Figure 1:
The Legendre transform is a valuable tool in classical mechanics and thermodynamics, and involves mapping a function of a coordinate to a function of a “derivative” of a coordinate. In some cases, the transform can be useful in converting a poorly behaved function into a well-behaved one. The Legendre transform of a functional f[x] is defined as f^{*}[q]=sup_{x}{〈x,q〉-f[x]}. The diagram exemplifies the concept for a function of single variable. Given a slope q, one determines the maximum value of qx-f(x) shown by the double arrows. This maximum value occurs at the point marked by the arrow, and gives the Legendre transform f^{*}(q). If f(x) is differentiable at the point where this maximum occurs, then f^{*}(q) is given by the intersection point between the tangent to the graph and the vertical axis, and q=df/dx. Eschrig used the Legendre transform to derive new properties of the density-functional theory for finite temperature applications.