3.1 Basic notions

We worked a very particular example to explain the concept of entropy. Specifically we considered $N$ quantum harmonic oscillators sharing (or partitioning) the $q$ units of energy $E=q\mathrm{\hslash }{\omega }_0$ amongst themselves. (The graphical presentation of this problem is given in the lecture notes.) The total number of ways the system can partition the available energy is $\mathrm{\Omega }(E)$ and the entropy is

The system will evolve until all possible partitions of the energy are equally likely. The probability of a specific partition is

If there are six possible outcomes $\mathrm{\Omega }(E)=6$ (like a regular die) then the probability of an outcome is $1/6$. $\mathrm{\Omega }$ is an exponentially large number and is of order $e^N$. A precise computation done in homework gives $\mathrm{\Omega }=e^{555}$ for $N=q=400$. The entropy is (up to a constant) the log of the number of partitions

Thus the entropy is of order $\sim N{k}_B$.