Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 3.1.
The concept of entropy as , where is the multiplicity of the macrostate in which the system is found, can be used to define the temperature of the system. Schroeder gives a good explanation in his section 3.1 so I’ll summarize the argument here.
We’ll use two interacting Einstein solids with and , with total energy quanta. The multiplicity function for each solid is
and the total multiplicity of the combined system is
The entropies are therefore
where we pick the subscript for the system we’re interested in. We can plot the three entropy curves as functions of (remember ), to get
Here the top red curve is , the violet curve is and the turquoise curve is . reaches a maximum at which is the macrostate where the energy is evenly distributed among all the oscillators. At this point, therefore
Since , this implies that
Since , so we can write this as
In the small scale system here, it’s not quite right to consider and as continuous variables, so the partial derivatives are an approximation. In very large systems, however, the quanta merge into a continuous energy variable , so we can write
As the units of entropy are and of are , this derivative has the dimensions of or the reciprocal of temperature. We can therefore define temperature to be
where the partial derivative implies that we hold everything else apart from the energy of the system constant while taking the derivative. Thus two systems that can exchange energy until they reach their most probable macrostate will end up with the same temperature, so they are in thermal equilibrium.
The relation 6 states that the slopes of the entropy-versus-energy curves are equal for the most probable macrostate. In the graph above, this means that the slope of the turquoise line is the negative of the slope of the violet line at , which looks about right if you eyeball the graph. We could prove it by taking the actual derivatives, but we’ll make do with a numerical example.
Suppose each energy quantum has a value of . We can then estimate the temperatures of the two solids at by calculating the slope of the line connecting the points for and . For solid , we use so the two energy points are and . We get
Thus the two temperatures are indeed equal at .
For we can use the slope between and for solid and and for solid . We get
Here, solid is much hotter than solid so if they are interacting, there would be a strong tendency for to transfer some of its energy to to bring the solids into thermal equilibrium.