Here also we will dispense with a full proof and content ourselves with the main steps which lead to the desired expression. We will follow PLANCK'S elegant presentation on pp. 136-148 of his Wärmestrahlung. On p. 22 we have dwelt on the usefulness and the necessity for the probability idea in general physics and in this particular case. We can start, therefore, with PLANCK'S theorem:
"The entropy of a physical system in a particular state depends solely on the probability of this state."
No rigorous proof is here attempted, nor any numerical computations; for present purposes it will suffice to fix in a general way the kind of dependence of entropy on probability.
Let designate the entropy and the probability of a physical system in a particular state, then the above theorem enunciates that where signifies a universal function of the argument . Now, however may be defined we can certainly infer from the Calculus of Probabilities that the probability of a system, composed of two entirely independent systems, is equal to the product of the separate probabilities of the individual systems. For example, if we take for the first system any terrestrial body whatever and for the second system any hollow space on Sirius, which is traversed by radiations, then the probability , that simultaneously the terrestrial body will be in a particular state 1 and said radiation in a particular state 2, will be given by where , respectively represent the separate probabilities of said two states. Now let , respectively represent the entropies of the separate systems corresponding to said states 1 and 2, then according to Eq. 8, we have But, according to the Second Law of Thermodynamics, the total entropy of two independent systems is , and consequently according to (8) and (9),
From this functional equation may be determined. After successive differentiation there is obtained a differential equation of the second order and its general integral is which determines the general dependence of entropy on probability. The universal integration constant is the same for a terrestrial system as for a cosmical system, and when its numerical value is known for either system it will be known for the other; indeed, this constant is the same for physically unlike systems, as above, where concurrence between a molecular and a radiating system was assumed. The last, additive, constant has no physical significance because entropy has an arbitrary additive constant and therefore this constant in (10) may be omitted at pleasure.
Relation (10) contains a general method of computing the entropy from probability considerations. But the relation becomes of practical value only when the magnitude of the probability of a system for a certain state can be given numerically. The most general and precise definition of this magnitude is an important physical problem and first of all demands closer insight into the details of what constitutes the "state" of a physical system. [This has been adequately done in the earlier part of this presentation. Later on pp. 27, 28, permutation considerations led us to define the probability W of a state as the number of complexions included in the given state.]