geometrothermodynamics

{{complex|section|date=June 2018}}

The main ingredient of GTD is a (2n + 1)-dimensional manifold $\backslash mathcal\{T\}$

with coordinates $Z^A=\backslash \{\backslash Phi,E^a,I^a\backslash \}$, where $\backslash Phi$ is an arbitrary thermodynamic potential, $E^a$, $a=1,2,\backslash ldots,n$, are the

extensive variables, and $I^a$ the intensive variables. It is also

possible to introduce in a canonical manner the fundamental

one-form $\backslash Theta\; =\; d\backslash Phi\; -\; \backslash delta\_\{ab\}I^a\; d\; E^b$ (summation over repeated indices) with $\backslash delta\_\{ab\}=\{\backslash rm$

diag}(+1,\ldots,+1), which satisfies the condition $\backslash Theta\; \backslash wedge$

(d\Theta)^n \neq 0, where $n$ is the number of thermodynamic

degrees of freedom of the system, and is invariant with respect to

Legendre transformations{{Cite book|last=Arnold|first=V.I.|title=Mathematical Methods of Classical Mechanics|publisher=Springer Verlag|year=1989|isbn=0-387-96890-3|url-access=registration|url=https://archive.org/details/mathematicalmeth0000arno}}

: $$

\{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 $i\backslash cup\; j$ is any disjoint decomposition of the set of indices $\backslash \{1,\backslash ldots,n\backslash \}$,

and $k,l=\; 1,\backslash ldots,i$. In particular, for $i=\backslash \{1,\backslash ldots,n\backslash \}$ and $i=\backslash emptyset$ we obtain

the total Legendre transformation and the identity, respectively.

It is also assumed that in $\backslash mathcal\{T\}$

there exists a metric $G$ which is also

invariant with respect to Legendre transformations. The triad

$(\backslash mathcal\{T\},\backslash Theta,G)$ 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 $\backslash mathcal\{E\}\backslash subset\; \backslash mathcal\{T\}$

induced by a smooth map $\backslash varphi:\backslash mathcal\{E\}\backslash rightarrow\backslash mathcal\{T\}$,

i.e. $\backslash varphi:\backslash \{E^a\backslash \}\; \backslash mapsto\; \backslash \{\backslash Phi,E^a,I^a\backslash \}$, with $\backslash Phi=\backslash Phi(E^a)$

and $I^a=\; I^a(E^a)$, such that $\backslash varphi^*(\backslash Theta)=\backslash varphi^*(d\backslash Phi$

- \delta_{ab}I^{a}dE^{b})=0 holds, where $\backslash varphi^*$ is the

pullback of $\backslash varphi$. The manifold $\backslash mathcal\{E\}$ is naturally equipped

with the Riemannian metric $g=\backslash varphi^*(G)$. The purpose of GTD is

to demonstrate that the geometric properties of $\backslash mathcal\{E\}$ are

related to the thermodynamic properties of a system with fundamental

thermodynamic equation $\backslash Phi=\backslash Phi(E^a)$.

The condition of invariance with respect total Legendre transformations leads to the metrics

: $$

G^I = (d\Phi- \delta_{ab}I^a d E^b)^2 + \Lambda\, (\xi_{ab} E^a I^b)\left( \delta_{cd} dE^c d I^d\right)\ ,\quad

\delta_{ab}={\rm diag}(1,\ldots,1)

: $$

G^{II} = (d\Phi- \delta_{ab}I^a d E^b)^2 + \Lambda\, (\xi_{ab} E^a I^b)\left( \eta_{cd} dE^c d I^d\right)\ ,\quad

\eta_{ab}={\rm diag}(-1,1,\ldots,1)

where $\backslash xi\_\{ab\}$ is a constant diagonal matrix that can be expressed in terms of $\backslash delta\_\{ab\}$ and

$\backslash eta\_\{ab\}$, and $\backslash Lambda$ is an arbitrary Legendre invariant function of $Z^A$. The metrics $G^I$ and $G^\{II\}$ 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

: $$

G^{III} = (d\Phi- \delta_{ab}I^a d E^b)^2 + \Lambda\, ( E_a I_a)^{2k+1} \left( dE^a d I^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 $\{\backslash mathcal\; E\}$ can be computed as

: $$

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,{{cite journal|last1=Quevedo | first1=H.|last2=Sanchez | first2=A. |last3=Taj | first3=S.|last4=Vazquez | first4=A. |year=2011|title=Phase transitions in Geometrothermodynamics|journal=Gen. Rel. Grav. |volume=43| issue=4|pages=1153–1165|arxiv=1010.5599 |bibcode=2011GReGr..43.1153Q|doi=10.1007/s10714-010-0996-2| s2cid=119152990}} in the context of relativistic cosmology,{{cite journal|last=Aviles | first=A. |title=Extending the generalized Chaplygin gas model by using geometrothermodynamics |year=2012

|journal=Phys. Rev. D |volume=86| issue=6 |pages=063508|arxiv=1203.4637|bibcode=2012PhRvD..86f3508A|doi=10.1103/PhysRevD.86.063508| s2cid=119185894 }} and to describe chemical reactions

.{{cite journal|last=Tapias | first=D. |year=2013|title=Geometric description of chemical reactions

|arxiv=1301.0262|bibcode=2013arXiv1301.0262Q}}

Category:Branches of thermodynamics

Category:Geometry