Souriau Gibbs state on a symplectic G-space

Intuition [Beginner]

In statistical mechanics, a system at thermal equilibrium is described by a probability distribution that tells you how likely each microscopic state is. The classical Gibbs distribution assigns probabilities proportional to $e^{-\beta E}$, where $E$ is the energy and $\beta$ is the inverse temperature. High-energy states are exponentially suppressed.

Souriau asked: what if the system has more conserved quantities than just energy? A rotating gas, for instance, conserves angular momentum as well as energy. The Gibbs distribution should reflect all these conserved quantities, not just energy. On a symplectic manifold with a symmetry group $G$, the conserved quantities are encoded in the moment map $\mu$, a function from the manifold to the dual of the Lie algebra.

The Souriau Gibbs state replaces $e^{-\beta E}$ with $e^{-⟨\beta ,\mu (x)⟩}$, where $\beta$ now lives in the Lie algebra (not just a scalar) and the pairing $⟨\beta ,\mu (x)⟩$ generalises "energy at inverse temperature" to "all conserved quantities weighted by their conjugate thermodynamic variables." Temperature is just one component; angular velocity and chemical potential are others. The symplectic structure ensures this construction is geometrically natural.

Visual [Beginner]

A symplectic manifold drawn as a curved surface with a Hamiltonian group action visualised as flow lines. The moment map $\mu$ is shown as a projection downward onto a lower-dimensional space (the dual Lie algebra). In the lower space, the coadjoint orbits are drawn as concentric shapes. A density on the surface is shown, peaked where $⟨\beta ,\mu (x)⟩$ is smallest (colored red) and decaying exponentially away from this region (fading to blue).

The key geometric insight is that the Gibbs state is a function on the symplectic manifold that is pulled back from a function on the dual Lie algebra via the moment map.

Worked example [Beginner]

The rigid rotor. Consider a free rigid body rotating about its centre of mass. The phase space is the cotangent bundle of the rotation group SO(3), and the moment map for the SO(3) action is the angular momentum $L=(L_1,L_2,L_3)$. The Souriau Gibbs state at "inverse temperature vector" $\beta =({\beta }_1,{\beta }_2,{\beta }_3)$ assigns probability density proportional to $e^{-⟨\beta ,L⟩}=e^{-({\beta }_1{L}_1+{\beta }_2{L}_2+{\beta }_3{L}_3)}$ on the angular momentum sphere.

When $\beta$ points along the $z$-axis, this becomes $e^{-{\beta }_3{L}_3}$, favouring states with $L_3$ aligned along $\beta$. This is the rotational analogue of the canonical ensemble: the "temperature vector" direction determines the preferred axis of rotation, and its magnitude controls how strongly the body prefers alignment. The partition function is the integral of $e^{-⟨\beta ,L⟩}$ over the angular momentum sphere.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Definition (Souriau Gibbs state). Let $(M,\omega )$ be a symplectic manifold with a Hamiltonian action of a Lie group $G$ and moment map $\mu :M\to {\mathfrak{g}}^{\ast }$. For a regular value $\beta \in \mathfrak{g}$ such that the function $x\mapsto ⟨\beta ,\mu (x)⟩$ is bounded below, the Souriau Gibbs state is the probability measure on $M$ with density

$${\rho }_{\beta }(x)=\frac{1}{P(\beta )}\exp (-⟨\beta ,\mu (x)⟩)$$

with respect to the Liouville measure ${\omega }^n/n!$, where $P(\beta )={\int }_M{e}^{-⟨\beta ,\mu (x)⟩}\frac{{\omega }^n}{n!}$ is the Souriau partition function.

Definition (Souriau thermodynamic functions). The free energy is $\Psi (\beta )=-\log P(\beta )$. The mean moment (expected value of the moment map) is $\bar{\mu }(\beta )=-d\Psi (\beta )/d\beta$, and the thermodynamic entropy is $S(\beta )=-⟨\beta ,\bar{\mu }(\beta )⟩-\Psi (\beta )$.

Definition (KKS form and coadjoint orbits). The Kirillov-Kostant-Souriau symplectic form on a coadjoint orbit ${\mathcal{O}}_{\lambda }\subset {\mathfrak{g}}^{\ast }$ is defined by ${\omega }_{\xi }^{\mathrm{KKS}}(\hat{X},\hat{Y})=⟨\xi ,[X,Y]⟩$ for $\xi \in {\mathcal{O}}_{\lambda }$ and $X,Y\in \mathfrak{g}$. The Gibbs density is constant on each coadjoint orbit, depending only on the orbit through $\mu (x)$.

