The Canonical Ensemble: Notion of Distribution

Abstract

The goal of statistical thermodynamics is to permit to appreciate the significance of the thermodynamic functions in terms of molecular parameters. Firstly, this chapter illustrates this point with the aid of the study of the canonical ensemble. It deals with the obtaining of the probabilities of the systems constituting the canonical ensemble to be in some energy states. It provides a description of the canonical ensemble and describes the followed strategy to calculate the average of the mechanical properties such as the pressure and energy with the help of the reasoning based on the fact that the mechanical variables have well-definite values in a given quantum state. It leads to the notion of distribution of the systems in the ensemble. It is the set of the numbers of systems found in well-defined energy states exhibiting the same composition (in one or several compounds) and the same volume. There can exist several distributions. Calculations, exemplified in the chapter, permit to obtain the elementary and global probabilities that a system of the ensemble would be in a definite energetic state. Once the probabilities are obtained, it becomes possible to calculate the canonical partition function.

The goal of statistical thermodynamics is to permit to appreciate the significance of the thermodynamic functions in terms of molecular parameters. Firstly, we choose to illustrate this point with the aid of the study of the canonical ensemble.

Actually, this chapter is necessary to introduce this theory. It deals with the obtaining of the probabilities of the systems constituting the canonical ensemble to be in some energy states. Obtaining these probabilities is the first necessary condition in order to be able, later, to specify the significance of some thermodynamic quantities.

The problem of the obtention of the probabilities is essentially not different from that of the determination of the distribution of the systems constituting the ensemble in the different possible energetic states. (To aim at the same goal, later, we shall consider the handling of other ensembles.)

1 Description of the Canonical Ensemble (N, V, T Imposed)

The canonical ensemble is constituted by a very large number ℵ (ℵ → ∞) of systems replicating the thermodynamic system (under study) which, by definition, possesses the fixed volume V, the number of molecules N (there can be several types of molecules, the numbers of which N₁, N₂ … are then constant), and the temperature T uniform and constant (viz. Fig. 22.1). The partitions between the different systems are thermal conductors but do not allow the crossing of the particles through them. The ensemble is placed in a heat bath granting an equal temperature in the whole systems. The partitions of the systems are not distorting excluding, hence, no work exchange between them.

Fig. 22.1

Canonical ensemble

If one places an isolating membrane outside the ensemble and the whole device (ensemble + membrane) located outside the heat bath, the ensemble, now, constitutes an isolated system of volume ℵV and of number of molecules ℵN and with a total energy E _t . This isolated system is called a supersystem.

2 Strategy

Let us recall that, finally, the goal is to find the meaning of some quantities of classical thermodynamics with the help of a reasoning of statistical thermodynamics, the meaning of which being searched for in the conditions which prevail in the canonical ensemble (constant composition, temperature, volume). According to what is preceding, the problem is to calculate the average of the mechanical properties such as the pressure and the energy with the help of this reasoning. Let us notice, indeed, that since the thermodynamic system is not isolated (it is in contact with other systems of the same ensemble), its energy fluctuates.

The process entails to know the value of the quantity under study in each quantum state and to determine the number of systems of the ensemble exhibiting this quantum state. The mechanical variables, indeed, have well-definite values in a given quantum state. Hence, the problem is to determine the fraction of the systems of the ensemble possessing a given quantum state.

These considerations are equivalent to say that the probability P _j that a system of the ensemble is in the state of energy E _j must be known. Once known, the values of the energy $\overline{E}$ and of the pressure $\overline{p}$ can be calculated through the following expressions:

$$ \begin{array}{l}\overline{E}={\displaystyle \sum_j{P}_j\;{E}_j}\\ {}\overline{p} = {\displaystyle \sum_j{P}_j\;{p}_j}\end{array} $$

p _j is the pressure in the energetic state E _j ; it is defined by the expression

$$ {p}_j=-{\left(\partial {E}_j/\partial V\right)}_N $$

−p _j dV = dE _j is the work that has to be done on the system (with a constant number of species N) in the energetic state E _j in order to increase its volume by dV. This expression is found by virtue of the quality of state function of E (viz. Appendix A). One can write, indeed,

$$ dE={\left(\partial E/\partial V\right)}_{N,T}dV+{\left(\partial E/\partial T\right)}_{N,\ V}dT $$

where by hypothesis dT = 0 (T imposed).

