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,...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
where G is the geometric mean. It follows that T > G.
I believe this argument was first made by P.T. Landsberg (no relation!).