Key theorem with proof [Intermediate+]

Theorem (Entropy maximisation). Among all probability densities on $(M,\omega )$ with a given expected value $\bar{\mu} \in \mathfrak{g}^$ofthemomentmap,theSouriauGibbsstate$\rho_\beta$uniquelymaximisestheGibbs-Boltzmannentropy$S[\rho] = -\int_M \rho \log \rho , \frac{\omega^n}{n!}$.*

Proof sketch. Use the method of Lagrange multipliers for the constrained optimisation problem: maximise $S[\rho ]$ subject to $\int \rho {\omega }^n/n!=1$ and $\int \mu (x)\rho (x){\omega }^n/n!=\bar{\mu }$. The Lagrangian functional is $\mathcal{L}[\rho ]=-\int \rho \log \rho +(\log Z-1)\int \rho -⟨\beta ,\int \mu \cdot \rho ⟩$. The variational derivative $\delta \mathcal{L}/\delta \rho =-\log \rho -1+\log Z-⟨\beta ,\mu ⟩$ vanishes when $\rho =Z^{-1}{e}^{-⟨\beta ,\mu ⟩}$, which is exactly the Souriau Gibbs state. Strict concavity of the entropy functional guarantees uniqueness. $\square$

Bridge. The entropy maximisation theorem places the Souriau Gibbs state at the junction of symplectic geometry and information theory; the moment map constraint $⟨\mu ⟩=\bar{\mu }$ is the symplectic-geometric refinement of the energy constraint in the classical canonical ensemble 11.04.01 pending, where the single scalar $\beta$ is replaced by a full element of the Lie algebra encoding all conserved quantities, while the Liouville measure ${\omega }^n/n!$ plays the role of the uniform reference measure, analogous to how the counting measure serves as the reference for discrete Gibbs distributions 08.01.03. The KKS symplectic form on coadjoint orbits ensures that the Gibbs state respects the symplectic geometry of the phase space, just as the canonical ensemble respects the energy shell structure.

Exercises [Intermediate+]

Advanced results [Master]

Souriau-Noether theorem. Souriau extended the classical Noether theorem to the statistical setting: the Gibbs state ${\rho }_{\beta }$ is invariant under the one-parameter subgroup generated by $X\in \mathfrak{g}$ if and only if $⟨\beta ,X⟩=0$. Equivalently, the symmetry directions conjugate to the thermodynamic variables encoded in $\beta$ are exactly the directions that leave the Gibbs state invariant. This provides a geometric criterion for which symmetries are "broken" by the thermal state.

Souriau's affine action. Souriau showed that the partition function $P(\beta )$ defines an affine action of $G$ on ${\mathfrak{g}}^{\ast }$ via $\Theta (g)(\beta )={\mathrm{Ad}}_g^{\ast }(\beta )+\theta (g)$, where $\theta (g)=d_{\mathrm{eq}}\Psi ({\mathrm{Ad}}_g^{\ast }\beta )$ is the Souriau cocycle. This cocycle measures the failure of the Gibbs state to be $G$-invariant and vanishes exactly when the moment map is equivariant. The cohomology class $[\theta ]\in H^1(G,{\mathfrak{g}}^{\ast })$ is a symplectic invariant.

Synthesis. The Souriau Gibbs state unifies symplectic geometry, statistical mechanics, and representation theory within a single geometric framework; the moment map $\mu :M\to {\mathfrak{g}}^{\ast }$ parametrises the conserved quantities as coordinates on the dual Lie algebra ^{[Souriau 1970]}, while the KKS symplectic form on coadjoint orbits provides the natural measure with respect to which thermal averages are computed, paralleling how the Liouville measure on the energy surface underpins the microcanonical ensemble in 11.04.01 pending. The Souriau cocycle $\theta$ measures the obstruction to global equivariance of the moment map, connecting back to the cohomological invariants of Hamiltonian group actions studied in the AGS convexity theorem, and the entropy maximisation principle identifies the Gibbs state as the unique equilibrium, generalising the Boltzmann distribution of 08.01.03 from a single energy constraint to a full Lie-algebra of constraints parametrised by the moment map.

