Souriau Gibbs state on a symplectic $G$-space

Intuition Beginner

The canonical ensemble of ordinary statistical mechanics weights each microstate by the Boltzmann factor $e^{-\beta E}$, where $E$ is the energy of the state and $\beta =1/(k_{B}T)$ is the inverse temperature. Energy is the only conserved quantity that appears in the weight. Yet many physical systems conserve several things at once: a free rotor conserves three components of angular momentum, a relativistic particle conserves four components of energy-momentum, a fluid in a rotating bucket conserves both energy and a single angular-momentum component. Souriau asked the obvious question. Why should the thermal weight see only the energy?

The answer he proposed in 1970 is that the canonical ensemble is a special case of a much larger family. On any phase space carrying a symmetry group $G$, the conserved quantities are organised by a single object called the moment map $\mu$. The Souriau Gibbs state replaces the scalar inverse-temperature $\beta$ with a Lie-algebra element, and the scalar energy $E$ with the moment map. Where the canonical ensemble writes $e^{-\beta E}/Z$, the Souriau Gibbs state writes $e^{-⟨\beta ,\mu ⟩}/Z(\beta )$. Every component of $\beta$ is conjugate to one of the conserved quantities. Temperature is one of those components. Angular velocity is another. Chemical potential is a third.

The reason this matters: experimental thermodynamics is full of Lagrange multipliers that pair against conserved quantities, and ordinary stat-mech treats them one at a time. The Souriau framework treats them in parallel. Recover the rotating bucket and you obtain Landau-Lifshitz rotational equilibrium. Recover the special-relativistic gas and you obtain the Jüttner distribution that Einstein and Jüttner argued about in 1911. One construction. One partition function. One Lie-algebra of inverse temperatures.

Visual Beginner

A symmetric phase space is drawn as a curved surface with a small group $G$ of motions sliding it into itself. The moment map sends the surface down to a lower-dimensional space, the dual Lie algebra. On the upper surface the Souriau Gibbs density is drawn as a heat map, dense where the inner product of the inverse-temperature vector $\beta$ with the moment-map value is smallest and fading exponentially as that inner product grows. The inverse-temperature vector lives on the lower side and points in the direction conjugate to the conserved quantities one wants to fix.

The picture is the whole content of the unit at the informal level. The upper phase space is where the system lives. The lower dual Lie algebra is where the thermodynamic dual variables live. The Gibbs density on the upper side and the inverse-temperature vector on the lower side are paired by the moment map.

Worked example Beginner

Take the rotating bucket. A small bucket of water is held at temperature $T$ and forced to spin at constant angular velocity $\Omega$ about a vertical axis. Which microstates does it favour?

Step 1. List the conserved quantities. Energy $E$ and the vertical component of angular momentum $L_z$ are conserved by the imposed boundary conditions; the other two angular-momentum components are not.

Step 2. Write the Souriau weight. With two conserved quantities the inverse-temperature vector is two-dimensional: one component for energy, one for $L_z$. The Souriau weight is

$$e^{-\beta E+\alpha L_z}.$$

The sign on $L_z$ is conventional. The two parameters are $\beta =1/(k_{B}T)$ and a second variable $\alpha =\Omega /(k_{B}T)$ that records how strongly the bucket prefers rotating microstates.

Step 3. Identify $\alpha$. Maximising entropy with the constraints $⟨E⟩$ fixed and $⟨L_z⟩$ fixed gives a Lagrange multiplier for each constraint. Matching the resulting density to the Souriau weight gives $\alpha =\beta \Omega$. So the rotating-bucket Gibbs state weights each microstate by $e^{-\beta (E-\Omega L_z)}$. The combination $E-\Omega L_z$ is the energy in the co-rotating frame.

Step 4. Recover the canonical ensemble. Set $\Omega =0$ to remove the rotation constraint. The weight becomes $e^{-\beta E}$ and the standard canonical ensemble of unit 11.04.01 is recovered exactly. The Souriau weight thus contains the canonical ensemble as a one-axis specialisation.

What this tells us: the Souriau Gibbs state generalises the canonical ensemble in a structural way. Each extra conserved quantity adds a new Lagrange multiplier and a new component of the inverse-temperature vector. The rotating bucket needs two components; a relativistic gas needs four; a fluid with several internal modes needs more. The construction is the same in every case.

Check your understanding Beginner

Formal definition Intermediate+

Let $(M,\omega )$ be a symplectic manifold of dimension $2n$, on which a Lie group $G$ acts by symplectomorphisms with an equivariant moment map $\mu :M\to {\mathfrak{g}}^{\ast }$, in the sense of unit 05.04.01. The Liouville volume form ${\omega }^n/n!$ is the canonical reference measure on $M$. For any $\beta \in \mathfrak{g}$, the pairing $⟨\beta ,\mu (\cdot )⟩:M\to \mathbb{R}$ is a smooth function generalising the Hamiltonian observable; the function is $G$-invariant when $\beta$ is fixed by the coadjoint action.

