Chapter IV

Direct Calculation of the Entropy in The Case of Thermodynamic Equilibrium

141. In the calculation of the entropy of an ideal gas and of a system of resonators, as carried out in the preceding chapters, we proceeded in both cases, by first determining the entropy for an arbitrarily given state, then introducing the special condition of thermodynamic equilibrium, i.e., of the maximum of entropy, and then deducing for this special case an expression for the entropy.

If the problem is only the determination of the entropy in the case of thermodynamic equilibrium, this method is a roundabout one, inasmuch as it requires a number of calculations, namely, the determination of the separate distribution densities $w_{1}$ , $w_{2}$ , $w_{3}, \dots$ which do not enter separately into the final result. It is therefore useful to have a method which leads directly to the expression for the entropy of a system in the state of thermodynamic equilibrium, without requiring any consideration of the state of thermodynamic equilibrium. This method is based on an important general property of the thermodynamic probability of a state of equilibrium.

We know that there exists between the entropy $S$ and the thermodynamic probability $W$ in any state whatever the general relation (164). In the state of thermodynamic equilibrium both quantities have maximum values; hence, if we denote the maximum values by a suitable index:

(229)

$S_{m} = k log W_{m} .$

It follows from the two equations that:

$\frac{W_{m}}{W} = e^{\frac{S_{m} - S}{k}} .$

Now, when the deviation from thermodynamic equilibrium is at all appreciable, $\frac{S_{m} - S}{k}$ is certainly a very large number. Accordingly $W_{m}$ is not only large but of a very high order large, compared with $W$ , that is to say: The thermodynamic probability of the state of equilibrium is enormously large compared with the thermodynamic probability of all states which, in the course of time, change into the state of equilibrium.

This proposition leads to the possibility of calculating $W_{m}$ with an accuracy quite sufficient for the determination of $S_{m}$ , without the necessity of introducing the special condition of equilibrium. According to Sec. 123, et seq., $W_{m}$ is equal to the number of all different complexions possible in the state of thermodynamic equilibrium. This number is so enormously large compared with the number of complexions of all states deviating from equilibrium that we commit no appreciable error if we think of the number of complexions of all states, which as time goes on change into the state of equilibrium, i.e., all states which are at all possible under the given external conditions, as being included in this number. The total number of all possible complexions may be calculated much more readily and directly than the number of complexions referring to the state of equilibrium only.

142. We shall now use the method just formulated to calculate the entropy, in the state of equilibrium, of the system of ideal linear oscillators considered in the last chapter, when the total energy $E$ is given. The notation remains the same as above.

We put then $W_{m}$ equal to the number of complexions of all stages which are at all possible with the given energy $E$ of the system. Then according to (219) we have the condition:

(230)

$E = h ν \sum_{n = 1}^{\infty} (n - \frac{1}{2}) N_{n} .$

Whereas we have so far been dealing with the number of complexions with given $N_{n}$ , now the $N_{n}$ are also to be varied in all ways consistent with the condition (230).

The total number of all complexions is obtained in a simple way by the following consideration. We write, according to (165), the condition (230) in the following form:

$\frac{E}{h ν} - \frac{N}{2} = \sum_{n = 1}^{\infty} (n - 1) N_{n}$

(231)

$0 \cdot N_{1} + 1 \cdot N_{2} + 2 \cdot N_{3} + \dots + (n - 1) N_{n} + \dots = \frac{E}{h ν} - \frac{N}{2} = P .$

$P$ is a given large positive number, which may, without restricting the generality, be taken as an integer.

According to Sec. 123 a complexion is a definite assignment of every individual oscillator to a definite region element $1$ , $2$ , $3, \dots$ of the state plane $(f, ψ)$ . Hence we may characterize a certain complexion by thinking of the $N$ oscillators as being numbered from $1$ to $N$ and, when an oscillator is assigned to the $n$ th region element, writing down the number of the oscillator $(n - 1)$ times. If in any complexion an oscillator is assigned to the first region element its number is not put down at all. Thus every complexion gives a certain row of figures, and vice versa to every row of figures there corresponds a certain complexion. The position of the figures in the row is immaterial.

What makes this form of representation useful is the fact that according to (231) the number of figures in such a row is always equal to $P$ . Hence we have “combinations with repetitions of $N$ elements taken $P$ at a time,” whose total number is

(232)

$\frac{N (N + 1) (N + 2) \dots (N + P - 1)}{123 \dots P} = \frac{(N + P - 1)!}{(N - 1)! P!} .$

If for example we had $N = 3$ and $P = 4$ all possible complexions would be represented by the rows of figures:

$\begin{matrix} 1111 & 1133 & 2222 \\ 1112 & 1222 & 2223 \\ 1113 & 1223 & 2233 \\ 1122 & 1233 & 2333 \\ 1123 & 1333 & 3333 \end{matrix}$

The first row denotes that complexion in which the first oscillator lies in the $5$ th region element and the two others in the first. The number of complexions in this case is $15$ , in agreement with the formula.

143. For the entropy $S$ of the system of oscillators which is in the state of thermodynamic equilibrium we thus obtain from equation (229) since $N$ and $P$ are large numbers:

$S = k log \frac{(N + P)!}{N! P!}$

and by making use of Stirling’s formula (171) 49

$S = k N \{(\frac{P}{N} + 1) log (\frac{P}{N} + 1) - \frac{P}{N} log \frac{P}{N}\} .$

If we now replace $P$ by $E$ from (231) we find for the entropy exactly the same value as given by (222) and thus we have demonstrated in a special case both the admissibility and the practical usefulness of the method employed.50

Chapter IV

49Compare footnote to page 491. See also page 473.

50A complete mathematical discussion of the subject of this chapter has been given by H. A. Lorentz. Compare, e.g., Nature, 92, p. 305, Nov. 6, 1913. (Tr.)