Legendre-Invariant Metric

• Legendre-invariant metric is a Riemannian metric on thermodynamic phase-space that remains unchanged under Legendre transformations, ensuring consistency across thermodynamic potentials.
• Its construction relies on preserving the contact structure using conditions like infinitesimal Legendre-invariance and K-contact criteria, which align with both geometric and statistical formulations.
• These metrics are key in geometrothermodynamics and extend into statistical and Kähler settings, providing representation-invariant insights into thermodynamic interactions and phase transitions.

A Legendre-invariant metric is a Riemannian metric on a thermodynamic phase-space or related geometric context possessing invariance under Legendre transformations—mappings that interchange thermodynamic potentials and their conjugate pairs. Such metrics play a central role in the formalism of Geometrothermodynamics (GTD), ensuring that geometric and physical properties derived from the metric are independent of the choice of thermodynamic potential. Legendre-invariance manifests both in the classical GTD phase-space and, in generalizations, in statistical and complex (Kähler) geometric settings.

1. Legendre Transformations and Thermodynamic Phase-Space Structure

In the GTD framework, a thermodynamic system with $n$ degrees of freedom is represented by a $(2n+1)$-dimensional contact manifold $\mathcal{T}$. The standard coordinates are $Z^A = (\Phi, q^a, p_a)$, with $\Phi$ the thermodynamic potential (e.g., entropy, internal energy), and $q^a$, $p_a$ the conjugate extensive and intensive variables. The contact structure is encoded by a 1-form $\eta = d\Phi - p_a\,dq^a$, which enforces the First Law of thermodynamics via the non-integrability condition $\eta\wedge(d\eta)^n \neq 0$.

Legendre transformations act on $\mathcal{T}$ by interchanging the role of $\Phi$ and some or all pairs $(q^a, p_a)$. A total Legendre transformation is given by: $$\tilde{\Phi} = \Phi - p_a\,q^a,\qquad \tilde{q}^a = -p_a,\qquad \tilde{p}_a = q^a,$$ leaving the contact structure invariant. Infinitesimal Legendre symmetry is generated by the contact Hamiltonian $h(q,p) = \frac12\sum_{a=1}^n[(q^a)^2+(p_a)^2]$ and its associated Hamiltonian vector field $X_L$ (Garcia-Pelaez et al., 2014).

2. Legendre-Invariant Metrics: Construction and Criteria

A Riemannian metric $G$ on $\mathcal{T}$ is Legendre-invariant if it is preserved under the action of Legendre transformations: $\mathcal{L}_{X_L}G = 0,$ where $\mathcal{L}_{X_L}$ denotes the Lie derivative with respect to the infinitesimal Legendre generator. In addition, K-contact metrics demand invariance under the Reeb vector field $R = \partial/\partial \Phi$, i.e., $\mathcal{L}_R G=0$.

For $n=2$ (five-dimensional phase space), imposing these requirements uniquely determines $G$ up to a single Legendre-invariant function $\Omega(q,p)$: $$G = (d\Phi - p_a\,dq^a)\otimes(d\Phi - p_b\,dq^b) + 2\,\Omega(q,p)\,\epsilon_{a}{}^{b}\,q^a\,dp_b\otimes dq^b,$$ where $\epsilon_{a}{}^{b}$ is the canonical symplectic matrix. Legendre-invariance further requires $X_L[\Omega]=0$, so $\Omega$ is constant along Legendre orbits (Garcia-Pelaez et al., 2014).

An alternative construction widely used in GTD is the $G^{III}$-type metric: $$G^{III} = (dS - I_a\,dE^a)^2 + (E^a I_a)^{2K+1} dE^a\,dI_a,\quad K\in\mathbb{Z},$$ which is manifestly invariant under arbitrary Legendre transformations, including partial ones (Pineda-Reyes et al., 2018, Bravetti et al., 2013).

3. Statistical and Geometric Origins

The statistical origin of Legendre-invariant metrics arises from the decomposition of the differential of the microscopic entropy $s(\Gamma) = \phi - I_a H^a(\Gamma)$. The first moment $\langle ds\rangle = d\phi - E^a dI_a$ yields the Gibbs one-form, and the variance $$\langle(ds - \langle ds\rangle)^2\rangle = dE^a\overset{s}{\otimes} dI_a$$ encodes equilibrium fluctuations. The canonical Legendre-invariant Riemannian metric in this formulation is: $$G = \langle ds\rangle\otimes\langle ds\rangle + \langle(ds-\langle ds\rangle)^2\rangle ,$$ whose pull-back to the equilibrium manifold produces standard thermodynamic metrics (Weinhold, Ruppeiner) and, upon appropriate reparametrization of variables, leads directly to the $G^{III}$ family (Pineda-Reyes et al., 2018).