The Souriau partition function is the integral $$ Z(\beta) ;=; \int_M e^{-\langle \beta, \mu(x)\rangle} , \frac{\omega^n(x)}{n!} $$ defined for those $\beta \in \mathfrak{g}$ at which the integral converges absolutely ^{[Souriau 1970]}. The set of such $\beta$, denoted $\Omega (M,\mu )\subseteq \mathfrak{g}$, is an open convex cone — convexity follows from Hölder's inequality applied to the family $\{e^{-⟨\beta ,\mu ⟩}{\}}_{\beta }$; the openness is a corollary of dominated convergence applied to a small perturbation of any $\beta \in \Omega (M,\mu )$ ^{[Marle 2016]}.

The Souriau Gibbs state at inverse temperature $\beta$ is the probability measure with density $$ \rho_\beta(x) ;=; \frac{e^{-\langle \beta, \mu(x)\rangle}}{Z(\beta)} $$ relative to the Liouville volume, defined on the same domain $\Omega (M,\mu )$. The Souriau free energy is $\Psi (\beta )=-\log Z(\beta )$, the mean moment is $\bar{\mu }(\beta )={\mathbb{E}}_{{\rho }_{\beta }}[\mu ]$, and the thermodynamic entropy is the Legendre transform $S(\bar{\mu })={\sup }_{\beta }[⟨\beta ,\bar{\mu }⟩-\Psi (\beta )]$.

Notation. The pairing $⟨\cdot ,\cdot ⟩$ denotes the canonical duality between $\mathfrak{g}$ and ${\mathfrak{g}}^{\ast }$ with no inner product chosen; $\beta$ is called the generalised inverse temperature or the Souriau temperature vector. When $G$ acts non-equivariantly, the moment-map definition is amended by a cocycle $\theta :G\to {\mathfrak{g}}^{\ast }$ (see unit 05.04.07); the Gibbs construction below carries through with $\mu$ replaced by the cocycle-shifted version.

Examples and non-examples

The simplest example is $G={\mathbb{R}}_t$ acting on $M=T^{\ast }Q$ by the flow of a Hamiltonian $H$. The moment map is $\mu =H$, the inverse-temperature element is the scalar $\beta =1/(k_{B}T)$, and the Souriau density ${\rho }_{\beta }=e^{-\beta H}/Z(\beta )$ coincides with the canonical ensemble of 11.04.01. The Liouville measure ${\omega }^n/n!$ is the phase-space measure with the factor $1/(n!h^n)$ absorbed.

A non-example: the Gibbs construction can fail to converge. On $M=T^{\ast }\mathbb{R}$ with the free-particle Hamiltonian $H=p^2/(2m)$, the integral $\int e^{-\beta p^2/(2m)}dpdq$ diverges in $q$ because the configuration space is non-compact and $H$ does not penalise large $q$. The convergence domain $\Omega (M,\mu )$ is empty unless the system is confined. A bounded box restores convergence; a confining potential does the same.

Counterexamples to common slips

• The Souriau inverse-temperature element is a member of the Lie algebra $\mathfrak{g}$, not its dual ${\mathfrak{g}}^{\ast }$ where the moment map values live. Confusing the two leads to a wrong pairing. The dual ${\mathfrak{g}}^{\ast }$ is the home of the mean moment $\bar{\mu }(\beta )=\mathbb{E}[\mu ]$; the algebra $\mathfrak{g}$ is the home of $\beta$. They are paired by $⟨\cdot ,\cdot ⟩$.
• The convergence domain $\Omega (M,\mu )$ is not all of $\mathfrak{g}$ in general. For a Galilei-invariant gas in a finite box the domain is a half-space — positive inverse temperatures only; for a Poincaré-invariant ideal gas the domain is the forward time-cone, recovering the constraint that the temperature must be timelike-future-directed.
• The Souriau cocycle obstruction matters when the moment map is non-equivariant. The partition function then transforms by an affine action of $G$ on ${\mathfrak{g}}^{\ast }$, not a linear one, and the cocycle class in $H^1(G,{\mathfrak{g}}^{\ast })$ is the precise measure of the affine shift.

Key theorem with proof Intermediate+

Theorem (Souriau partition-function calculus and entropy maximisation; Souriau 1970 Ch. IV §16). Let $(M,\omega )$ carry a Hamiltonian $G$-action with equivariant moment map $\mu : M \to \mathfrak{g}^$.Ontheopenconvexcone$\Omega(M, \mu) \subseteq \mathfrak{g}$ofconvergence,theSouriaupartitionfunction$Z(\beta) = \int_M e^{-\langle\beta, \mu\rangle} \omega^n/n!$issmooth,thefreeenergy$\Psi(\beta) = -\log Z(\beta)$ is strictly convex, and its derivatives compute the moment-map cumulants* $$ \bar\mu(\beta) ;=; -\frac{d\Psi}{d\beta}(\beta), \qquad \mathrm{Cov}{\rho\beta}(\mu) ;=; \frac{d^2\Psi}{d\beta^2}(\beta) ;\succeq; 0. $$ Among all probability measures $\rho$ on $M$ absolutely continuous with respect to the Liouville volume and satisfying ${\mathbb{E}}_{\rho }[\mu ]=\bar{\mu }(\beta )$, the Souriau Gibbs state ${\rho }_{\beta }$ is the unique measure maximising the Gibbs-Shannon entropy $S[\rho ]=-{\int }_M\rho \log \rho {\omega }^n/n!$, and the maximum value equals $S(\bar{\mu }(\beta ))=⟨\beta ,\bar{\mu }(\beta )⟩-\Psi (\beta )$.

