2.15: Appendix II- Legendre Transformations

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\( \newcommand\VLambda{\vec\Lambda}\)
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\(\newcommand \gtwid{\approx}\)

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\( \newcommand\bsqcap{\mbox{\boldmath{$\sqcap$}}}\)

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\( \newcommand\longrightleftharpoons{ \mathop{\vcenter{\hbox{\ooalign{\raise1pt\hbox{$\longrightharpoonup\joinrel$}\crcr  \lower1pt\hbox{$\longleftharpoondown\joinrel$}}}}}}\)
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\( \newcommand\rok{\Br\ns_1\,,\,\ldots\,,\,\Br\ns_k}\)
\( \newcommand\xhiOZ{\xhi^\ssr{OZ}}\)
\( \newcommand\xhihOZ

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\( \newcommand\nda{\nd_\alpha}\)
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A convex function of a single variable \(f(x)\) is one for which \(f''(x)>0\) 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

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

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 \(\{y\ns_1,\ldots,y\ns_n\}\).