o74 TRANSACTIONS OF SECTION A. 
2. The Structure of the Atom. 
By Professor Sir J. J. Toomson, F'.L.S. 
9 
3. The Relation between Entropy and Probability. 
By Professor H. A. Lorentz. 
The important problem of the interpretation of entropy in the terms of molecular 
theory may be considered as having been solved by the physicists who developed the 
methods of modern statistical mechanics. It is now universally recognised that the 
entropy of a body in a certain state is intimately connected with the probability of 
that state, the relation between the two being expressd by Boltzmann’s formula 
S= x ion 
where S is the entropy, P the probability, R the gas constant for a gramme-molecule 
and N the number of molecules in a gramme-molecule. 
The question is, however, in what way the probability is to be evaluated. 
In order to find P, and consequently 8, as a definite function of the energy E and 
appropriate geometrical parameters, such as the volume v in the case of a gas or a 
liquid, one may proceed as follows. Let ¢,, gq... . qs be the co-ordinates which 
determine the position of the particles of the body, p,, p, . . . ps the corresponding 
momenta. Conceive a polydimensional space (the ‘extension in phase’) in which 
the 2 § quantities g,. . . ps are taken as co-ordinates, so that the state of the body 
is represented by a single point. Consider further two surfaces in the space 
(q, - - - ps), the first of which is characterised by the condition that at each of its 
points the energy has a definite value E, whereas the second corresponds in the same 
way to the value E-++ dE. ‘Then the volume of the layer between these surfaces will 
be proportional to dE and may therefore be represented by PdE, the factor P being 
a function of E and also of the volume », the value of which is to be taken into account 
in the calculation. Now, if the quantity P, defined in this way, is substituted in 
Boltzmann’s formula, S becomes identical with the thermodynamical entropy. In 
the case of a mono-atomic gas, this may be shown by direct calculation and the 
proposition may be extended to other bodies by means of a mode of reasoning (based 
on a certain assumption) which cannot now be considered. 
The above method is closely connected with Gibbs’s microcanonical ensembles, and 
it is to be remarked that the introduction of canonical ensembles likewise enables us 
to determine a thermodynamical function. As is well known, an ensemble of the 
latter kind consists of a very great number of systems or bodies, whose representative 
points are distributed throughout the extension in phase with a density given by the 
expression 
Vv—-E 
Ae ® 
where A is the total number of systems, © a constant that plays the part of the tem- 
perature, and ¥ another constant that proves to be the ‘free energy.’ Now, if do 
is an element of the extension in phase, we have 
v—E 
fe @ an=1 
i) - 
(because the integral of Ac dQ must give the total number of systems) or 
4 E 
© ~o 
é =/e dn. 
By this equation one can calculate the free energy for a given temperature and a 
given value of the volume. The result will be connected with the value of the entropy 
deduced from Boltzmann’s formula, in the way that is known from thermodynamics. 
On account of the enormous number of molecules contained in a body, Boltzmann’s 
formula has a very remarkable property, namely that great changes in the value 
assigned to the probability P have no appreciable influence on the entropy S. 
Me 
