Souriau-like Thermodynamics

• Souriau-like thermodynamics is a geometric framework that generalizes Gibbs mechanics by replacing scalar temperature with a Lie-algebra-valued counterpart linked to momentum maps.
• It organizes thermodynamic variables through symplectic and Kähler geometries, using affine coadjoint actions and convergence conditions to define generalized Gibbs states.
• The framework integrates information geometry and nonlinear observables, paving the way for alternative formulations in cycle geometry and irreversible thermodynamics.

Searching arXiv for recent and foundational papers on Souriau-like thermodynamics and closely related geometric thermodynamics. arXiv query: "Souriau thermodynamics Kähler non compact symmetric spaces" Souriau-like thermodynamics denotes a family of geometric thermodynamic formalisms whose canonical reference point is Jean-Marie Souriau’s generalization of Gibbs equilibrium from a scalar inverse temperature to a Lie-algebra-valued generalized temperature conjugate to a momentum map on a symplectic manifold of motions. In that setting, equilibrium is not tied only to time translations, but to a Hamiltonian Lie-group action; thermodynamic variables are organized by symplectic geometry, affine coadjoint structures, and generalized Gibbs measures. In contemporary usage, the label also covers direct extensions to Kähler symmetric spaces and a wider penumbra of contact, symplectic, and information-geometric frameworks that preserve part of this structural program without reproducing all of Souriau’s machinery (Marle, 2016, Fré et al., 18 Dec 2025).

1. Foundational Souriau formalism

In Souriau’s formulation, the basic geometric object is the manifold of motions rather than a chosen time-slice of phase space. Starting from an evolution space such as $\mathbb{R}\times T^*N$, equipped with a presymplectic form derived from the Poincaré–Cartan 1-form, one quotients by the characteristic foliation to obtain a symplectic manifold parametrizing entire solutions. This shift is essential because it removes the privileged role of a single time parameter and makes room for more general symmetry groups than the one-dimensional group of time translations (Marle, 2016).

Given a Hamiltonian action of a Lie group $G$ with Lie algebra $\mathfrak g$ on a symplectic manifold $(M,\omega)$, a momentum map

$J:M\to\mathfrak g^*$

replaces the ordinary Hamiltonian as the fundamental thermodynamic observable. Souriau’s generalized Gibbs state is then defined, for $b\in\mathfrak g$, by

$$P(b)=\int_M e^{-\langle J,b\rangle}\,d\lambda_\omega,\qquad p_b=\frac{1}{P(b)}e^{-\langle J,b\rangle},$$

with mean momentum

$$E_J(b)=\frac{1}{P(b)}\int_M J\,e^{-\langle J,b\rangle}\,d\lambda_\omega.$$

The entropy takes the Legendre form

$S(b)=\log P(b)+\langle E_J(b),b\rangle.$

Ordinary Gibbs statistical mechanics is recovered when $G=\mathbb{R}$, $G$0, and $G$1 is scalar inverse temperature. In that precise sense, Souriau-like thermodynamics is a strict extension of Gibbs thermodynamics rather than an unrelated alternative (Marle, 2016).

A central consequence is that equilibrium is defined relative to the one-parameter subgroup generated by $G$2, not relative to a universal laboratory time flow. The generalized Gibbs state remains invariant under $G$3, and entropy retains a constrained-maximum characterization, but the constraint is now the full mean momentum map rather than mean energy alone. This replacement of $G$4 by $G$5 is the decisive conceptual move that distinguishes Souriau-like thermodynamics from more conventional geometric thermodynamics.

2. Affine equivariance, cocycles, and generalized temperature

A characteristic Souriau feature is that momentum maps need not be exactly coadjoint-equivariant. In the general case they transform by an affine coadjoint action corrected by a group 1-cocycle $G$6, satisfying

$G$7

with infinitesimal Lie-algebra cocycle

$G$8

This cocycle correction is not a technical embellishment: it governs how partition functions, mean momentum, and entropy transform under symmetry, and it is one of the most specific markers of Souriau’s own formalism (Marle, 2016).

Within this framework, the generalized inverse temperature is the Lie algebra element $G$9. Its components may encode not only thermal intensity but also kinematical data associated with the chosen symmetry subgroup. In Galilean examples, the entries of $\mathfrak g$0 can include rotation, acceleration, boosts, and time translation; the usual scalar inverse temperature appears as one component, with

$\mathfrak g$1

This geometrizes temperature by embedding it into the symmetry algebra rather than treating it as an isolated scalar parameter (Marle, 2016).