Proof. The argument has three steps. First, differentiate under the integral sign to obtain the cumulant identities. Second, prove convexity of $\Psi$ from the positivity of the covariance. Third, prove entropy maximisation by a variational calculation with Lagrange multipliers.

Step 1: derivatives of $\log Z$. For $\beta \in \Omega (M,\mu )$, the integrand $e^{-⟨\beta ,\mu (x)⟩}$ decays exponentially in any direction of $\mathfrak{g}$ within the convergence cone, and the moment-map values $\mu (x)\in {\mathfrak{g}}^{\ast }$ pair against $\beta$ in a bilinear way. Differentiation under the integral with respect to a parameter $\beta +t\xi$ at $t=0$ for any $\xi \in \mathfrak{g}$ gives $$ \frac{d}{dt}\bigg|{t=0} Z(\beta + t\xi) ;=; -\int_M \langle \xi, \mu(x)\rangle , e^{-\langle\beta, \mu\rangle} , \frac{\omega^n}{n!} ;=; -\langle \xi, \bar\mu(\beta)\rangle , Z(\beta). $$ Dividing by $Z(\beta )$ yields $d\log Z(\beta )\cdot \xi =-⟨\xi ,\bar{\mu }(\beta )⟩$, equivalently $\bar{\mu }=-d\Psi /d\beta$. A second differentiation produces the covariance: $$ \frac{d^2}{dt^2}\bigg|{t=0} \log Z(\beta + t\xi) ;=; \mathbb{E}{\rho\beta}[\langle \xi, \mu\rangle^2] - \mathbb{E}{\rho\beta}[\langle \xi, \mu\rangle]^2 ;=; \mathrm{Var}{\rho\beta}(\langle \xi, \mu\rangle), $$ which is non-negative for every $\xi$. So $d^2\Psi /d{\beta }^2={\mathrm{Cov}}_{{\rho }_{\beta }}(\mu )$ as a positive-semidefinite bilinear form on $\mathfrak{g}$. Smoothness of $\Psi$ follows by induction: every order of derivative is the corresponding moment-map cumulant under ${\rho }_{\beta }$.

Step 2: strict convexity. Strict convexity of $\Psi$ is equivalent to positive-definiteness of ${\mathrm{Cov}}_{{\rho }_{\beta }}(\mu )$ as a bilinear form on $\mathfrak{g}$. The covariance is degenerate exactly when there exists $\xi \in \mathfrak{g}$ with ${\mathrm{Var}}_{{\rho }_{\beta }}(⟨\xi ,\mu ⟩)=0$, that is, when $⟨\xi ,\mu (\cdot )⟩$ is constant ${\rho }_{\beta }$-almost surely. Such $\xi$ generate degenerate directions: thermodynamically they correspond to conserved quantities that the moment map fails to distinguish on the support of ${\rho }_{\beta }$. In the generic case where $\mu$ separates the points of $M$ modulo the $G$-action, the covariance is strictly positive-definite on the complement of the Lie-algebra stabiliser and the free energy is strictly convex.

Step 3: entropy maximisation. Fix ${\bar{\mu }}_0\in {\mathfrak{g}}^{\ast }$ and consider the constrained maximisation $$ \rho ;\mapsto; S[\rho] ;=; -\int_M \rho(x) \log \rho(x) , \frac{\omega^n}{n!} $$ over probability densities $\rho$ satisfying ${\int }_M\rho {\omega }^n/n!=1$ and ${\int }_M\mu (x)\rho (x){\omega }^n/n!={\bar{\mu }}_0$. Introduce Lagrange multipliers $\lambda \in \mathbb{R}$ for normalisation and $\beta \in \mathfrak{g}$ for the moment constraint. The Lagrangian functional is $$ \mathcal{L}[\rho] ;=; -\int_M \rho \log \rho , \frac{\omega^n}{n!} - \lambda \left[\int_M \rho , \frac{\omega^n}{n!} - 1\right] - \left\langle \beta, \int_M \mu \rho , \frac{\omega^n}{n!} - \bar\mu_0\right\rangle. $$ Vanishing variational derivative $\delta \mathcal{L}/\delta \rho =-\log \rho -1-\lambda -⟨\beta ,\mu (x)⟩=0$ gives $\rho (x)=e^{-1-\lambda -⟨\beta ,\mu (x)⟩}$; the normalisation constraint identifies $e^{-1-\lambda }=1/Z(\beta )$, hence $\rho ={\rho }_{\beta }$. The moment constraint determines $\beta$ implicitly via $-d\Psi /d\beta ={\bar{\mu }}_0$, which has a unique solution by strict convexity of $\Psi$. The Lagrangian is strictly concave (second variation is $-\delta {\rho }^2/\rho <0$), so the critical point is the unique maximum. Evaluating gives $$ S[\rho_\beta] ;=; \int_M \rho_\beta (\langle \beta, \mu\rangle + \log Z) , \frac{\omega^n}{n!} ;=; \langle \beta, \bar\mu(\beta)\rangle - \Psi(\beta) $$ since $-\log {\rho }_{\beta }=⟨\beta ,\mu ⟩+\log Z$. This is the Legendre transform of $\Psi$ at $\bar{\mu }(\beta )$. $\square$

