The first and second laws of thermodynamics allow you to recover the inequality between the arithmetic and the geometric means: Bring together n identical heat reservoirs with heat capacity $C$ and temperatures $T_1,\ldots,T_n$ and allow them to reach a final temperature $T$. The first law of thermodynamics tells you that $T$ is the arithmetic mean of the $T_i$. The second law of thermodynamics demands the non-negativity of the change in entropy, which is
$$ C_n \, log(T/G) $$
where $G$ is the geometric mean. It follows that $T > G$.
I believe this argument was first made by P.T. Landsberg (no relation!).