The physical content becomes especially clear in moving-frame examples. For a gas in a vessel transported by a one-parameter Galilean subgroup, the generalized Gibbs distribution reduces in the co-moving frame to an ordinary Gibbs distribution for relative kinetic energy plus an effective potential. Uniform translation reproduces the ideal gas in an inertial frame; uniform acceleration reproduces equilibrium in a constant gravity field; uniform rotation gives the centrifugal equilibrium relevant for centrifuges. Souriau-like thermodynamics therefore enlarges the very notion of equilibrium: equilibrium is symmetry-adapted, and different choices of $\mathfrak g$2 select different stationary sectors of the same dynamical theory (Marle, 2016).

This also explains an important limitation. Generalized Gibbs states exist only on an open domain $\mathfrak g$3 where the defining integrals converge. For the full Galilean group, $\mathfrak g$4 may be empty, so only suitable subgroups admit generalized equilibrium states. Souriau-like thermodynamics is therefore not a purely formal reparameterization; it is constrained by analytic existence conditions.

3. Kähler symmetric-space extensions

Recent work has extended Souriau-like thermodynamics from abstract symplectic manifolds to non-compact Kähler symmetric spaces

$\mathfrak g$5

motivated in part by Cartan Neural Networks and by the search for Gibbs distributions directly on curved event manifolds rather than on tangent bundles. In this setting, the microscopic observables are the Killing moment maps

$\mathfrak g$6

where $\mathfrak g$7 is the Kähler form and $\mathfrak g$8 are Killing vector fields. A global formula used in the Calabi–Vesentini construction is

$\mathfrak g$9

with Poisson realization

$(M,\omega)$0

The corresponding Gibbs state is

$(M,\omega)$1

A sharp theorem in this program states that, for non-compact symmetric spaces $(M,\omega)$2, Souriau-type Gibbs distributions on the manifold itself exist only when $(M,\omega)$3 is Kähler, equivalently when the maximal compact subalgebra contains a central $(M,\omega)$4 factor (Fré et al., 18 Dec 2025).

The same work determines the space of generalized temperatures by convergence of the partition function. The admissible set is the orbit under the adjoint action of $(M,\omega)$5 of a positivity domain in the compact Cartan subalgebra $(M,\omega)$6. Consequently, the independent temperature parameters can always be reduced to a minimal set of cardinality

$(M,\omega)$7

In explicit examples, this becomes a concrete cone condition. On the Poincaré plane, one obtains inequalities such as

$(M,\omega)$8

while on the Siegel plane the compact-Cartan reduction yields

$(M,\omega)$9

These are not heuristic stability conditions; they are the exact convergence domain of the generalized partition function (Fré et al., 18 Dec 2025).

A further development equips this microscopic Kähler construction with an exact macroscopic thermodynamic layer on Calabi–Vesentini manifolds

$J:M\to\mathfrak g^*$0

For the pure Souriau limit, exact partition functions are obtained for the entire Calabi–Vesentini Tits–Satake universality class: $J:M\to\mathfrak g^*$1 with

$J:M\to\mathfrak g^*$2

and convergence cone

$J:M\to\mathfrak g^*$3

The formalism then extends the temperature vector by additional components $J:M\to\mathfrak g^*$4 conjugate not to Killing moment maps but to nonlinear action variables built from square roots of Casimirs of nested compact subalgebras. The resulting “extended Souriau” Gibbs distributions introduce order parameters

$J:M\to\mathfrak g^*$5

and the papers explicitly propose an analogy with spontaneous magnetization when these expectation values remain nonzero as $J:M\to\mathfrak g^*$6 (Fré et al., 8 Jun 2026).

This suggests a significant broadening of Souriau’s original program. The Lie-group/moment-map structure remains central, but the observables need no longer be restricted to linear momentum-map components; nonlinear Casimir-derived actions may enter as additional thermodynamic channels.

4. Information geometry and macroscopic thermodynamic geometry

A recent strand of Souriau-like work makes the link between macroscopic thermodynamics and information geometry explicit. Starting from a Gibbs family

$J:M\to\mathfrak g^*$7

one introduces a macroscopic thermodynamic manifold

$J:M\to\mathfrak g^*$8

with coordinates $J:M\to\mathfrak g^*$9, contact form

$b\in\mathfrak g$0

and Reeb vector field

$b\in\mathfrak g$1

On the Reeb-transverse leaves, the induced symplectic form is

$b\in\mathfrak g$2

The crucial innovation is the introduction of a macroscopic metric

$b\in\mathfrak g$3

whose restriction to the Reeb-transverse symplectic leaves is Kähler, and whose pullback to the equilibrium Lagrangian submanifold is exactly the Fisher/Hessian metric

$b\in\mathfrak g$4

The sequence

$b\in\mathfrak g$5

is the paper’s geometric explanation of why information geometry and thermodynamic geometry coincide in Gibbs families (Fré et al., 8 Jun 2026).