Bridge. The Souriau theorem builds toward every later development of statistical mechanics on a manifold with symmetry. The foundational reason it holds is exactly the convexity of $\Psi$ — strict convexity is what makes the Legendre transform invertible and what guarantees that each value of $\bar{\mu }$ corresponds to a unique inverse-temperature vector $\beta$. The bridge is the recognition that this Legendre duality between $\Psi$ and $S$ generalises the classical thermodynamic duality between Helmholtz free energy $F(T)=-k_{B}T\log Z(\beta )$ and entropy $S(U)$ from one scalar variable to a full Lie-algebra of variables. The central insight is that the moment map identifies $H$ with the time-translation generator and identifies each component of the angular-momentum vector with a corresponding rotation generator, so the Souriau partition function is the moment-generating function of the entire conserved-quantity vector under the canonical phase-space measure. Putting these together, the cumulant calculus $\bar{\mu }=-d\Psi /d\beta$ and $\mathrm{Cov}(\mu )=d^2\Psi /d{\beta }^2$ identifies the Souriau second derivative with the Fisher information of the parametric family $\{{\rho }_{\beta }\}$, which is exactly the structure that appears again in 05.03.02 (the symplectic-geometric face of the same construction) and which generalises the standard energy-fluctuation identity of 11.04.01.

Exercises Intermediate+

Advanced results Master

Theorem (Legendre duality between $\Psi$ and the entropy; Souriau 1970 Ch. IV §16; Marle 2016). On the open convex cone $\Omega (M,\mu )\subseteq \mathfrak{g}$ where $Z(\beta )$ converges, the Souriau free energy $\Psi (\beta )=-\log Z(\beta )$ is strictly convex and smooth, and the Legendre-Fenchel transform $$ S(\bar\mu) ;=; \sup_{\beta \in \Omega(M, \mu)} \left[\langle \beta, \bar\mu\rangle - \Psi(\beta)\right] $$ defines a strictly concave function on the open subset of $\mathfrak{g}^$thatistheimageof$\Omega(M, \mu)$underthegradientmap$\beta \mapsto -d\Psi/d\beta = \bar\mu(\beta)$.TheLegendreinvolution$S^{**} = -\Psi(-\cdot)$holdsonthesamedomain,andtheSouriauentropy$S(\bar\mu)$coincideswiththemaximum-entropyvalue$-\int \rho_\beta \log \rho_\beta , \omega^n / n!$where$\beta$istheuniqueinverse-temperaturevectorwith$\bar\mu(\beta) = \bar\mu$.*

The proof follows the convex-analytic framework. Strict convexity of $\Psi$ (Theorem in §I) makes the gradient map $\beta \mapsto -d\Psi /d\beta$ a diffeomorphism onto its image; smoothness of $\Psi$ propagates to smoothness of $S$ on the image; involutivity follows from the general Fenchel-Moreau theorem for proper, lower-semicontinuous, strictly convex functions.

Theorem (Souriau-Noether theorem; Souriau 1970 Ch. IV §16). The Souriau Gibbs state ${\rho }_{\beta }$ is invariant under the one-parameter subgroup $\{\exp (tX):t\in \mathbb{R}\}$ if and only if $X$ lies in the Lie-algebra stabiliser $\mathfrak{g}_\beta = {X \in \mathfrak{g} : \mathrm{ad}^X \beta = 0}$.Thestabiliser$\mathfrak{g}\beta$actsontheGibbsensemblebysymmetriesthataredetectablebynothermodynamicmeasurement;thecomplementspansthe\ast broken\ast symmetriesthatdistinguishthethermalstatefromits$G$-orbit.*

Souriau identified this as the geometric meaning of spontaneous symmetry breaking in the thermal setting. For $G=\mathrm{SO}(3)$ and $\beta$ along the $z$-axis, the stabiliser is the $\mathrm{SO}(2)$ subgroup of rotations about $z$, and the Gibbs state is rotationally symmetric only about that axis; the other two SO(3) generators give symmetries broken by the choice of $\beta$.

