Methods. This Calculus works largely by the determination of averages and its results must be interpreted accordingly. Moreover, for the present we will take a popular, practical view of these results and consider a very great improbability as equivalent to an impossibility. Numerical computations are essential in most uses of this Calculus, but here they will be entirely omitted.
[7]The H-theorem considers a process (consisting of a number of separate, reversible processes) which is irreversible in the aggregate.
(2) What is Meant by the Probability of a State? Example
To come back to the matter in hand we will now show what is here meant by the probability of any state.
When we speak of the probability
of a particular "elementar-ungeordnete" state, we thereby imply that this state may be variously realized. For every state (which contains many like independent constituents) corresponds to a certain "distribution," namely, a distribution among the gas molecules of the location co-ordinates and of the velocity components. But such a distribution is a permutation problem, is always an assignment of one set of like elements (co-ordinates, velocity components) to a different set of like elements (molecules). So long as only a particular state is kept in view, it is of consequence as to how many elements of the two sets are thus interchangeably assigned to each other and not at all as to which individual elements of the one set are assigned to particular individual elements of the other set.[8] Then a particular state may be realized by a great number of assignments individually differing from one another, but all equally likely to occur.[9] If with PLANCK we call such an assignment a "complexion,"[10] we may now say that in general a particular state contains a large number of different, but equally likely, complexions. This number, i.e., the number of the complexions included in a given state can now be defined as the probability
of the state.[11] Let us present the matter in still another form. BOLTZMANN derives the expression for magnitude of the probability by at once distinguishing between a state of a considered system and the complexion of the considered system. A state of the system is determined by the law of locus and velocity distribution, i.e., by a statement of the number of particles which lie in each elementary district of space and the number of particles which lie in each elementary velocity realm, assuming that among themselves these districts and realms are alike and each such infinitesimal element still harbors very many particles. Accordingly a particular state of the system embraces a very large number of complexions. For if any two particles belonging to different regions swap their co-ordinates and velocities, we get thereby a new complexion, but still the same state. Now BOLTZMANN assumes all complexions to be equally probable and therefore the number of complexions included in a particular state furnishes at the same time the numerical value for the Probability of the state in question. Illustration taken from the simultaneous throwing of two, ordinary, cubical dice. Suppose that the sum is to be 4 for each throw, then this can be realized by the following three complexions:
First cube shows 1, the second cube shows 3;
First cube shows 2, the second cube shows 2;
First cube shows 3, the second cube shows 1.