Statistical Assemblies and the Entropy

Let us ask what the randomness that we associated with entropy in Chap. I means in terms of the assembly. A random system, or one of large entropy, is one in which the microscopic properties may be arranged in a great many different ways, all consistent with the same large-scale behavior. Many different assignments of velocity to individual molecules, for instance, can be consistent with the picture of a gas at high temperatures, while in contrast the assignment of velocity to molecules at the absolute zero is definitely fixed all the molecules are at rest. Then 

Then in a random assembly, describing a system of large entropy, there will be systems of the assembly distributed over a great many complexions, so that many /t s will be different from zero, each one of these fractions being necessarily small. On the other hand, in an assembly of low entropy, systems will be distributed over only a small number of complexions, so that only a few/t s will be different from zero, and these will be comparatively large. 

We shall now postulate a mathematical definition of entropy, in terms of the/t7s, wliicIT Is larfflfih the case of a random distribution small otherwise. This definition is 

Here k is a constant, called Boltzmann,s constant, which will appear frequently in our statistical work. It has the same dimensions as entropy, or specific heat, that is. energy divided bv temperature. Jts value in absolute units is 1.379 X 10 16 erg per degree. This value is derived indirectly using Eq. (1.2), for the mifropy, derive tlir porfftp.t 

Thn r ifrnnv LP m li a. js proportional to the logarithm of the number of complex nna in.whmb systems of the assembly can lie found. As this number of complexions increases, the distribution becomes more random or diffuse, and the entropy increases. 

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