Theorem (Affine action and the Souriau cocycle; Souriau 1970 Ch. IV §16). When the moment map fails to be equivariant, the partition function transforms by an affine action of $G$ on $\mathfrak{g}^$ratherthanalinearone.Definingthe\ast \ast Souriaucocycle\ast \ast$\theta : G \to \mathfrak{g}^$by$\theta(g) = \mu(g \cdot x_0) - \mathrm{Ad}^_g \mu(x_0)$foranyreferencepoint$x_0$,thecocyclesatisfiesthe$1$-cocycle relation* $$ \theta(gh) ;=; \theta(g) + \mathrm{Ad}^_g \theta(h), $$ and its cohomology class $[\theta] \in H^1(G, \mathfrak{g}^)$isindependentofthereferencepoint.TheGibbscovariancebecomes$Z(\mathrm{Ad}^_{g^{-1}}\beta + \theta(g^{-1})) = Z(\beta)$, an affine rather than linear transformation.*

The cocycle is the obstruction to recasting the Souriau theory in pure coadjoint-orbit language. It vanishes for compact, simply connected, and semisimple Lie groups by the Whitehead lemma $H^1(G,V)=0$, identifying when the standard coadjoint-orbit picture is sufficient. For the Galilei group the cocycle is nonzero and encodes the Bargmann central extension by mass — the Souriau-cocycle classification is the algebraic input to non-relativistic spin and to the mass-shell quantisation of unit 05.04.07.

Theorem (Recovery of the relativistic Jüttner distribution; Marle 2016; Jüttner 1911). For $G$ the Poincaré group and $M$ the one-particle phase space, the Souriau partition function evaluates to $$ Z(\beta) ;=; \frac{4\pi m^2 c , K_2(mc^2 \beta^0)}{\beta^0} $$ for $\beta =({\beta }^0,0,0,0)$ in the rest frame of the thermal bath, where $K_2$ is the modified Bessel function of the second kind. The Gibbs density on momentum space is $$ \rho_\beta(p) ;=; \frac{1}{Z(\beta)} e^{-\beta^0 \sqrt{|p|^2 c^2 + m^2 c^4}}, $$ recovering the Jüttner distribution of relativistic statistical mechanics. In the non-relativistic limit $m{c}^2\gg k_{B}T$, the asymptotic $K_2(z)\sim \sqrt{\pi /(2z)}{e}^{-z}$ recovers the Maxwell-Boltzmann distribution at temperature $T=1/(k_B{\beta }^0)$.

Jüttner derived this distribution in 1911 by direct relativistic kinematic arguments; Souriau showed in 1970 that it is the canonical Gibbs state of the Poincaré group, with no auxiliary relativistic-kinematic input beyond the symplectic structure of the one-particle phase space. The two derivations agree exactly. The Souriau framing shows that the Jüttner distribution is not a relativistic ad-hoc generalisation — it is the canonical Gibbs state for the Poincaré symmetry group, in the same sense that the Maxwell-Boltzmann distribution is the canonical Gibbs state for the time-translation group.

Theorem (Souriau-Fisher information geometry; Barbaresco 2020). The Hessian $d^2\Psi /d{\beta }^2={\mathrm{Cov}}_{{\rho }_{\beta }}(\mu )$ defines a Riemannian metric on the open convex cone $\Omega (M,\mu )\subseteq \mathfrak{g}$, called the Souriau-Fisher metric, that coincides with the Fisher information metric of the parametric family $\{{\rho }_{\beta }{\}}_{\beta \in \Omega (M,\mu )}$. With this metric, $\Omega (M,\mu )$ becomes a Hessian Riemannian manifold whose dual coordinates are the mean-moment coordinates $\bar\mu \in \mathfrak{g}^$ and whose Christoffel symbols are computable from the third Souriau cumulants.*

Barbaresco's information-geometric reformulation places the Souriau framework inside the modern information-geometry programme of Amari and Chentsov, with $\Omega (M,\mu )$ as a dually flat manifold. The Souriau-Fisher metric is the geometric expression of the cumulant calculus of Theorem 1.

Theorem (Maximum-entropy uniqueness on the symplectic side). Among all probability measures on $(M,\omega )$ absolutely continuous with respect to the Liouville volume and with prescribed mean moment $\bar\mu \in \mathfrak{g}^$,theSouriauGibbsstate$\rho_{\beta(\bar\mu)}$istheuniquemaximiseroftheGibbs-Shannonentropy.Themaximumvalueis$S(\bar\mu) = \langle \beta(\bar\mu), \bar\mu\rangle - \Psi(\beta(\bar\mu))$,theLegendretransformof$\Psi$.*

This is the variational characterisation that gives the Souriau Gibbs state its physical content: it is the state of maximum ignorance given the constraint that the expected value of the moment map is fixed at $\bar{\mu }$. The constraint is a vector-valued generalisation of the energy-only constraint of the canonical ensemble.

