# Legendre Transforms

The microcanonical ensemble involved the thermodynamic variables $$N$$, $$V$$ and $$E$$ as its variables. However, it is often convenient and desirable to work with other thermodynamic variables as the control variables. Legendre transforms provide a means by which one can determine how the energy functions for different sets of thermodynamic variables are related. The general theory is given below for functions of a single variable.

Consider a function $$f(x)$$ and its derivative

$y=f'(x) = {df \over dx} \equiv g(x)$

The equation $$y = g (x)$$ defines a variable transformation from $$x$$ to $$y$$. Is there a unique description of the function $$f(x)$$ in terms of the variable $$y$$? That is, does there exist a function $$\phi (y)$$ that is equivalent to $$f(x)$$?

Given a point $$x_0$$, can one determine the value of the function $$f (x_0)$$ given only $$f' (x_0)$$ ? No, for the reason that the function $$f (x_0) + c$$ for any constant $$c$$ will have the same value of $$f' (x_0 )$$ as shown in the figure below.

Figure 1: The Legendre transfer in action

However, the value $$f (x_0)$$ can be determined uniquely if we specify the slope of the line tangent to $$f$$ at $$x_0$$, i.e., $$f' (x_0)$$ and the $$y$$-intercept, $$b (x_0)$$ of this line. Then, using the equation for the line, we have

$f (x_0) = x_0 f' (x_0 ) + b ( x_0 )$

This relation must hold for any general $$x$$:

$f (x) = x f' (x) + b (x)$

Note that $$f' (x)$$ is the variable $$y$$, and $$x = g^{-1} (y)$$, where $$g_{-1}$$ is the functional inverse of $$g$$, i.e., $$g ( g^{-1} (x) ) = x$$. Solving for $$b (x) = b ( g^{-1} (y) )$$ gives

$b (g^{-1} (y)) = f ( g^{-1} (y)) - y g^{-1} (y) \equiv \phi (y)$

where $$\phi (y)$$ is known as the Legendre transform of $$f (x)$$. In shorthand notation, one writes

$\phi (y) = f (x) - xy$

however, it must be kept in mind that $$x$$ is a function of $$y$$.