3 The Mathematical Problem

Let us, now, consider one system of the canonical ensemble. It is a system obeying quantum mechanics. Its characteristics depend on the values N and V which constitute the limits entailing the energy quantification (viz. quantum mechanics). As a result, there exists the collection of the following possible (authorized) energetic states written by order of increasing energy: E ₁, E ₂, …, E _j . We must not forget that they are the energy states of the whole system, that is to say of a great number of particles, and not the energy states of one species. Let us recall that, for different reasons (some of which being mathematical ones), it is not possible to calculate the energy states E _j from the Schrödinger’s equation for a very large number of particles. Nevertheless, for the following reasoning, we suppose that we know them.

3.1 The Notion of Distribution

Since all the systems of the canonical ensemble have the same composition N (in one or several compounds) and the same volume V, everyone does possess the same quantified levels of energy E ₁, E ₂, …, E _j. (It is a consequence of the principles of quantum mechanics.) Let us suppose that we can simultaneously observe the energetic state of each system and that we are able to count the number of systems in every energetic state E ₁, …, E _j . Let n ₁, n ₂ … be the numbers of systems found in sates E ₁, E ₂, …. The set of values n ₁, n ₂ … is a distribution of the systems. For each distribution, the following relations are obligatorily satisfied:

$$ \begin{array}{l}{\displaystyle \sum_j{n}_{\mathrm{j}}=\mathrm{\aleph}}\\ {}{\displaystyle \sum_j{n}_j\;{E}_j={E}_t}\end{array} $$

where E _j is the energy of the considered system within the ensemble for the considered distribution. E _t is the energy of the ensemble (also named supersystem). (We shall see that it is not necessary to know the values ℵ and E _t because they disappear during the calculations.)

Let us suppose, in order to simplify, that the ensemble possesses four systems labelized A, B, C, and D and that the possible energy states of each system are E ₁, E ₂, and E ₃. Let us also suppose that the total energy (of the supersystem) is as follows:

$$ {E}_t={E}_1+2{E}_2+{E}_3 $$

(22.1)

that is to say n ₁ = 1, n ₂ = 2, and n ₃ = 1. These values (E _t , E ₁, E ₂, E ₃, n ₁, n ₂, n ₃) define the distribution.

3.2 The Notion of Sub-distribution

There are several possibilities of attribution of the energies E ₁, E ₂, and E ₃ to the systems A, B, C, and D in order that the distribution defined by relation (22.1) exists. They are those mentioned in Table 22.1. We call them “sub-distributions” (personal terminology).

Table 22.1 Sub-distributions corresponding to the distribution n ₁ = 1, n ₂ = 2, and n ₃ = 1; N = 4, labelized systems A, B, C, and D

We notice that there are 12 sub-distributions corresponding to the same distribution, labelized k. This result is no more than the solution of the classical problem of combinatory analysis which, in this case, can be presented by giving the answer to the following question: How many (number Ω) possibilities to group 4 objects by groups of 2, 1, and 1 do exist? The answer is

$$ \varOmega =\left(2+1+1\right)!/\left(2\ !1\ !1\ !\right)=12 $$

From the general viewpoint, the number Ω of possibilities to group (n ₁ + n ₂ + …n _j ) objects by groups of n ₁, n ₂, …, n _j objects is given by the relation

$$ \varOmega =\left({n}_1+{n}_2+\dots {n}_j\right)\ !\ /\ \left({n}_1\ !\ {n}_2\ !\dots {n}_j\ !\right) $$

(22.2)

Let us recall that all the sub-distributions have the same energy.

3.3 Case of Several Distributions

We must bear in mind that there are numerous distributions existing for the same set of parameters N, V, and T. For the same example as previously, let us suppose that it is the case for the distribution n ₁ = 2, n ₂ = 0, and n ₃ = 2, that is to say

$$ 2{E}_1+0{E}_2+2{E}_3={E}_t $$

where the energy E _t is the same as that of the preceding distribution. This new distribution exists under (2 + 0 + 2) !/(2 !0 !2 !) = 6 sub-distributions. Let us also suppose that only two distributions exist for the same total energy. Since they possess the same energy E _t , according to the second postulate, the sub-distributions of both distributions are equiprobable, whichever their origin.

What is being searched for is the probability to find a system of the ensemble in the energy state E _j , that is to say, remaining in the same example as previously, the probability to find the system A or B or C or D with the energy E ₁, E ₂, or E ₃. In this very simple example, the result can be found by a direct numbering by placing in the same table all the sub-distributions and by performing the numbering.

The direct numbering indicates that each system A, B, C, or D possesses 1/3 chance to possess the quantified energy levels E ₁, E ₂, and E ₃. (The fact that all these probabilities are all equal (1/3) must not be generalized. It results solely from the chosen numerical values. It must be considered as a numerical accident (Table 22.2).)