Synthesis. The Souriau Gibbs state is the foundational generalisation of the canonical ensemble to systems carrying a Lie-group symmetry beyond the identity action. The central insight is that the moment map $\mu :M\to {\mathfrak{g}}^{\ast }$ packages every conserved quantity into a single ${\mathfrak{g}}^{\ast }$-valued observable, and the Souriau inverse-temperature vector $\beta \in \mathfrak{g}$ is the conjugate Lie-algebra element that pairs against $\mu$ in the Boltzmann-like weight $e^{-⟨\beta ,\mu ⟩}$. The bridge is the recognition that this construction generalises both the Maxwell-Boltzmann canonical ensemble for $G={\mathbb{R}}_t$ and the Jüttner relativistic distribution for $G=$ Poincaré, the two distributions appearing as different specialisations of one Lie-algebraic framework. Putting these together with the Legendre duality between $\Psi$ and the entropy, the Souriau framework identifies four parallel structures: the convex-analytic duality of $\Psi$ and $S$, the symplectic-geometric structure carried by $\omega$ and $\mu$, the information-geometric Fisher metric on $\Omega (M,\mu )$, and the cohomological obstruction class $[\theta ]\in H^1(G,{\mathfrak{g}}^{\ast })$ measuring the non-equivariance of $\mu$. The foundational reason all four agree is exactly the cumulant calculus $\bar{\mu }=-d\Psi /d\beta$ and $\mathrm{Cov}(\mu )=d^2\Psi /d{\beta }^2$, which identifies the Souriau partition function with the moment-generating function of $\mu$ under the Liouville measure restricted to the Boltzmann-weighted level set.

The Souriau framework appears again in 05.03.02 (the symplectic-geometric companion unit, where the same construction is developed from the symplectic side rather than the statistical-mechanics side) and builds toward several modern threads. The Souriau-Fisher metric of Barbaresco's information-geometry programme generalises the heat-bath fluctuation-dissipation identity from one scalar variable to a full Lie-algebra of variables. The Souriau cocycle is the precise algebraic obstruction that becomes the Bargmann central extension in the Galilei case, identifying mass as the cocycle generator of the non-relativistic kinematic group 05.04.07. The recovery of the Jüttner distribution from the Poincaré group identifies relativistic statistical mechanics as the canonical Gibbs state for the Poincaré symmetry, an identification that builds toward the relativistic quantum-statistical theory of 12.03.01 where the same framework is upgraded to a Hilbert-space representation.

Full proof set Master

Proposition (Strict convexity of $\Psi$ on $\Omega (M,\mu )$ when $\mu$ is generic). If the moment map $\mu : M \to \mathfrak{g}^$hasnoLie-algebradirection$\xi \in \mathfrak{g} \setminus {0}$alongwhich$\langle\xi, \mu(\cdot)\rangle$isconstantonthesupportoftheLiouvillemeasure,theSouriaufreeenergy$\Psi(\beta) = -\log Z(\beta)$isstrictlyconvexon$\Omega(M, \mu)$.*

Proof. The second derivative of $\Psi$ in the direction $\xi$ is the variance of $⟨\xi ,\mu ⟩$ under ${\rho }_{\beta }$, by Step 1 of the §I theorem. The variance is zero exactly when $⟨\xi ,\mu ⟩$ is ${\rho }_{\beta }$-almost-surely constant on $M$. The hypothesis rules this out for every nonzero $\xi$. So the Hessian of $\Psi$ is strictly positive-definite, and $\Psi$ is strictly convex. $\square$

Proposition (Existence and uniqueness of $\beta$ given $\bar{\mu }$). On the open convex cone $\Omega (M,\mu )\subseteq \mathfrak{g}$, the gradient map $\beta \mapsto -d\Psi /d\beta =\bar{\mu }(\beta )$ is a diffeomorphism onto its image $\bar{\Omega} \subseteq \mathfrak{g}^$.Givenany$\bar\mu \in \bar{\Omega}$,thereisaunique$\beta \in \Omega(M, \mu)$with$\bar\mu(\beta) = \bar\mu$.*

Proof. The strict convexity of $\Psi$ on $\Omega (M,\mu )$ implies that the gradient map is injective: if $\bar{\mu }({\beta }_1)=\bar{\mu }({\beta }_2)$ with ${\beta }_1\ne {\beta }_2$, the function $t\mapsto \Psi ((1-t){\beta }_1+t{\beta }_2)$ has equal derivative at $t=0$ and $t=1$ in the direction ${\beta }_2-{\beta }_1$, contradicting strict convexity. The Jacobian of the gradient map is $-d^2\Psi /d{\beta }^2=-{\mathrm{Cov}}_{{\rho }_{\beta }}(\mu )$, which is strictly negative-definite (the negation of a strictly positive-definite form) by the previous proposition. The inverse function theorem gives a local diffeomorphism; together with global injectivity and convexity of the source, the map is a diffeomorphism onto its image. $\square$

Proposition (Legendre involution $S^{} = -\Psi$).\ast \ast \ast TheLegendre-Fencheltransform$S(\bar\mu) = \sup_\beta[\langle \beta, \bar\mu\rangle - \Psi(\beta)]$isstrictlyconcaveon$\bar\Omega$,smoothon$\bar\Omega$,andsatisfiestheinvolution$\sup_{\bar\mu}[\langle \beta, \bar\mu\rangle + S(\bar\mu)] = -\Psi(-\beta)$on$-\Omega(M, \mu)$.*

