which is much larger than the effective degeneracy $\displaystyle \omega (\varepsilon_0, d \varepsilon)$ .

Partition function, thermal de Broglie wavelength

Using Eq.(82), the partition function $\displaystyle q$ in Eq.(71) for a particle in a box can now be evaluated as follows (Hill (1986), p.77)

$\displaystyle q = \frac {\pi} {4} \left( \frac {8 m} {h^2} \right)^{3/2} V \int\limits_0^\infty \sqrt{\varepsilon} \exp (- \beta \varepsilon) \ d \varepsilon = \left( \frac {2 \pi m k_B T} {h^2} \right)^{3/2} V = \frac {V} {\Lambda^3}$

(91)

where $\displaystyle \Lambda$ , 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:

$\displaystyle \Gamma (x) := \int\limits_{t=0}^{t=\infty} t^{x-1} e^{-t} dt \ {\rm with} \ x > 0 \Longrightarrow \Gamma (\frac{3}{2}) = \int\limits_{t=0}^{t=\infty} \sqrt{t} e^{-t} dt \ {\rm and} \ \int\limits_{t=0}^{t=\infty} \sqrt{t} e^{-at} dt = \frac {1} {a^{3/2}} \Gamma (\frac{3}{2})$

(92)

Integrating by parts the Gamma function in Eq.(92)₁, we obtain:

$\displaystyle \Gamma (x) = \frac{1}{x} \Gamma (x+1) \Longrightarrow \Gamma (x+1) = x \Gamma (x) \Longrightarrow \Gamma (\frac{3}{2}) = \frac{1}{2} \Gamma (\frac{1}{2})$

(93)

Next, by changing the variable $\displaystyle t = y^2$ , we obtain

$\displaystyle \Gamma (\frac{1}{2}) = \int\limits_{t=0}^{t=\infty} t^{-1/2} e^{-t} dt = 2 \int\limits_{y=0}^{y=\infty} e^{- y^2} dy = \sqrt{\pi}$

(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 $\displaystyle \Lambda$ has the dimension of length (of course): In the numerator of $\displaystyle \Lambda$ , the Planck constant $\displaystyle h$ has the dimension of energy times time, i.e., force $\displaystyle F$ times length $\displaystyle L$ times time $\displaystyle T$ . In the denominator of $\displaystyle \Lambda$ , the term $\displaystyle k_B T$ has the dimension of energy, i.e., force $\displaystyle F$ times length $\displaystyle L$ , or equivalently mass $\displaystyle m$ times velocity squared $\displaystyle v^2$ . Thus, the denominator has the dimension of mass $\displaystyle m$ times velocity $\displaystyle v$ , or momentum. The dimension of $\displaystyle \Lambda$ is then

$\displaystyle [\Lambda] = \frac {[h]} {[m] [v]} = \frac {FLT} {(F L^{-1} T^2) \cdot (L T^{-1})} = L$

(95)

See further reading on $\displaystyle \Lambda$ .

N particles in a box, partition function

The partition function of a system with $\displaystyle N$ independent, identical, indistinguishable particles can now be written as

$\displaystyle Q = \frac {q^N} {N!} = \frac {1} {N!} \left( \frac {V} {\Lambda^3} \right)^{N}$

(101)

It can be verified that in the absence of a potential energy, i.e., $\displaystyle U = 0$ (since the particles are independent; there is no interparticle forces), the configuration integral $\displaystyle Z_N = V^N$ , and the partition function $\displaystyle Q_{class}$ in Eq.(6) is reduced to Eq.(101).

Helmholtz energy

From Eq.(9), the Helmholtz energy^[10]^[26] $\displaystyle A$ of this system can now be written as

$\begin{align} \displaystyle A & = - k_B T \ \log Q = - k_B T \ \left( - \log N! + N \, \log q \right) \approx k_B T \left( N \log N - N - N \, \log q \right) \\ & = - k_B T N \ \log \left( \frac {q e} {N} \right) \end{align}$

by using the Stirling approximation for $\displaystyle \log N!$ , i.e.,

$\displaystyle \log N! \approx N \log N - N$

and $\displaystyle \log e = 1$ . Thus,

$\displaystyle A = - k_B T N \log \left[ \frac {V e} {\Lambda^3 N} \right] = - k_B T N \log \left[ \frac {(2 \pi m k_B T)^{3/2} V e} {h^3 N} \right]$

(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 $\displaystyle A (T,V,N)$ is available, one can then obtain the expressions for the thermodynamic properties of the canonical ensemble, i.e., entropy $\displaystyle S$ , pressure $\displaystyle p$ , chemical potentials $\displaystyle \mu_\alpha$ . In addition, since $\displaystyle A = U - TS$ , we can also obtain the expression for the total internal energy $\displaystyle U$ of the system.