Full proof set [Master]

Proposition (Ad-invariance of the partition function). The Souriau partition function satisfies $P(\mathrm{Ad}^_g \beta) = P(\beta)$forall$g \in G$.*

Proof. By the equivariance of the moment map, $\mu (g\cdot x)={\mathrm{Ad}}_{g^{-1}}^{\ast }\mu (x)$. Therefore $⟨{\mathrm{Ad}}_g^{\ast }\beta ,\mu (g\cdot x)⟩=⟨\beta ,{\mathrm{Ad}}_{g^{-1}}^{\ast }\mu (g\cdot x)⟩=⟨\beta ,\mu (x)⟩$. The change of variables $x\mapsto g\cdot x$ preserves the Liouville measure (by the Liouville theorem applied to the Hamiltonian $G$-flow). Hence $P({\mathrm{Ad}}_g^{\ast }\beta )=\int e^{-⟨{\mathrm{Ad}}_g^{\ast }\beta ,\mu (x)⟩}{\omega }^n/n!=\int e^{-⟨\beta ,\mu (g^{-1}\cdot x)⟩}{\omega }^n/n!=\int e^{-⟨\beta ,\mu (y)⟩}{\omega }^n/n!=P(\beta )$. $\square$

Proposition (Free energy convexity). The free energy $\Psi (\beta )=-\log P(\beta )$ is convex on its domain, and strictly convex whenever the moment map is not supported on a single coadjoint orbit.

Proof. Compute $\frac{d^2}{d{\beta }^2}(-\log P)=⟨{\mu }^2{⟩}_{\beta }-⟨\mu {⟩}_{\beta }^2={\mathrm{Var}}_{\beta }(\mu )$, which is positive semidefinite as a covariance matrix. Strict convexity at $\beta$ requires ${\mathrm{Var}}_{\beta }(\mu )\ne 0$, which fails exactly when the Gibbs measure is supported on a single level set of $\mu$, i.e., a single coadjoint orbit. $\square$

Connections [Master]

• The Souriau Gibbs state on a symplectic G-space generalises the canonical ensemble of 11.04.01 pending; when $G=\mathbb{R}$ (time translations), the moment map is the Hamiltonian $H$ and the Souriau state reduces to $\rho \propto e^{-\beta H}$, recovering the standard Gibbs distribution.

• The coadjoint-orbit structure underlying the Souriau construction is the same structure that classifies irreducible unitary representations via the orbit method, connecting the thermodynamic Gibbs state to the representation-theoretic framework of 07.01.12 and the geometric quantisation program.

• The Souriau cocycle measuring the failure of Gibbs-state invariance under the group action is the infinitesimal version of the Duistermaat-Heckman variation of the reduced symplectic form, linking the thermodynamic formalism here to the symplectic reduction and wall-crossing phenomena studied in the Duistermaat-Heckman theorem.

Bibliography [Master]

@book{souriau1970-thermo,
  author = {Souriau, Jean-Marie},
  title = {Structure des Syst\`emes Dynamiques},
  publisher = {Dunod},
  year = {1970}
}

@book{kirillov2004,
  author = {Kirillov, Alexandre A.},
  title = {Lectures on the Orbit Method},
  publisher = {AMS},
  year = {2004}
}

@article{marle2016,
  author = {Marle, Charles-Michel},
  title = {Gibbs states on symplectic manifolds},
  journal = {Preprint},
  year = {2016}
}

@book{marsden-ratiu1999,
  author = {Marsden, Jerrold E. and Ratiu, Tudor S.},
  title = {Introduction to Mechanics and Symmetry},
  publisher = {Springer},
  year = {1999}
}

Historical & philosophical context [Master]

Souriau introduced the Gibbs state on a symplectic G-space in his 1970 monograph ^{[Souriau 1970]}, Chapter IV, as part of his systematic programme to express all of mechanics in symplectic-geometric language. His insight was that statistical mechanics requires no additional structure beyond the symplectic manifold and the moment map: the Liouville measure provides the reference, and the moment map provides the constraints.

Kirillov's 1962 and 1976 work on the orbit method ^{[Kirillov 2004]} provided the representation-theoretic perspective: the Souriau Gibbs state on a coadjoint orbit is the geometric incarnation of the character formula, and the partition function is the Fourier transform of the orbital measure. This identification unified statistical mechanics with geometric quantisation, a programme carried forward by Guillemin and Sternberg in the 1980s.

Philosophically, the Souriau construction demonstrates that thermodynamic equilibrium is a symplectic-geometric notion, not merely a probabilistic one. The symplectic form determines the volume element, the moment map determines the constraints, and the Gibbs state emerges as the unique maximum-entropy distribution. Temperature, angular velocity, and chemical potential are all coordinates on the same Lie-algebraic space, and the thermodynamic dualities between them (e.g., the magnetisation-temperature relation) are shadows of the coadjoint-orbit geometry.