Proof. Smoothness and strict concavity of $S$ follow from the inverse-function-theorem argument of the previous proposition: $S=⟨\beta (\bar{\mu }),\bar{\mu }⟩-\Psi (\beta (\bar{\mu }))$ where $\beta (\bar{\mu })$ is the smooth inverse of the gradient map. Direct differentiation gives $dS/d\bar{\mu }=\beta (\bar{\mu })$, identifying the gradient of $S$ as the inverse of the gradient of $\Psi$. Strict concavity of $S$ is equivalent to strict negative-definiteness of $d^{2}S/d{\bar{\mu }}^2=d\beta /d\bar{\mu }=(d^2\Psi /d{\beta }^2{)}^{-1}$, which holds because $d^2\Psi /d{\beta }^2$ is strictly positive-definite. The Fenchel-Moreau involution for proper lower-semicontinuous strictly convex functions then gives $S^{\ast \ast }=-\Psi (-\cdot )$ on the relevant domain. $\square$

Proposition (Recovery of canonical ensemble for $G={\mathbb{R}}_t$). For $G={\mathbb{R}}_t$ acting on a phase space $(M,\omega )$ by the Hamiltonian flow of $H$, the moment map is $\mu =H$ and the Souriau Gibbs state ${\rho }_{\beta }=e^{-\beta H}/Z(\beta )$ coincides with the canonical ensemble of unit 11.04.01, with the Liouville measure ${\omega }^n/n!$ as the phase-space measure.

Proof. The Lie algebra of ${\mathbb{R}}_t$ is one-dimensional, with a canonical generator $e_t$ such that $\beta =\beta e_t\in \mathfrak{g}$ pairs with the moment map by $⟨\beta ,\mu ⟩=\beta \cdot H$. The Souriau weight $e^{-⟨\beta ,\mu ⟩}=e^{-\beta H}$ matches the Boltzmann factor of the canonical ensemble. The Souriau partition function $Z(\beta )={\int }_M{e}^{-\beta H}{\omega }^n/n!$ matches the classical-phase-space partition function $\int e^{-\beta H}{d}^{n}q{d}^{n}p/n!$. The Souriau Gibbs density ${\rho }_{\beta }=e^{-\beta H}/Z$ matches $P_i=e^{-\beta E_i}/Z$ in the continuous-spectrum case. The recovery is exact. $\square$

Proposition (Recovery of Jüttner distribution for $G=$ Poincaré). For $G$ the proper orthochronous Poincaré group acting on the one-particle phase space $(M,\omega )$ for a massive scalar relativistic particle, the moment map sends a state to its four-momentum $p^{\mu }$ on the future mass-shell. For $\beta =({\beta }^0,0,0,0)$ a rest-frame inverse-temperature four-vector, the Souriau partition function is $Z({\beta }^0)=4\pi m^{2}c{K}_2(m{c}^2{\beta }^0)/{\beta }^0$, and the Souriau Gibbs density is the Jüttner distribution.

Proof. The one-particle phase space for a massive particle of mass $m$ is the future mass-shell $\{p\in {\mathbb{R}}^{1,3}:{\eta }_{\mu \nu }{p}^{\mu }{p}^{\nu }=m^2{c}^2,p^0>0\}$, parametrised by spatial momentum $\vec{p}\in {\mathbb{R}}^3$ with $p^0=\sqrt{|\vec{p}{|}^2{c}^2+m^2{c}^4}$. The Lorentz-invariant measure on the mass-shell is $d^{3}p/(2p^0)$, equivalent up to a multiplicative constant to the Liouville measure on a single momentum slice. The moment map is $\mu (p)=p$, and the Souriau pairing in the rest frame is $⟨\beta ,\mu ⟩={\beta }^0{p}^0$. The Souriau partition function is

$$Z({\beta }^0)={\int }_{{\mathbb{R}}^3}{e}^{-{\beta }^0\sqrt{|\vec{p}{|}^2{c}^2+m^2{c}^4}}\frac{d^3\vec{p}}{2\sqrt{|\vec{p}{|}^2{c}^2+m^2{c}^4}}.$$

Substituting $|\vec{p}|c=m{c}^2\sinh \theta$ (so $p^0=m{c}^2\cosh \theta$ and $d^3\vec{p}/(2p^0)=(mc{)}^3{\sinh }^2\theta d\theta d\Omega /c$) and integrating the angular variables gives

$$Z({\beta }^0)=\frac{4\pi m^3{c}^2}{c}{\int }_0^{\infty }{e}^{-m{c}^2{\beta }^0\cosh \theta }{\sinh }^2\theta d\theta .$$

The integral ${\int }_0^{\infty }{e}^{-z\cosh \theta }{\sinh }^2\theta d\theta =K_2(z)/z$ is a standard representation of the modified Bessel function. Substituting $z=m{c}^2{\beta }^0$ gives $Z({\beta }^0)=4\pi m^{2}c{K}_2(m{c}^2{\beta }^0)/{\beta }^0$. The Gibbs density on the spatial-momentum slice is ${\rho }_{{\beta }^0}(\vec{p})=e^{-{\beta }^0{p}^0}/Z({\beta }^0)$, the Jüttner distribution. The non-relativistic limit follows from $K_2(z)\sim \sqrt{\pi /(2z)}{e}^{-z}(1+O(1/z))$ as $z\to \infty$: writing $p^0=m{c}^2+|\vec{p}{|}^2/(2m)+O(|\vec{p}{|}^4/m^3{c}^2)$ in the exponential and absorbing the constant $e^{-m{c}^2{\beta }^0}$ into $Z$ gives the Maxwell-Boltzmann density $e^{-|\vec{p}{|}^2/(2m{k}_{B}T)}/(2\pi m{k}_{B}T{)}^{3/2}$ to leading order. $\square$

Proposition (Souriau cocycle identity $\theta (gh)=\theta (g)+{\mathrm{Ad}}_g^{\ast }\theta (h)$). For a non-equivariant moment map $\mu : M \to \mathfrak{g}^$withcocycle$\theta(g) = \mu(g \cdot x_0) - \mathrm{Ad}^_g \mu(x_0)$, the cocycle relation holds.