Recall that the independent variables of the internal energy $\displaystyle U$ for the canonical ensemble are entropy $\displaystyle S$ , volume $\displaystyle V$ , and the particle numbers $\displaystyle \{ N_\alpha \}$ for different components (or species); we write $\displaystyle U(S,V,\{ N_\alpha \})$ . We have (McQuarrie (2000), p.17)

$\displaystyle dU = T dS - p dV + \sum_{\alpha} \frac {\partial U} {\partial N_\alpha} d N_\alpha = T dS - p dV + \sum_{\alpha} \mu_\alpha d N_\alpha , \ {\rm with} \ \mu_\alpha := \frac {\partial U} {\partial N_\alpha}$

(111)

being the chemical potential for species $\displaystyle \alpha$ .

With the Legendre transformation

$\displaystyle A = U - TS$

(112)

we have

$\displaystyle dA = dU - TdS - SdT = - SdT - p dV + \sum_{\alpha} \mu_\alpha d N_\alpha$

(113)

making $\displaystyle (T,V,\{N_\alpha\})$ the independent variables for the Helmholtz energy $\displaystyle A$ . From Eq.(113) and using Eq.(9), we have (cf. Hill (1986), p.19)

$\displaystyle S = - \left. \frac {\partial A} {\partial T} \right|_{V,N_\alpha} = k_B \log Q + k_B T \left( \left. \frac {\partial \log Q} {\partial T} \right|_{V,N_\alpha} \right)$

(114)

$\displaystyle p = - \left. \frac {\partial A} {\partial V} \right|_{T,N_\alpha} = k_B T \left( \left. \frac {\partial \log Q} {\partial V} \right|_{T,N_\alpha} \right)$

(115)

$\displaystyle \mu_\alpha = \left. \frac {\partial A} {\partial N_\alpha} \right|_{T,V,\{N_\gamma, \gamma \ne \alpha\}} = - k_B T \left( \left. \frac {\partial \log Q} {\partial N_\alpha} \right|_{T,V,\{N_\gamma, \gamma \ne \alpha\}} \right)$

(116)

Finally, using Eq.(114) in Eq.(9), we obtain the expression for the total internal energy $\displaystyle U$

$\displaystyle U = A + TS = k_B T^2 \left( \left. \frac {\partial \log Q} {\partial T} \right|_{V,\{N_\alpha\}} \right) = - \left. \frac {\partial \log Q} {\partial \beta} \right|_{V,\{N_\alpha\}}$

(117)

with $\displaystyle \beta = 1/(k_B T)$ defined in Eq.(2).

Now using the expression for the Helmholtz energy $\displaystyle A$ 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 $\displaystyle N$ distinguishable, identical, and independent particles):

$\displaystyle S = - \left. \frac {\partial A} {\partial T} \right|_{V,N_\alpha} = \frac {\partial [bT \log (a T^{3/2})]} {\partial T} = b \log (a T^{3/2}) + b T \frac{3}{2} T^{-1} = b \log (a T^{3/2} e^{3/2})$

where $\displaystyle a$ and $\displaystyle b$ are constants with respect to $\displaystyle T$ ; see Eq.(102). Thus, the entropy for ideal monoatomic gas takes the form (cf. Hill (1986), p.79)

$\displaystyle S = k_B N \log \left[ \frac {(2 \pi m k_B T)^{3/2} V e^{5/2}} {h^3 N} \right]$

(118)

Similarly, the pressure takes the familiar form of the ideal gas law:

$\displaystyle p = - \left. \frac {\partial A} {\partial V} \right|_{T,N_\alpha} = \frac {N k_B T} {V}$

(119)

Chemical potential for the case with one species:

$\displaystyle \mu = \left. \frac {\partial A} {\partial N} \right|_{T,V} = - k_B T \log \left[ \frac {(2 \pi m k_B T)^{3/2} V} {h^3 N} \right] = - k_B T \log \left[ \frac {(2 \pi m k_B T)^{3/2} k_B T} {h^3 p} \right]$

(120)

where Eq.(119) had been used in the last equation. Thus, (cf. Hill (1986), p.80)

$\displaystyle \mu = \mu_0 (T) + k_B T \log p , \ {\rm with} \ \mu_0 (T) = - k_B T \log \left[ \frac {(2 \pi m k_B T)^{3/2} k_B T} {h^3} \right]$

(121)

The internal energy $\displaystyle U$ of an ideal monoatomic gas follows from the expression for $\displaystyle A$ in Eq.(102) and Eq.(112), and the expression for $\displaystyle S$ in Eq.(118)

$\displaystyle U = A + TS = N k_B T \ \log e^{3/2} = \frac {3} {2} N k_B T$

(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 $\displaystyle \frac{k_B T}{2}$ (there are $\displaystyle 3N$ momentum dofs)^[44]. Of course, the same result shown in Eq.(122) can be obtained by differentiating the partition function $\displaystyle Q$ using Eq.(117)₂ or Eq.(117)₃. Eq.(119) for pressure $\displaystyle p$ and Eq.(122) for internal energy $\displaystyle U$ are called the "classical results" since they can be derived from using only the kinetic theory, without a need for a quantum-mechanical setting. Here, we recover the classical results starting from a quantum-mechanical 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 $\displaystyle j$ is^[24]

$\displaystyle \varphi_j = \frac {\exp (-\beta \varepsilon_j)} {q} = \frac {\exp (-\beta \varepsilon_j)} { \displaystyle \sum_{\stackrel{j}{{\rm (states)}}} \exp (- \beta \varepsilon_j) } \ \mathrm{such \ that} \ \sum_{\stackrel{j}{{\rm (states)}}} \varphi_j = 1$

(123)

and the average energy of a single particle is (cf. Hill (1986), p.12)

$\displaystyle \langle \varepsilon \rangle = \sum_{\stackrel{j}{{\rm (states)}}} \varepsilon_j \varphi_j = \frac {1}{q} \sum_{\stackrel{j}{{\rm (states)}}} \varepsilon_j {\exp (-\beta \varepsilon_j)}$

(124)

For the continous energy case, we obtain from Eq.(71) the probability for a single particle to be within the energy band $\displaystyle [\varepsilon , \varepsilon + d \varepsilon]$ as (Hill (1986), p.77)

$\displaystyle \varphi (\varepsilon) d \varepsilon = \frac { \overline \omega(\varepsilon) \exp (- \beta \varepsilon) d \varepsilon } {q} = \frac { \overline \omega(\varepsilon) \exp (- \beta \varepsilon) d \varepsilon } { \displaystyle \int\limits_0^\infty \overline \omega(\varepsilon) \exp (- \beta \varepsilon) d \varepsilon }$

(125)

and the average energy of a single particle as

$\displaystyle \langle \varepsilon \rangle = \int\limits_0^\infty \varepsilon \varphi (\varepsilon) d \varepsilon = \frac {1}{q} \int\limits_0^\infty \varepsilon \overline \omega(\varepsilon) \exp (- \beta \varepsilon) d \varepsilon$

(126)

Next, using Eq.(91), i.e., the expression for $\displaystyle q$ , in Eq.(126), we have the average energy of a single particle as (after cancelling out the common factor)

$\displaystyle \langle \varepsilon \rangle = \frac { \int\limits_0^\infty \varepsilon^{3/2} \exp (- \beta \varepsilon) \ d \varepsilon } { \int\limits_0^\infty \varepsilon^{1/2} \exp (- \beta \varepsilon) \ d \varepsilon } = \frac {\Gamma (5/2) / \beta^{5/2}} {\Gamma (3/2) / \beta^{3/2}} = \frac {(3/2) \Gamma (3/2) / \beta} {\Gamma (3/2)} = \frac{3}{2} k_B T$

(127)

where we have made use of Eq.(92)₃ and Eq.(93)₂. For a system with $\displaystyle N$ 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.

