122.

If now the calculation of the probability $W$ of a state is in question, it is evident that the state is to be thought of in the macroscopic sense.

The first and most important question is now: How is a macroscopic state defined? An answer to it will dispose of the main features of the whole problem.

For the sake of simplicity, let us first consider a special case, that of a very large number, $N$, of simple similar molecules. Let the problem be solely the distribution of these molecules in space within a given volume, $V$, irrespective of their velocities, and further the definition of a certain macroscopic distribution in space.

The latter cannot consist of a statement of the coordinates of all the separate molecules, for that would be a definite microscopic distribution.

We must, on the contrary, leave the positions of the molecules undetermined to a certain extent, and that can be done only by thinking of the whole volume $V$ as being divided into a number of small but finite space elements, $G$, each containing a specified number of molecules.

By any such statement a definite macroscopic distribution in space is defined.

The manner in which the molecules are distributed within every separate space element is immaterial, for here the hypothesis of elemental chaos ([sect:117.] Sec. 117) provides a supplement, which insures the unambiguity of the macroscopic state, in spite of the microscopic indefiniteness.

If we distinguish the space elements in order by the numbers $1$, $2$, $3,\dots$ and, for any particular macroscopic distribution in space, denote the number of the molecules lying in the separate space elements by $N_1$, $N_2$, $N_3\dots$, then to every definite system of values $N_1$, $N_2$, $N_3\dots$, there corresponds a definite macroscopic distribution in space. We have of course always: $N_1+N_2+N_3+\dots =N\mathrm{\backslash Label}[eqn](165)\text{upshape (165)}$ or if

$$\begin{gathered} \frac{N_1}{N}=w_1,\frac{N_2}{N}=w_2,\dots \mathrm{\backslash Label}[eqn](166)\text{upshape (166)} \\ {w}_1+w_2+w_3+\dots =1.\mathrm{\backslash Label}[eqn](167)\text{upshape (167)} \end{gathered}$$

The quantity $w_i$ may be called the density of distribution of the molecules, or the mathematical probability that any molecule selected at random lies in the $i$th space element.

If we now had, e.g., only $10$ molecules and $7$ space elements, a definite space distribution would be represented by the values:

$N_1=1,N_2=2,N_3=0,N_4=0,N_5=1,N_6=4,N_7=2,\mathrm{\backslash Label}[eqn](168)\text{upshape (168)}$

which state that in the seven space elements there lie respectively $1$, $2$, $0$, $0$, $1$, $4$, $2$ molecules.