A complementary one-parameter equilibrium construction appears in the Fisher-based “dual structure” of thermodynamics. For the canonical Gibbs ensemble, the Fisher information with respect to $b\in\mathfrak g$6 is promoted to a thermodynamic state function

$b\in\mathfrak g$7

with Legendre-dual companion

$b\in\mathfrak g$8

The same quantity is the reciprocal of entropy curvature,

$b\in\mathfrak g$9

so it simultaneously measures fluctuation magnitude, heat capacity, temperature-estimation precision, and local thermodynamic stability. This framework is not Souriau’s Lie-theoretic thermodynamics, but it develops a parallel Hessian/Legendre geometry in which Fisher information becomes a bona fide thermodynamic potential (Porporato, 2013).

Taken together, these results support a broad interpretation of Souriau-like thermodynamics in which symmetry, convexity, and information are not competing languages but different projections of a common geometric infrastructure. This suggests an overview, although the available papers stop short of a single unified formalism.

5. Adjacent geometric frameworks

Several papers are closely related to Souriau-like thermodynamics while remaining formally distinct from it. One major line uses symplectization of contact thermodynamics. In this approach, the thermodynamic phase space is the projectivized cotangent bundle $$P(b)=\int_M e^{-\langle J,b\rangle}\,d\lambda_\omega,\qquad p_b=\frac{1}{P(b)}e^{-\langle J,b\rangle},$$0, while its homogeneous symplectic lift is $$P(b)=\int_M e^{-\langle J,b\rangle}\,d\lambda_\omega,\qquad p_b=\frac{1}{P(b)}e^{-\langle J,b\rangle},$$1. Equilibrium states are represented by homogeneous Lagrangian submanifolds, energy and entropy representations become different dehomogenizations of the same global object, and thermodynamic processes are generated by Hamiltonians homogeneous of degree one in the co-extensive variables. The resulting framework unifies local Weinhold and Ruppeiner metrics as restrictions of a global degenerate metric on the homogeneous Lagrangian manifold and develops open-system dynamics via port-thermodynamic systems. It is strongly parallel to the homogeneous symplectic side of Souriau-like thermodynamics, but it does not use Lie-group actions, momentum maps, or generalized Gibbs states (Schaft et al., 2018).

A second nearby development concerns cycle geometry on equilibrium thermodynamic manifolds. For a simple compressible system, one introduces the contact form

$$P(b)=\int_M e^{-\langle J,b\rangle}\,d\lambda_\omega,\qquad p_b=\frac{1}{P(b)}e^{-\langle J,b\rangle},$$2

and on the equilibrium Legendre submanifold obtains the intrinsic two-form

$$P(b)=\int_M e^{-\langle J,b\rangle}\,d\lambda_\omega,\qquad p_b=\frac{1}{P(b)}e^{-\langle J,b\rangle},$$3

This identifies the familiar work and reversible-heat area laws as two coordinate expressions of a single underlying geometric object: $$P(b)=\int_M e^{-\langle J,b\rangle}\,d\lambda_\omega,\qquad p_b=\frac{1}{P(b)}e^{-\langle J,b\rangle},$$4 Infinitesimal cycle work is then governed by the local coefficient $$P(b)=\int_M e^{-\langle J,b\rangle}\,d\lambda_\omega,\qquad p_b=\frac{1}{P(b)}e^{-\langle J,b\rangle},$$5, interpreted in the paper as a local work density or curvature density. This is not Souriau thermodynamics proper, but it is a clear example of representation-invariant thermodynamic geometry built from canonical forms and Legendre submanifolds (Bittner, 23 Mar 2026).