One-dimensional harmonic oscillator

Energy

Classical case (continuous energy)

The potential energy $\displaystyle U$ in a classical harmonic oscillator (spring-mass system) with linear force-displacement relationship is a quadratic form in the coordinate $\displaystyle x$

$\displaystyle U = \frac{1}{2} k x^2$

(131)

where $\displaystyle k$ is the spring stiffness coefficient ^[45], and $\displaystyle x$ the displacement.

From Eq.(21), the classical Hamiltonian is now written as

$\displaystyle H(x,p) = \frac {p^2} {2 m} + \frac {k x^2} {2}$

(132)

The classical frequency $\displaystyle \nu$ of this simple oscillator takes the form ^[46]

$\displaystyle \nu = \frac{1}{2} \sqrt{\frac{k}{m}}$

(133)

Thus, the Hamiltonian can be written in terms of the frequency $\displaystyle \nu$ as follows (cf. Hill (1986), p.113)

$\displaystyle H = \frac {p^2} {2 m} + 2 \pi^2 m \nu^2 x^2$

(134)

The Hamiltonian $\displaystyle H$ in Eq.(134) is the total internal energy of the system, and is continuous in terms of the phase-space variables $\displaystyle (x,p)$ .

Quantum case (discrete energy)

The energy levels of a quantum harmonic oscillator, obtained by solving Eq.(27a), are discrete and non-degenerate

$\displaystyle \varepsilon_n = \left( n + \frac{1}{2} \right) h \nu = \left( n + \frac{1}{2} \right) \hbar \omega , \ {\rm for} \ n = 0, 1, 2, \cdots$

(141)

where $\displaystyle h$ is the Planck constant, $\displaystyle \nu$ the classical frequency in Eq.(133), $\displaystyle \hbar = \frac{h}{2 \pi}$ , and $\displaystyle \omega = \frac{2 \pi}{\nu}$ 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

$\displaystyle q = \sum_{n=0}^{\infty} \exp (- \beta \varepsilon_n) = \exp \left( - \frac{h \nu}{2 k_B T} \right) \sum_{n=0}^{\infty} \exp \left( - \frac{n h \nu}{k_B T} \right) = \exp \left( - \frac{h \nu}{2 k_B T} \right) \sum_{n=0}^{\infty} \left[ \exp \left( - \frac{h \nu}{k_B T} \right) \right]^n$

$\displaystyle = \frac {\exp (- h \nu / 2 k_B T)} {1 - \exp( - h \nu / k_B T)} = \frac {\exp (- u/2)} {1 - \exp( - u)} \ {\rm with} \ u := \frac {h \nu} {k_B T} = \frac {\hbar \omega} {k_B T}$

(151)

As temperature rises, i.e., $\displaystyle T \rightarrow \infty$ , thus $\displaystyle u \rightarrow 0$ ; a development in Taylor series leads to an asymptotic function in term of $\displaystyle T$ for the partition function $\displaystyle q$ (cf. Hill (1986), p.89)

$\displaystyle q = \frac {1 - \frac{u}{2} + \frac{1}{2} \left(\frac{u}{2}\right)^2 + \cdots} {1- \left( 1 - u + \frac{1}{2} u^2 + \cdots \right)} \rightarrow \frac {1} {u} = \frac {k_B T} {h \nu} = \frac {k_B T} {\hbar \omega} \ {\rm as} \ u \rightarrow 0$

(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 $\displaystyle u \rightarrow 0$ is very small, $\displaystyle \exp (-u)$ is nearly equal to 1, and thus $\displaystyle [\exp(-u)]^n$ would be essentially continuous with $\displaystyle n$ , i.e., the increment

$\displaystyle [\exp(-u)]^{n+1} - [\exp(-u)]^{n}$

is very small. In this case, the summation in Eq.(151) can be approximated by an integration, i.e.,

$\displaystyle q \approx \exp \left( - \frac{h \nu}{2 k_B T} \right) \int\limits_{n=0}^{\infty} \left[ \exp \left( - \frac{h \nu}{k_B T} \right) \right]^n d n = \exp \left( - \frac{h \nu}{2 k_B T} \right) \frac {1} {\log \exp (h \nu / k_B T)} \rightarrow \frac {k_B T} {h \nu}$

(153)

since

$\displaystyle \int\limits_{n=0}^{\infty} \left[ \exp \left( - \frac{h \nu}{k_B T} \right) \right]^n d n = \int\limits_{n=0}^{n=\infty} a^{-n} d n = - \int\limits_{y=1}^{y=0} y \frac {d y} {y \log a} = \frac {1} {\log a}$

with

$\displaystyle a := \exp \left( \frac{h \nu}{k_B T} \right) \ {\rm and} \ y := a^{-n}$

Average quantum energy, failure of equipartition

Fig.3. Quantum correction to equipartition at low temperature: Quantum harmonic oscillator (red), equipartition (blue), electromagnetic oscillator (green).