Table 22.2 Sub-distributions of the same total energy E and, hence, of the same probability stemming from two distributions (see text)

The direct numbering is not, of course, envisageable in statistical thermodynamics, given the huge number of the existing distributions and sub-distributions. Fortunately, there exists a useful mathematical relation which generalizes what is preceding. It results from the following reasoning:

• The elementary probability prob₁ (1 because it concerns the first distribution) in order that one of the systems A, B, C, or D possesses the energy E ₂ in the first distribution is 2/4 since n ₂ = 2 and since there are four systems. The number of times that one of the systems in the first distribution is endowed with the energy E ₂ is 12 × 2/4 = 6, that is to say by generalizing Ω₁ • prob₁.

• The elementary probability prob₂ in order that one of the systems possesses the energy E ₂ in the distribution 2 is 6 × 0/4 = 0, that is to say Ω₂ • prob₂.

• The total number of possibilities that a system would be in an ordinary state of energy is this example 12 + 6 = 18, that is, Ω₁ + Ω₂. The global probability (and not elementary) P ₂ that a system would be in the energetic state E ₂ is as follows:

  $$ {P}_2=\left(12\times 2/4+6\times 0/4\right)\ /\ \left(12+6\right)=1/\ 3 $$

  and by generalizing

  $$ {P}_j=\left({\displaystyle \sum_j{\mathrm{prob}}_j\;{\varOmega}_k}\right)/{\displaystyle \sum_k{\varOmega}_k} $$

  (22.3)

  where j marks the authorized state of energy of the system. prob _j is the elementary probability in order that in the distribution k, the energy be E _j .

• The probability P ₂ can also be written (in a strictly equivalent manner) as

  $$ {P}_2=\left(1/4\right)\left(2\times 12+0\times 6\right)/\left(12+6\right) $$

  where 4 is the number of systems and 2 × 12 and 0 × 6 are the numbers of times that the state of energy E ₂, respectively, appears in the first and second distribution.

The general relation (22.3) can also be written according to

$$ {P}_j=\left(1/\mathrm{\aleph}\right)\ \left({\displaystyle \sum_k{n}_j{\varOmega}_k}\right)\ /{\displaystyle \sum_k{\varOmega}_k} $$

(22.4)

This expression is a generalization of the preceding which gave P ₂.

4 Obtention of P _j

4.1 Great Number of Distributions: Method of the Maximal Term

The obtaining of P _j is performed in a mathematical way. It is based on the fact that there exist numerous possible distributions obeying the constraints of the problem. The latter ones are

• The number ℵ of systems of the ensemble

• The temperature T

• The different possible energies E _j of every system. (They depend on the total number of particles N and of the volume V, according to the principles of quantum mechanics.)

Given the very large number ℵ, one demonstrates that one distribution weighs much more and even quasi-infinitely more than other ones. Therefore, one can make the assumption that it entails its repartition of the systems in the ensemble, and it is done as a function of the energies E _j . The hypothesis is entitled “method of the maximal term.” From the mathematical standpoint, it consists in replacing the logarithm of a sum by the logarithm of the highest term of the sum, when the latter is constituted of very numerous terms. The expression giving the probability P _j to find a system of the ensemble in the energetic state E _j is constituted of very numerous terms. Taking only into account the largest term seems to be an approximation. It is the case, but it does not lead to any detectable error. (viz. Appendix A).

By applying the hypothesis, the relation (22.4) reduces to

$$ {P}_j={n}_j^{*}/\mathrm{\aleph} $$

where n ^* _j is the number of times that the quanto-energetic state E _j appears in the most probable distribution. Of course, there are as many n ^* _j to calculate as quanto-energetic E _j levels do exist.

Hence, the most probable distribution must be found.

4.2 Calculations

The calculations are performed by starting from ln Ω rather than from Ω. It is easier to process in such a manner and it does not change anything concerning the result since ln x varies as x.

According to the expression (22.2), we obtain

$$ \ln \kern0.24em \varOmega = \ln\ \left[\left({n}_1+{n}_2+\dots {n}_j\right)!\right] - \ln\ {n}_1\ ! - \ln \kern0.37em {n}_2\ ! - \dots \ln\;{n}_{\mathrm{j}}! $$

Then, they are performed by using Stirling’s approximation which is written as

$$ \ln \kern0.37em y\ !\approx y\kern0.37em \ln \kern0.37em y - y $$

The use of this approximation is all the more justified as y is a large number. This is the case here. With this approximation, ln Ω becomes

$$ \begin{array}{c} \ln \kern0.24em \varOmega =\left({n}_1+{n}_2+ \dots {n}_j\right)\ \ln\ \left({n}_1+{n}_2+ \dots {n}_j\right)\ \hbox{--}\ \left({n}_1+{n}_2+ \dots {n}_j\right)\hbox{--} {n}_1\; \ln\;{n}_1\\ {} + {n}_1 - {n}_2\; \ln \kern0.37em {n}_2+{n}_2\dots \dots - {n}_j\; \ln \kern0.37em {n}_j + {n}_j\end{array} $$

