Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problems 3.17 – 3.18.
We can apply the formulas for entropy, temperature and heat capacity to a real-life system by looking at a two-state paramagnet. This is a system of magnetic dipoles which, when placed in a magnetic field , align themselves so that their magnetic moment points either parallel or antiparallel to the field. The energy of a dipole that is aligned with the field is lower, and we’ll call it , so that the antiparallel dipole has energy , and the total energy of the system is
where an up arrow indicates parallel alignment and a down arrow antiparallel.
The net magnetization is then
The multiplicity of states is the same as a set of coins with heads, so
For small systems, we can find the entropy directly as
For , we get , , and .
For each value of from 0 up to , we can evaluate and from the formulas above and then plot versus (I used Maple for the plot):
For , the curve is a ‘normal’ entropy curve in that the entropy increases with increasing energy, and the curve is concave down.
From this we can get the temperature
Thus the temperature increases with energy in the region . At , however, the derivative is zero implying an infinite temperature, and for , the slope is negative, indicating a negative temperature. Since, in this case, negative temperatures occur at higher energies than positive temperatures, we have to interpret a negative temperature as actually being higher than a positive one, in fact, higher than an ‘infinite’ positive temperature.
In this case, it’s probably better to use the entropy of the system as a physical measure of what’s going on, since the second law implies that the system will tend to the energy with the greatest entropy, which is . At this energy, there are equal numbers of up and down dipoles, so the system is maximally randomized.
For a system with , we can approximate 5 by taking finite differences. Thus for a given value of we can estimate the temperature as
For , we get
This allows us to plot temperature versus energy:
We can see the flip over from to as the energy increases through zero.
The heat capacity can be obtained similarly as
where we use 7 to calculate the temperatures. For we have
The plot of versus temperature is
[The plot does actually extend down to 0 at if we use the analytic solution, but because we’re dealing with discrete values of , it cuts out early.] This curve is similar in shape to that of an Einstein solid at low temperatures.
Finally, we can plot the magnetization as a function of temperature:
If we start off with and lower the temperature towards zero, the magnetization gradually increases until at , the system is frozen into the state where all dipoles are parallel to the field. As we increase to , we approach maximum randomness with equal numbers of dipoles pointing up and down so . Increasing the temperature beyond so it becomes negative (starting at ) the magnetization again increases, but now the dipoles are aligned antiparallel to the field, eventually saturating as from the negative side.