Geometrothermodynamics

The main ingredient of GTD is a (2n + 1)-dimensional manifold ${\displaystyle {\mathcal {T}}}$ with coordinates ${\displaystyle Z^{A}=\{\Phi ,E^{a},I^{a}\}}$, where ${\displaystyle \Phi }$ is an arbitrary thermodynamic potential, ${\displaystyle E^{a}}$, ${\displaystyle a=1,2,\ldots ,n}$, are the extensive variables, and ${\displaystyle I^{a}}$ the intensive variables. It is also possible to introduce in a canonical manner the fundamental one-form ${\displaystyle \Theta =d\Phi -\delta _{ab}I^{a}dE^{b}}$ (summation over repeated indices) with ${\displaystyle \delta _{ab}={\rm {diag}}(+1,\ldots ,+1)}$, which satisfies the condition ${\displaystyle \Theta \wedge (d\Theta )^{n}\neq 0}$, where ${\displaystyle n}$ is the number of thermodynamic degrees of freedom of the system, and is invariant with respect to Legendre transformations[3]

${\displaystyle \{Z^{A}\}\longrightarrow \{{\widetilde {Z}}^{A}\}=\{{\tilde {\Phi }},{\tilde {E}}^{a},{\tilde {I}}^{a}\}\ ,\quad \Phi ={\tilde {\Phi }}-\delta _{kl}{\tilde {E}}^{k}{\tilde {I}}^{l},\quad E^{i}=-{\tilde {I}}^{i},\quad E^{j}={\tilde {E}}^{j},\quad I^{i}={\tilde {E}}^{i},\quad I^{j}={\tilde {I}}^{j}\ ,}$

where ${\displaystyle i\cup j}$ is any disjoint decomposition of the set of indices ${\displaystyle \{1,\ldots ,n\}}$, and ${\displaystyle k,l=1,\ldots ,i}$. In particular, for ${\displaystyle i=\{1,\ldots ,n\}}$ and ${\displaystyle i=\emptyset }$ we obtain the total Legendre transformation and the identity, respectively. It is also assumed that in ${\displaystyle {\mathcal {T}}}$ there exists a metric ${\displaystyle G}$ which is also invariant with respect to Legendre transformations. The triad ${\displaystyle ({\mathcal {T}},\Theta ,G)}$ defines a Riemannian contact manifold which is called the thermodynamic phase space (phase manifold). The space of thermodynamic equilibrium states (equilibrium manifold) is an n-dimensional Riemannian submanifold ${\displaystyle {\mathcal {E}}\subset {\mathcal {T}}}$ induced by a smooth map ${\displaystyle \varphi$ :{\mathcal {E}}\rightarrow {\mathcal {T}}} , i.e. ${\displaystyle \varphi$ :\{E^{a}\}\mapsto \{\Phi ,E^{a},I^{a}\}} , with ${\displaystyle \Phi =\Phi (E^{a})}$ and ${\displaystyle I^{a}=I^{a}(E^{a})}$, such that ${\displaystyle \varphi ^{*}(\Theta )=\varphi ^{*}(d\Phi -\delta _{ab}I^{a}dE^{b})=0}$ holds, where ${\displaystyle \varphi ^{*}}$ is the pullback of ${\displaystyle \varphi }$. The manifold ${\displaystyle {\mathcal {E}}}$ is naturally equipped with the Riemannian metric ${\displaystyle g=\varphi ^{*}(G)}$. The purpose of GTD is to demonstrate that the geometric properties of ${\displaystyle {\mathcal {E}}}$ are related to the thermodynamic properties of a system with fundamental thermodynamic equation ${\displaystyle \Phi =\Phi (E^{a})}$. The condition of invariance with respect total Legendre transformations leads to the metrics

${\displaystyle G^{I}=(d\Phi -\delta _{ab}I^{a}dE^{b})^{2}+\Lambda \,(\xi _{ab}E^{a}I^{b})\left(\delta _{cd}dE^{c}dI^{d}\right)\ ,\quad \delta _{ab}={\rm {diag}}(1,\ldots ,1)}$

${\displaystyle G^{II}=(d\Phi -\delta _{ab}I^{a}dE^{b})^{2}+\Lambda \,(\xi _{ab}E^{a}I^{b})\left(\eta _{cd}dE^{c}dI^{d}\right)\ ,\quad \eta _{ab}={\rm {diag}}(-1,1,\ldots ,1)}$

where ${\displaystyle \xi _{ab}}$ is a constant diagonal matrix that can be expressed in terms of ${\displaystyle \delta _{ab}}$ and ${\displaystyle \eta _{ab}}$, and ${\displaystyle \Lambda }$ is an arbitrary Legendre invariant function of ${\displaystyle Z^{A}}$. The metrics ${\displaystyle G^{I}}$ and ${\displaystyle G^{II}}$ have been used to describe thermodynamic systems with first and second order phase transitions, respectively. The most general metric which is invariant with respect to partial Legendre transformations is

${\displaystyle G^{III}=(d\Phi -\delta _{ab}I^{a}dE^{b})^{2}+\Lambda \,(E_{a}I_{a})^{2k+1}\left(dE^{a}dI^{a}\right)\ ,\quad E_{a}=\delta _{ab}E^{b}\ ,\quad I_{a}=\delta _{ab}I^{b}\ .}$

The components of the corresponding metric for the equilibrium manifold ${\displaystyle {\mathcal {E}}}$ can be computed as

${\displaystyle g_{ab}={\frac {\partial Z^{A}}{\partial E^{a}}}{\frac {\partial Z^{B}}{\partial E^{b}}}G_{AB}\ .}$

GTD has been applied to describe laboratory systems like the ideal gas, van der Waals gas, the Ising model, etc., more exotic systems like black holes in different gravity theories,[4] in the context of relativistic cosmology,[5] and to describe chemical reactions .[6]