120.

The logarithmic connection between entropy and probability was first stated by L. Boltzmann L. Boltzmann, Vorlesungen über Gastheorie, 1, [sect:6.] Sec. 6. in his kinetic theory of

gases. Nevertheless our equation [eqn:(164)] (164) differs in its meaning from the corresponding one of Boltzmann in two essential points.

Firstly, Boltzmann's equation lacks the factor $k$, which is due to the fact that Boltzmann always used gram-molecules, not the molecules themselves, in his calculations. Secondly, and this is of greater consequence, Boltzmann leaves an additive constant undetermined in the entropy $S$ as is done in the whole of classical thermodynamics, and accordingly there is a constant factor of proportionality, which remains undetermined in the value of the probability $W$.

In contrast with this we assign a definite absolute value to the entropy $S$. This is a step of fundamental importance, which can be justified only by its consequences. As we shall see later, this step leads necessarily to the "hypothesis of quanta" and moreover it also leads, as regards radiant heat, to a definite law of distribution of energy of black radiation, and, as regards heat energy of bodies, to Nernst's heat theorem.

From [eqn:(164)] (164) it follows that with the entropy $S$ the probability $W$ is, of course, also determined in the absolute sense. We shall designate the quantity $W$ thus defined as the "thermodynamic probability," in contrast to the "mathematical probability," to which it is proportional but not equal. For, while the mathematical probability is a proper fraction, the thermodynamic probability is, as we shall see, always an integer.