First, using Eq.(70)₁, we can rewrite Eq.(124) as follows

$\displaystyle \langle \varepsilon \rangle = \frac{1}{q} \sum_{n} \varepsilon_n \exp (- \beta \varepsilon_n) = \frac{1}{q} \left( - \frac {\partial q} {\partial \beta} \right) = - \frac {\partial \log q} {\partial \beta}$

(156)

Compare Eq.(156)₃ to Eq.(117)₃. Next, with the partition function for a quantum harmonic oscillator $\displaystyle q$ 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)

$\displaystyle \langle \varepsilon \rangle = \frac {h \nu} {2} \left[ \frac {1 + \exp (- \beta h \nu)} {1 - \exp (- \beta h \nu)} \right] \rightarrow k_B T \ {\rm as} \ T \rightarrow \infty$

(157)

with the limit of $\displaystyle \langle \varepsilon \rangle$ as $\displaystyle T \rightarrow \infty$ being the result of using the same method as in Eq.(152). Thus we have equipartition of energy for high temperature $\displaystyle T$ , half from the kinetic energy and half from the potential energy (cf. Eq.(127) for a particle in a 3-D box where there was only kinetic energy, with zero potential energy). At low temperature $\displaystyle T$ , 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 black-body radiation)^[47].

Classical case, phase integral

A logical "generalization" of the quantum partition function for discrete energy levels in Eq.(151)₁, i.e.,

$\displaystyle q = \sum_{n=0}^{\infty} \exp (- \beta \varepsilon_n)$

is to replace the quantum summation with an integral for continuous energy to obtain the classical partition function $\displaystyle q_{class}$ . But it is not that simple; to make sure that the classical partition function $\displaystyle q_{class}$ agrees with the quantum partition function in the limit when $\displaystyle T \rightarrow \infty$ , i.e.,

$\displaystyle \frac {k_B T} {h \nu} = \frac {k_B T} {\hbar \omega}$

in Eqs.(152)-(153), we need to add a proportionality constant $\displaystyle c$ in the expression for $\displaystyle q_{class}$ :

$\displaystyle q_{class} = c \iint\limits_{-\infty}^{+\infty} \exp[ - \beta H (x,p)] dx dp = c \iint\limits_{-\infty}^{+\infty} \exp \left[ - \beta \left( \frac {p^2} {2 m} + 2 \pi^2 m \nu^2 x^2 \right) \right] dx dp$

$\displaystyle = c \left( \int\limits_{-\infty}^{+\infty} \exp \left[ - \beta \frac {p^2} {2 m} \right] dp \right) \left( \int\limits_{-\infty}^{+\infty} \exp \left[ - \beta 2 \pi^2 m \nu^2 x^2 \right] dx \right) = c \sqrt{ \frac {\pi} {\beta / 2m} } \sqrt{ \frac {\pi} {\beta 2 \pi^2 m \nu^2} }$

$\displaystyle = \frac {c} {\beta \nu} = \frac {c k_B T} {\nu} = \frac {k_B T} {h \nu} \Longrightarrow c = \frac {1} {h}$

(161)

where we made use of the integration results

$\displaystyle \left( \int\limits_{x=-\infty}^{x=+\infty} \exp (-x^2) dx \right)^2 = \left( \int\limits_{x=-\infty}^{x=+\infty} \exp (-x^2) dx \right) \left( \int\limits_{y=-\infty}^{y=+\infty} \exp (-y^2) dy \right) = \iint\limits_{-\infty}^{+\infty} \exp [-(x^2 + y^2)] dx dy$

$\displaystyle = \int\limits_{r=0}^{+\infty} \int\limits_{\theta=0}^{2 \pi} \exp (- r^2) r dr d\theta = 2 \pi \int\limits_{u=0}^{u=+\infty} \exp (-u) \frac {du} {2} = \pi$

