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Two crucial concepts of Thermodynamics that spring directly from our work in the previous section are entropy and temperature. Here we define both and discuss how they relate to their more common definitions.
We begin by revisiting the multiplicity function we looked at earlier. Let us modify the function slightly, so that instead of being a function of N and Nup, the total number of particles and the number of up magnets, let us generalize and let g now be a function of N and U, the energy of the system at hand. Now, this does not alter the definition at all; g still represents the number of states of the system with the same value of a particular variable, though in this case that variable is the energy U.
The entropy is defined as:
Notice that entropy is unitless. (Here, log is used to represent the natural logarithm, ln.) You might wonder why the entropy is defined this way. We will get at the answer via a short discussion of thermal equilibrium.
Suppose that we have two isolated thermal systems. The first has energy U1 and the second energy U2. Let the total energy between the two systems be constant, namely U. Then we can express the energy in the second system as U - U1. Furthermore, let the number of particles in the first system be N1 and that in the second N2, with the total number of particles N kept constant (so we can write N2 = N - N1).
Now suppose that the two systems are brought into thermal contact with each other, meaning that they can exchange energy but not number of particles. Then the total multiplicity function is given by:
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