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2.15: Appendix II- Legendre Transformations

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A convex function of a single variable f(x) is one for which f everywhere. The Legendre transform of a convex function f(x) is a function g(p) defined as follows. Let p be a real number, and consider the line y=px, as shown in Figure \PageIndex{1}. We define the point x(p) as the value of x for which the difference F(x,p)=px-f(x) is greatest. Then define g(p)=F\big(x(p),p\big).26 The value x(p) is unique if f(x) is convex, since x(p) is determined by the equation

f'\big(x(p)\big)=p\ .

Note that from p=f'\big(x(p)\big) we have, according to the chain rule,

{d\over dp}\,f'\big(x(p)\big)=f''\big(x(p)\big)\,x'(p)\qquad\Longrightarrow\qquad x'(p)=\Big[f''\big(x(p)\big)\Big]^{-1}\ .

From this, we can prove that g(p) is itself convex:

\begin{split} g'(p)&={d\over dp} \Big[ p\,x(p)-f\big(x(p)\big)\Big]\\ &=p\,x'(p) + x(p) - f'\big(x(p)\big)\,x'(p)= x(p)\ , \end{split}

hence

g''(p)=x'(p)=\Big[f''\big(x(p)\big)\Big]^{-1}>0\ .

clipboard_e1024ab614a8717f55b8bbba63a60dfd5.png
Figure \PageIndex{1}: Construction for the Legendre transformation of a function f(x).

In higher dimensions, the generalization of the definition f''(x)>0 is that a function F(x\ns_1,\ldots,x\ns_n) is convex if the matrix of second derivatives, called the Hessian,

H_{ij}(\Bx)={\pz^2 \!F\over\pz x\ns_i\,\pz x\ns_j}

is positive definite. That is, all the eigenvalues of H_{ij}(\Bx) must be positive for every \Bx. We then define the Legendre transform \BG(\Bp) as

\BG(\Bp)=\Bp\cdot\Bx-F(\Bx)

where

\Bp=\bnabla F\ .

Note that

dG=\Bx\cdot d\Bp + \Bp\cdot d\Bx - \bnabla F\cdot d\Bx = \Bx\cdot d\Bp\ ,

which establishes that G is a function of \Bp and that

{\pz G\over\pz p\ns_j}=x\ns_j\ . \label{LTcond}

Note also that the Legendre transformation is self dual, which is to say that the Legendre transform of G(\Bp) is F(\Bx): F\to G\to F under successive Legendre transformations.

We can also define a partial Legendre transformation as follows. Consider a function of q variables F(\Bx,\By), where \Bx=\{x\ns_1,\ldots,x\ns_m\} and \By=\{y\ns_1,\ldots,y\nd_n\}, with q=m+n. Define \Bp=\{p\ns_1,\ldots,p\ns_m\}, and

G(\Bp,\By)=\Bp\cdot\Bx-F(\Bx,\By)\ ,

where

p\ns_a={\pz F\over \pz x\ns_a}\qquad (a=1,\ldots,m)\ .

These equations are then to be inverted to yield

x\ns_a=x\ns_a(\Bp,\By)={\pz G\over \pz p\ns_a}\ .

Note that

p\ns_a={\pz F\over \pz x\ns_a}\,\big(\Bx(\Bp,\By),\By\big)\ .

Thus, from the chain rule,

\delta\ns_{ab}={\pz p\ns_a\over\pz p\ns_b}={\pz^2\!F\over\pz x\ns_a\,\pz x\ns_c}\,{\pz x\ns_c\over\pz p\ns_b} ={\pz^2\!F\over\pz x\ns_a\,\pz x\ns_c}\,{\pz^2\!G\over\pz p\ns_c\,\pz p\ns_b}\ ,

which says

{\pz^2 \! G\over\pz p\ns_a\,\pz p\ns_b}={\pz x\ns_a\over\pz p\ns_b}=\SK^{-1}_{ab}\ ,

where the m\times m partial Hessian is

{\pz^2\!F\over\pz x\ns_a\,\pz x\ns_b}={\pz p\ns_a\over\pz x\ns_b}=\SK\ns_{ab}\ .

Note that \SK\ns_{ab}=\SK\ns_{ba} is symmetric. And with respect to the \By coordinates,

{\pz^2 \! G\over\pz y\ns_\mu\,\pz y\ns_\nu}=-{\pz^2 \! F\over\pz y\ns_\mu\,\pz y\ns_\nu}=-\SL\ns_{\mu\nu}\ ,

where

\SL\ns_{\mu\nu}={\pz^2\!F\over\pz y\ns_\mu\,\pz y\ns_\nu}

is the partial Hessian in the \By coordinates. Now it is easy to see that if the full q\times q Hessian matrix H\ns_{ij} is positive definite, then any submatrix such as \SK\ns_{ab} or \SL\ns_{\mu\nu} must also be positive definite. In this case, the partial Legendre transform is convex in \{p\ns_1,\ldots,p\ns_m\} and concave in \{y\ns_1,\ldots,y\ns_n\}.


This page titled 2.15: Appendix II- Legendre Transformations is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Daniel Arovas.

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