Talk:Legendre transformation

Possible copyright problem

The exposition quite closely follows the reference Rockafellar. I hope this is not a copyright problem. —the preceding unsigned comment is by 84.239.128.9 (talk • contribs) 09:26, 31 October 2005 (UTC1)

Name for this property?

Does anybody know how the property

$f(x_1,\dots ,x_N)=g(x_1)+\cdots +g(x_N)\Rightarrow f^{\star }(p_1,\dots ,p_N)=g^{\star }(p_1)+\cdots +g^{\star }(p_N)$

is refered to in the literature? —the preceding unsigned comment is by Tobias Bergemann (talk • contribs) 12:12, 13 December 2005 (UTC1)

$x^{*}\to \phi$ for clarity

In the definition, I would switch the $x^{*}$ to $\phi$ (or something else) for clarity. Hvstms (talk) 10:48, 19 September 2023 (UTC)

Circularity in definition

The definitions provided are phrased in a confusing way, with the domain of the function $f_{*}$ depending on $f_{*}$ itself. One can avoid this by saying, for example, 'defined where this supremum is finite', or simply by repeating the function's definition in the definition for $I_{*}$, ie $I_{*}=\{y\in \mathbb{ℝ}:\underset{x\in \mathbb{ℝ}}{\sup }\{xy-f(x)\}<\mathrm{\infty }\}$. Happy to make this edit myself if others agree, but this is not my area of expertise and if this circularity is common in related texts and papers it probably makes sense to keep it here. Corlio5994 (talk) 06:16, 16 March 2024 (UTC)

Legendre transform in thermodynamics

In thermodynamics, the Legendre transform is used in a special simple context. It is that of homogeneous functions of degree 1. It is not usually considered in other thermodynamic contexts.

Thermodynamics refers to functions of several variables, the number of the latter depending on the physical scenario.

The cardinal function is often such as $U(S,V,\{N_j\})$, with, for example

$\mathrm{\Delta }U=T\mathrm{\Delta }S+P\mathrm{\Delta }V+{\mu }_1\mathrm{\Delta }{N}_1+{\mu }_2\mathrm{\Delta }{N}_2$.

This refers to a thermodynamic process of finite changes of extensive variables $U$, $S$, $V$, $\mathrm{\Delta }{N}_1$, and $\mathrm{\Delta }{N}_2$, with the intensive variables $T$, $P$, ${\mu }_1$, and ${\mu }_2$, constrained to be respectively the same at the beginning and end of the process (though not necessarily constant during the finite time course of the process).

In this case, the cardinal function $U(S,V,\{N_j\})$ is considered as a function of the extensive arguments $S$, $V$, and $\{N_j\}$. The variables $T$, $P$, ${\mu }_1$, ${\mu }_2$ are considered as parameters, not as arguments of the cardinal function. They happen to be defined as partial derivatives of the cardinal function with respect to its arguments. They are all experimentally measurable in the system itself, or in the surroundings of the system when the system is connected to the surroundings by suitable selectively 'permeable' walls. For example, the suitable selectively 'permeable' wall for the intensive variable pressure $P$ can be a piston that moves freely in a suitable cylinder that contains the system, and defines its conjugate extensive volume $V$, which is also experimentally measurable in the system or in its surroundings. The free movement of the piston ensures that the pressure within the system is the same as that in the relevant part of the surroundings.

One is then interested in some other homogeneous function such as $H(S,P,\{N_j\})$, with

$\mathrm{\Delta }H=T\mathrm{\Delta }S+V\mathrm{\Delta }P+{\mu }_1\mathrm{\Delta }{N}_1+{\mu }_2\mathrm{\Delta }{N}_2$.

The function $H(S,P,\{N_j\})$ of the arguments $S$, $P$, $N_1$, and $N_2$ is the Legendre transform of the cardinal function $U(S,V,\{N_j\})$ with respect to its extensive argument $V$. The effect of the transform is to replace the extensive argument $V$ with its conjugate intensive variable $P$, which has now been transformed from a parameter to an argument. In this case, the new function has a special name, the enthalpy. The inverse Legendre transform takes one back to the cardinal function, and swaps back argument and parameter.

Such Legendre transforms can be performed with respect to any argument of the above form. Some of the transforms are blessed with special names, but there are many other possibilities that are sometimes useful but lack the blessings of special names, as set out conveniently by Callen.

In effect, this scheme defines the canonical formalism of thermodynamics.

Template:Ref list Template:Ref end Chjoaygame (talk) 08:54, 6 November 2024 (UTC)