(162)

and

$\displaystyle \int\limits_{-\infty}^{+\infty} \exp(- a x^2) dx = \sqrt{ \frac {\pi} {a} }$

(163)

Thus (cf. Hill (1986), p.113)

$\displaystyle q_{class} = \frac {1} {h} \iint\limits_{-\infty}^{+\infty} \exp[ - \beta H (x,p)] dx dp$

(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

$\displaystyle q = \sum_{\stackrel{l_x, l_y, l_z}{{\rm (states)}}} \exp (- \beta \varepsilon_{l_x, l_y, l_z})$

At sufficiently high temperature, $\displaystyle T \rightarrow \infty$ , the condition in Eq.(73) is satisfied, i.e.,

$\displaystyle \frac {\Delta \varepsilon} {k_B T} \ll 1$

and the quantum partition function can be approximated by an integration, leading to the expression in Eq.(91), i.e.,

$\displaystyle q = \left( \frac {2 \pi m k_B T} {h^2} \right)^{3/2} V = \frac {V} {\Lambda^3}$

Consider the following "generalization" of Eq.(72) to the case with continuous energy

$\displaystyle q_{class} = c^\prime \int\limits_{-\infty}^{+\infty} \int\limits_{V} \exp (- \beta H) \; d^3 x \; d^3 p$

(171)

with the Hamiltonian (or energy) being simply the kinetic energy (the potential energy is zero)

$\displaystyle H = \frac {p_x^2 + p_y^2 + p_z^2} {2 m}$

(172)

and

$\displaystyle d^3 x = d x d y d z \ {\rm and} \ d^3 p = d p_x d p_y d p_z$

(173)

Using the integration result in Eq.(163), and the fact that the classical partition function $\displaystyle q_{class}$ is the limiting case of the quantum partition function $\displaystyle q$ as $\displaystyle T \rightarrow \infty$ , we have

$\displaystyle q_{class} = c^\prime V \left( \int\limits_{-\infty}^{+\infty} \exp (- \frac{p_x^2}{2m k_B T}) d p_x \right)^3 = c^\prime V \left( 2 \pi m k_B T \right)^{3/2} = \left( \frac {2 \pi m k_B T} {h^2} \right)^{3/2} V$

(174)

Thus,

$\displaystyle c^\prime = \frac {1} {h^3}$

(175)

Hence, for a particle in a box with 3 momentum dofs and 3 position dofs^[44] (see Hill (1986), p.115):

$\displaystyle q_{class} = \frac {1} {h^3} \int\limits_{-\infty}^{+\infty} \int\limits_{V} \exp (- \beta H) \; d^3 x \; d^3 p$

(176)

System of indistinguishable particles

Zero potential energy (without interparticle forces)

In general, for a particle system in a box with $\displaystyle n$ momentum dofs and with zero potential energy, the Hamiltonian is purely kinetic energy

$\displaystyle H = \frac {1} {2m} \sum_{i=1}^{n} p_i^2$

(181)

and the classical partition function takes the form

$\displaystyle Q_{class} = \frac {1} {h^n} \int\limits_{-\infty}^{+\infty} \int\limits_{V} \exp (- \beta H) \; d^n x \; d^n p$

(182)

For a system of $\displaystyle N$ particles in 3-D space $\displaystyle \mathbb R^3$ , we have $\displaystyle n = 3N$ , which represents both $\displaystyle 3N$ momentum dofs and $\displaystyle 3N$ position dofs^[44]. Without a zero potential energy, the integration can be carried out to yield

$\displaystyle Q_{class} = \left( \frac {V} {\Lambda^3} \right)^N$

(183)

The above partition function is for distinguishable particles. For indistinguishabe particles, a division by $\displaystyle N!$ yields the classical partition function

$\displaystyle Q_{class} = \frac{1}{N! h^{3N}} \int\limits_{-\infty}^{+\infty} \int\limits_{V} \exp (- \beta H) \; d^{3N} x \; d^{3N} p = \frac{1}{N!} \left( \frac {V} {\Lambda^3} \right)^N$

(184)

which is the same as in Eq.(101).

Fig.4. Double counting in phase integral: Two indistinguishable particles swapping locations in phase space; no new state.