4. Legendre-Invariant Metrics on the Equilibrium Manifold

The space of equilibrium states $\mathcal{E}$ is defined as the maximal integral submanifold where the contact form vanishes, $\eta=0$, which enforces $p_a = \partial \Phi/\partial q^a$. The induced metric $g = \varphi^* G$ (where $\varphi$ is the embedding of $\mathcal{E}$ into $\mathcal{T}$) inherits Legendre-invariance if $G$ is so. For the $G^{III}$ metrics, the induced metric reads: $$g = \left(E^a \frac{\partial S}{\partial E^a}\right)^{2K+1} \frac{\partial^2 S}{\partial E^a \partial E^b} dE^a dE^b.$$ This structure ensures that scalar curvature and other geometric invariants are independent of the thermodynamic potential (representation) chosen, provided the underlying potential is homogeneous of a fixed degree (Bravetti et al., 2013).

5. Applications and Constraints in Thermodynamic Geometry

The principal motivation for Legendre-invariant metrics is to provide geometric invariants—especially the scalar curvature—encoding information about thermodynamic interaction and phase transitions that do not depend on arbitrary choices of thermodynamic potential. For example, in models with homogeneous fundamental relations, the scalar curvature computed from the induced Legendre-invariant metric remains invariant under changes of representation (e.g., internal energy, entropy, Helmholtz free energy) (Bravetti et al., 2013, Pineda-Reyes et al., 2018).

A significant constraint is encountered when imposing stricter symmetry, such as infinitesimal Legendre-invariance (i.e., $X_L[\Omega]=0$ for the metric function $\Omega$) in conjunction with the requirement that the scalar curvature vanishes for non-interacting systems (such as the ideal gas). In this regime, no nontrivial solution exists reconciling both conditions, so regular interpretations of curvature as a direct indicator of interactions may fail. This suggests potential need for either relaxing the K-contact hypothesis, broadening the metric ansatz, or reconsidering the geometric interpretation of curvature (Garcia-Pelaez et al., 2014).

6. Extensions: Complex Legendre-Invariance and Kähler Geometry

In complex geometry, Legendre-invariant metrics appear as fixed points of complex Legendre duality on spaces of Kähler potentials. On a compact Kähler manifold $(X, \omega_0)$, the space of real-analytic Kähler potentials $\mathcal{H}$ admits an involutive transformation $L$ (complex Legendre duality), which is a local isometry for the Mabuchi–Semmes–Donaldson metric: $$g_\varphi(\delta\varphi_1, \delta\varphi_2) = \int_X \delta\varphi_1 \delta\varphi_2\, \omega_\varphi^n/n!.$$ A potential $\varphi\in\mathcal{H}$ is "Legendre-invariant" if $L(\varphi) = \varphi$ up to an additive constant, characterized by the reflection property $$\varphi(z)+\varphi(\zeta) = \text{Re}\langle z, \zeta\rangle$$ for $\zeta = \partial\varphi(z)$. These fixed points correspond to real-analytic Kähler metrics. Classical examples include the Euclidean and Fubini–Study metrics, both of which are Legendre-invariant in this sense (Lempert, 2017).

7. Comparative Table: Key Properties of Legendre-Invariant Metrics in GTD

Property	K-contact Infinitesimal (Garcia-Pelaez et al., 2014)	$G^{III}$ Family (Pineda-Reyes et al., 2018)	Conformal Metrics (Bravetti et al., 2013)
Phase-space dimension	$2n+1$	$2n+1$	$2n+1$
Metric form	$\eta\otimes\eta + 2\Omega\,\epsilon q dp\otimes dq$	$(dS-I_a dE^a)^2 + (E^a I_a)^{2K+1} dE^a dI_a$	$\Theta\otimes\Theta+(E^a I_a)dE^a dI^a$
Legendre invariance type	Infinitesimal (Lie derivative vanishes)	Full (partial and total transformations)	Total Legendre transformations
Free functions	$\Omega(q,p)$, $X_L[\Omega]=0$	$K\in\mathbb{Z}$	Choice of conformal factor
Curvature invariance	Not generally compatible with $R=0$	Representation-invariant for homogeneous potentials	Representation-invariant for homogeneous potentials

Legendre-invariant metrics thus represent a cornerstone of geometric thermodynamics, ensuring that physical and geometric features extracted from the manifold structure are robust under the fundamental symmetry of Legendre transformation. The interplay of geometric, statistical, and algebraic requirements shapes both the formalism and potential limitations of such metrics across thermodynamics and Kähler geometry (Garcia-Pelaez et al., 2014, Pineda-Reyes et al., 2018, Lempert, 2017, Bravetti et al., 2013).