Member-only story
The One Operation Hiding Inside Free Energy, the Hamiltonian, and the Price of Bread
A graduate student in physics meets the same piece of mathematics at least three times, and is rarely told it is the same piece. It appears first in thermodynamics, when the internal energy of a gas gets traded for its free energy. It appears again in classical mechanics, when the Lagrangian is exchanged for the Hamiltonian — the formulation that quantum mechanics would later be built on top of. It appears a third time in optimization, where it is the engine that turns every convex minimization problem into a twin problem with the same answer. In statistics it governs the probability of rare events; in economics it converts a firm’s costs into its profits. Each field gives it a different name and a different notation, and each presents it as a local trick of the trade.
It is one operation, the Legendre transform: a way of rewriting a convex function in terms of its slopes rather than its points, so that the same information is carried by a new function of a new variable. The question worth holding onto is whether its recurrence is a coincidence of notation — five fields that happened to write down the same algebra — or a clue that they are all, underneath, doing the same thing for the same reason. The answer turns out to be a single picture, and the picture is almost embarrassingly simple. It is a…