A third cluster addresses irreversible finite-dimensional thermodynamics through contact or almost-Poisson dynamics. For simple thermodynamical systems with friction on $$P(b)=\int_M e^{-\langle J,b\rangle}\,d\lambda_\omega,\qquad p_b=\frac{1}{P(b)}e^{-\langle J,b\rangle},$$6, the evolution field

$$P(b)=\int_M e^{-\langle J,b\rangle}\,d\lambda_\omega,\qquad p_b=\frac{1}{P(b)}e^{-\langle J,b\rangle},$$7

satisfies

$$P(b)=\int_M e^{-\langle J,b\rangle}\,d\lambda_\omega,\qquad p_b=\frac{1}{P(b)}e^{-\langle J,b\rangle},$$8

for natural mechanical Hamiltonians, thereby encoding exact total-energy conservation together with entropy growth. Closely related skew-bracket formulations describe isolated irreversible systems through evolution vector fields generated by a single skew structure and compatible discrete-gradient schemes that preserve $$P(b)=\int_M e^{-\langle J,b\rangle}\,d\lambda_\omega,\qquad p_b=\frac{1}{P(b)}e^{-\langle J,b\rangle},$$9 exactly while maintaining $$E_J(b)=\frac{1}{P(b)}\int_M J\,e^{-\langle J,b\rangle}\,d\lambda_\omega.$$0-production under appropriate hypotheses (Simoes et al., 2020, Simoes et al., 2020).

A further extension concerns noncanonical Souriau forms on velocity phase space. Noncommutative Souriau 2-forms on $$E_J(b)=\frac{1}{P(b)}\int_M J\,e^{-\langle J,b\rangle}\,d\lambda_\omega.$$1 produce deformed Poisson brackets, presymplectic reductions at singular loci, and invariant volume-preserving flows. Thermodynamics is not developed there, but the preserved volume and reduced symplectic structures suggest a possible route toward generalized statistical mechanics on noncanonical state spaces (Cariñena et al., 2015).

These adjacent frameworks share canonical forms, Lagrangian or Legendre submanifolds, and representation changes. The crucial distinction is that they typically lack Souriau’s defining triad: Hamiltonian Lie-group action, momentum map, and generalized Gibbs state.

6. Boundaries, alternatives, and current directions

A recurring misconception is that any geometric or variational thermodynamics is thereby Souriau-like. The recent literature argues against that identification. A notable example is the Path-Dependent Energy Lagrangian for irreversible thermomechanical systems,

$$E_J(b)=\frac{1}{P(b)}\int_M J\,e^{-\langle J,b\rangle}\,d\lambda_\omega.$$2

with upper-limit/tangential variation rule

$$E_J(b)=\frac{1}{P(b)}\int_M J\,e^{-\langle J,b\rangle}\,d\lambda_\omega.$$3

This framework unifies mechanics, internal-variable dynamics, and entropy/heat balance in a single action and routes the same channel power into both dissipative forces and heat production. Yet it explicitly omits Lie-group action, momentum maps, coadjoint orbits, and symplectic/contact statistical mechanics, so it is best classified as a neighboring irreversible variational formalism rather than Souriau thermodynamics (Ren, 1 Nov 2025).

An even more radical alternative is constructor-theoretic thermodynamics, where the primitive objects are substrates, attributes, variables, and tasks rather than manifolds and differential forms. Its central notions are possible and impossible tasks, work media, heat media, and adiabatic accessibility defined by side-effects on work media alone. Within that framework, thermodynamics becomes a scale-independent theory of accessibility and impossibility, not a geometric theory of equilibrium states or symmetries. This makes it highly relevant as a foundational comparison point, but not as a Souriau-like framework in any formal sense (Marletto, 2016).

There is also a foundational equilibrium line based on measured state spaces, two orderings, and Samuelson’s area condition. In that setting one derives empirical temperature and entropy from the two orderings, then recalibrates them to absolute variables so that

$$E_J(b)=\frac{1}{P(b)}\int_M J\,e^{-\langle J,b\rangle}\,d\lambda_\omega.$$4

from which Maxwell relations and the potentials $$E_J(b)=\frac{1}{P(b)}\int_M J\,e^{-\langle J,b\rangle}\,d\lambda_\omega.$$5 follow. This produces a minimal area-preserving equilibrium geometry that is strongly suggestive of symplectic structure, but it is order-theoretic rather than Lie-theoretic (Cooper et al., 2011).

The present landscape therefore has a clear internal stratification. At its core, Souriau-like thermodynamics remains the symmetry-driven thermodynamics of momentum maps, affine coadjoint structures, and generalized Gibbs states. Around that core lie Kähler symmetric-space extensions with exact partition functions, Fisher/Hessian reinterpretations of equilibrium geometry, contact and homogeneous symplectic formulations of thermodynamic processes, and variational or operational alternatives for irreversibility. A plausible implication is that “Souriau-like” now names not a single closed doctrine but a research program: to identify which parts of thermodynamics are best understood through symmetry, which through contact or Hessian geometry, and which through broader structural notions of accessibility, response, and statistical inference.