The mathematical process coming immediately in mind is to have to successively vanish the partial derivatives (∂ ln Ω/∂n ₁), (∂ ln Ω/∂n ₂) … (∂ ln Ω/∂n _j ) and, from this process, to extract the values n ₁, n ₂, …, n _j leading to this result. But, there is a difficulty: the mathematical system is submitted to the following constraints:

$$ \begin{array}{l}{n}_1+{n}_2 + \dots {n}_j=\mathrm{\aleph}\\ {}{n}_1{E}_1 + {n}_2{E}_2 + \dots {n}_j{E}_j = {E}_t\end{array} $$

The smartest means permitting this process of maximalization taking into account these constraints is to use the method of Lagrange’s multipliers (viz. Appendix A) which, in this case, translates itself into the successive vanishing of the partial derivatives with respect to n ₁, n ₂, …, n _j of function F, and no longer of function ln Ω:

$$ F= \ln\;\varOmega -\alpha \left({n}_1+{n}_2+\dots {n}_j\right)-\beta \left({n}_1{E}_1+{n}_2{E}_2+\dots {n}_j{E}_j\right) $$

where α and β are two constants, the physical meaning of which will appear in the following calculations.

When the calculation of the derivatives is performed, we obtain the following relations:

$$ \begin{array}{l}{n}_1=\mathrm{\aleph}\kern0.24em {\mathrm{e}}^{-\alpha -\beta {E}_1}\\ {}{n}_2=\mathrm{\aleph}\kern0.24em {\mathrm{e}}^{-\alpha -\beta {E}_2}\\ {}{n}_j=\mathrm{\aleph}\kern0.24em {\mathrm{e}}^{-\alpha -\beta {E}_j}\end{array} $$

(22.5)

These relations are very important. We can deduce the following points from them:

• The signification of the constant e^α.

Since Σ_j n _j  = ℵ, the addition of relations (22.5) leads to

$$ {\mathrm{e}}^{\alpha } = {\mathrm{e}}^{-\beta {E}_1}+{\mathrm{e}}^{-\beta {E}_2} + \dots {{\mathrm{e}}^{-}}^{\beta {E}_j} $$

• The mean energy $\overline{E}$ of each system.

Since E _t  = ℵ $\overline{E}$,

$$ {\displaystyle \sum_j{n}_j{E}_j}=\mathrm{\aleph}\overline{E} $$

By replacing the n _j by their expressions (22.5) and e^−α by the above expression, we obtain

$$ \overline{E}={\displaystyle \sum_j{E}_j{\mathrm{e}}^{-\beta {E}_j}}/{\displaystyle \sum_j{\mathrm{e}}^{-\beta {E}_j}} $$

(22.6)

It is important to notice that, according to the expression (22.6), the parameter β appears as being an implicit function of the mean energy $\overline{E}$ and also, therefore, of the composition N and of the volume V which govern the quantum levels E _j . It is the same for α which depends on the same parameters. But, actually, the studied ensemble is that defined by the macroscopic parameters N, V, and T and not by N, V, and $\overline{E}$. However, as we shall see, $\overline{E}$ depends on T. Let us anticipate what is following by mentioning that β is inversely proportional to the absolute temperature. More precisely, β = 1/kT where k is Boltzmann’s constant and T the absolute temperature.

• The expression giving the probability P _j to find a system of the ensemble in the energetic state E _j is constituted of very numerous terms. Taking only into account the largest term seems to be an approximation. It is the case, but it does not lead to any detectable error.

It is calculated by applying the general definition of a probability, through the relation

$$ {P}_j = {n}_j/\mathrm{\aleph} $$

By replacing n _j by its expression (22.5) and by introducing the above expression e^−α, we find

$$ {P}_j = {\mathrm{e}}^{-\beta {E}_j}/{\displaystyle \sum_i{\mathrm{e}}^{-\beta {E}_i}} $$

(22.7)

We shall see in the following chapter that these expressions permit to grasp the meaning at the molecular scale of the great thermodynamic functions.

The expressions (22.6) and (22.7) call for the great importance of the sum Σ _i ${\mathrm{e}}^{-\beta {E}_i}$. Indeed, it will play a considerable part. In statistical thermodynamics, such a function is called partition function. As it happens here, it is the partition function of the canonical ensemble. It is symbolized by Q:

$$ Q={\displaystyle \sum_i{\mathrm{e}}^{-\beta {E}_i}} $$