An interpretation for the division by $\displaystyle N!$ in Eq.(184) can be given as follows. Consider the case with two particles moving in a 1-D space, with a 2-D phase space. The phase integral in Eq.(184) can be approximated by a quadruple summation

$\displaystyle \sum_{x_1 \in \mathcal X} \sum_{p_1 \in \mathcal P} \sum_{x_2 \in \mathcal X} \sum_{p_2 \in \mathcal P} \exp[-\beta H (x_1, p_1, x_2, p_2)] (d \tau)^2$

(185a)

where

$\displaystyle d\tau = dx \; dp$

(185b)

is an infinitesimal area in the phase space, as represented by a green square in Fig.4. The sets $\displaystyle \mathcal X$ and $\displaystyle \mathcal P$ are sets of discrete values of the continuous variables $\displaystyle x_i$ and $\displaystyle p_i$ , for $\displaystyle i=1,2$ :

$\displaystyle \mathcal X = \left\{ x_a , x_b, \cdots \right\} \ {\rm and} \ \mathcal P = \left\{ p_a , p_b, \cdots \right\}$

(186)

The phase-space coordinates of particle $\displaystyle i=1,2$ are denoted by $\displaystyle (x_i, p_i)$ . In the summation in Eq.(185a), there are two terms that correspond respectively to (i) particle 1 (red dot in Fig.4) at position $\displaystyle a$ in phase space, i.e., $\displaystyle (x_1, p_1) = (x_a, p_a)$ , and particle 2 (blue dot in Fig.4) at position $\displaystyle b$ in phase space, i.e., $\displaystyle (x_2, p_2) = (x_b, p_b)$ , and (ii) particle 2 (blue dot in Fig.4) at position $\displaystyle a$ in phase space, i.e., $\displaystyle (x_2, p_2) = (x_a, p_a)$ , and particle 1 (red dot in Fig.4) at position $\displaystyle b$ in phase space, i.e., $\displaystyle (x_1, p_1) = (x_b, p_b)$ . 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., $\displaystyle N=2$ .

For a system with $\displaystyle N$ indistinguishable particles, each state is counted $\displaystyle N!$ times, and thus a division by $\displaystyle N!$ in Eq.(184) (cf. Hill (1986), p.117). The above argument to justify for the factor $\displaystyle 1/N!$ 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 quantum-mechanical approach to obtain the factor $\displaystyle 1/N!$ ; 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 $\displaystyle 1/N!$ 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 $\displaystyle 1/h^{3N}$ —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 $\displaystyle 1/h^{3N}$ , with or without the correction factor $\displaystyle 1/N!$ , as the "classical partition function".

Non-zero potential energy (with interparticle forces)

When the potential energy is non-zero, i.e., there are interparticle forces, which is the more interesting and general case, the Hamiltonian is written as

$\displaystyle H = \frac {1} {2m} \sum_{i=1}^{N} \sum_{j=1}^{3} (p_i^j)^2 + U (x_1 , \cdots , x_N)$

(201)

where $\displaystyle p_i^j$ is the momentum of particle $\displaystyle i$ along the $\displaystyle j$ th coordinate direction, for $\displaystyle i=1, \cdots , N$ and $\displaystyle j=1,2,3$ , $\displaystyle U$ the potential energy, and $\displaystyle x_i = (x^1_i, x^2_i, x^3_i)$ $\displaystyle \equiv (x_i, y_i, z_i)$ the position of particle $\displaystyle i$ in 3-D space. In this case, the classical partition function $\displaystyle Q_{class}$ in Eq.(184)₁ becomes Eq.(6), after integrating out the kinetic energy part (first term in Eq.(201)), i.e.,

$\displaystyle Q_{class} = \frac{1}{N! h^{3N}} \int\limits_{-\infty}^{+\infty} \int\limits_{V} \exp (- \beta H) \; d^{3N} x \; d^{3N} p = \frac {1} {N! \Lambda^{3N}} \int\limits_{V} \exp (- \beta U) \; d^{3N} x = \frac {Z_N} {N! \Lambda^{3N}}$

(6)

When there the potential function $\displaystyle U = 0$ , we obtain $\displaystyle Z_N = V^N$ , thus recovering Eq.(184)₂; see Hill (1986), p.118.

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 up-to-date). Jones, E.R., Fahrenheit and Celsius, a history, The Physics Teacher, Nov. 1980, Vol.18, No.8, pp.594-595. 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.259-280.
↑ 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. HIV-1 evades antibody-mediated neutralization through conformational masking of receptor-binding sites, Nature 420, 678-682 (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, 648-659 (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 1876-1878, and by Lewis and Randall, Thermodynamics and the Free Energy of Chemical Substances, McGraw-Hill, 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": $\displaystyle \bigstar$ "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 $\displaystyle F$ for the two different functions," p.350 in M.W. Zemansky, Fashions in Thermodynamics, Am. J. Phys., Vol.25, No.6, pp.349-351, 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" $\displaystyle A$ is Huang (1987), p.22 and Index p.489. $\displaystyle \bigstar$ "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.753-755, 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 & Boerio-Goates (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 statistical-thermodynamic basis for computation of binding affinities: A critical review. Biophysical Journal 72: 1047-1069 (1997).
↑ M. K. Gilson and H.-X. Zhou, Calculation of Protein-Ligand Binding Affinities, Annual Review of Biophysics and Biomolecular Structure Vol. 36: 21-42 (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 "sum-over-states". 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.60-78), where his work in 1922 on the partition of energy was mentioned on p.69; his 1922 papers were co-authored 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 "Darwin-Fowler method" without mentioning the original papers. The Darwin-Fowler method is an exact derivation of the Maxwell-Boltzmann statistics without the need to appeal to the Stirling approximation of $\displaystyle N!$ . Tolman (1979), footnote p.567, also referred to the treatments of partition function by "Darwin and Fowler" without citing the original papers. Since the Darwin-Fowler 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.361-375. 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.53-57, 2002. For more historical details and the relationship between Fowler, Lennard-Jones, 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.187-212, 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.415-417, 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 English-German 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 "sum-over-states".
↑ For this reason ("Zustandssumme"), some authors use the letter $\displaystyle Z$ to denote the partition function, and the letter $\displaystyle Q$ the configuration integral; see, e.g., Reichl (1998). Kirkwood (1933) used $\displaystyle \sigma$ for "sum-over-states".
↑ ^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.860-861. 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. 164-167. 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 "sum-over-states" 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 $\displaystyle F$ , which stands for "Free", instead of $\displaystyle A$ , 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 $\displaystyle F$ 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 $\displaystyle H$ for enthalpy in "agreement with almost universal usage". The IUPAC (also from Wikipedia) recommends the symbol $\displaystyle A$ 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.2690-2693 (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.1285-1287.
↑ 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 Addison-Wesley 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 $\displaystyle \varepsilon$ is the multiplicity of the eigenvalue $\displaystyle \varepsilon$ obtained when solving the eigenvalue problem in Eq.(61).
↑ In Hill (1986), p.77, the order in Eq.(76) was $\displaystyle \mathcal O (10^{-14})$ instead of $\displaystyle \mathcal O (10^{-15})$ ; it is not clear which value of $\displaystyle N$ 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 $\displaystyle N$ particles in 3-D space $\displaystyle \mathbb R^3$ has $\displaystyle 6N$ dofs, $\displaystyle 3N$ momentum dofs and $\displaystyle 3N$ 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 $\displaystyle f$ .
↑ In classical-mechanics literature, the frequency is usually denoted by $\displaystyle f$ ; in quantum-mechanics literature, the frequency is denoted by $\displaystyle \nu$ .
↑ 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 $\displaystyle (n h \nu)$ (missing $\displaystyle (h \nu / 2)$ ) 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 $\displaystyle \langle \varepsilon \rangle$ as $\displaystyle T \rightarrow \infty$ 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 $\displaystyle T$ ; see Fig.3. On the other hand, the problem can be fixed based on a léger-de-main feat of Planck to explain black-body radiation: Here, just arbitrarily remove the problematic term $\displaystyle (h \nu)/2$ from the energy of the quantum harmonic oscillator (Tolman (1979), p.381); what is left, i.e., $\displaystyle (n h \nu)$ 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.51-53, Jan 1974.
↑ R.W. Zwanzig, Transition from Quantum to "Classical" Partition Function, Phys. Rev. 106, 13 - 15 (1957).