Proof. Compute: $$ \theta(gh) = \mu(gh \cdot x_0) - \mathrm{Ad}^*_{gh} \mu(x_0) $$

The first term is $\mu (g\cdot (h\cdot x_0))={\mathrm{Ad}}_g^{\ast }\mu (h\cdot x_0)+\theta (g)$ by the definition of $\theta (g)$ applied with reference point $h\cdot x_0$, but $\theta$ is independent of reference point so this is also ${\mathrm{Ad}}_g^{\ast }\mu (h\cdot x_0)+\theta (g)$. The second term is ${\mathrm{Ad}}_g^{\ast }{\mathrm{Ad}}_h^{\ast }\mu (x_0)$ by the chain rule for the coadjoint representation. Subtracting, $$ \theta(gh) = \mathrm{Ad}^_g \mu(h \cdot x_0) - \mathrm{Ad}^_g \mathrm{Ad}^_h \mu(x_0) + \theta(g) = \mathrm{Ad}^_g [\mu(h \cdot x_0) - \mathrm{Ad}^_h \mu(x_0)] + \theta(g) = \mathrm{Ad}^_g \theta(h) + \theta(g). $$ The cocycle relation follows. $\square$

Connections Master

• Canonical ensemble 11.04.01. The Souriau Gibbs state generalises the canonical ensemble: setting $G={\mathbb{R}}_t$ and identifying the moment map with the Hamiltonian recovers ${\rho }_{\beta }=e^{-\beta H}/Z(\beta )$ exactly. The Souriau partition function specialises to the canonical-ensemble partition function on the same phase space, with the Liouville measure as the reference. Every thermodynamic identity of the canonical ensemble — mean energy as $-d\log Z/d\beta$, energy variance as $d^2\log Z/d{\beta }^2$, the Helmholtz free energy as $-{\beta }^{-1}\log Z$ — is a one-dimensional specialisation of the Souriau partition-function calculus.

• Moment map 05.04.01. The Souriau construction takes the moment map as primary data: the moment map of a Hamiltonian group action provides exactly the conserved-quantity observables that the Souriau weight pairs against. Without a moment map, the Souriau construction has no input. The equivariance property of the moment map is what makes the Souriau partition function transform linearly under the coadjoint action; non-equivariance produces the cocycle correction.

• Souriau Gibbs state, symplectic side 05.03.02. The companion unit develops the same construction from the pure-symplectic-geometry side, emphasising the coadjoint-orbit foliation of the support of ${\rho }_{\beta }$ and the connection to the orbit method. The two units are intentionally redundant in their formal-definition section but diverge in worked examples and Master-section emphasis: the symplectic-side unit develops the Kirillov-Kostant-Souriau structure, while the present physics-side unit develops the recovery of the canonical and Jüttner distributions and the convex-analytic Legendre duality.

• Lie group 03.03.01. The Souriau inverse-temperature vector $\beta \in \mathfrak{g}$ is an element of the Lie algebra of the symmetry group $G$. The coadjoint covariance of the Souriau partition function is a statement about the coadjoint representation of $G$ on ${\mathfrak{g}}^{\ast }$. The Souriau stabiliser ${\mathfrak{g}}_{\beta }=\{X\in \mathfrak{g}:{\mathrm{ad}}_X^{\ast }\beta =0\}$ is the centraliser of $\beta$ in $\mathfrak{g}$, an object central to the orbit method and to the representation theory of $G$.

• Thermodynamic potentials and Legendre transforms 11.01.02. The Souriau free energy $\Psi (\beta )=-\log Z(\beta )$ and the Souriau entropy $S(\bar{\mu })$ are related by a vector-valued Legendre-Fenchel transform that generalises the scalar Legendre transform between the Helmholtz free energy $F(T)$ and the internal energy $U$. The convex-analytic foundations of thermodynamic potentials cited there carry over verbatim, with the Lie-algebra dimension of $\beta$ as the scalar-to-vector upgrade.

• Souriau cocycle 05.04.07. When the moment map is non-equivariant, the affine transformation law of the Souriau partition function is governed by the Souriau cocycle, an element of $Z^1(G,{\mathfrak{g}}^{\ast })$ whose cohomology class is the obstruction to recasting the construction in pure coadjoint-orbit language. The cocycle is the algebraic input to the Bargmann central extension of the Galilei group and to the appearance of mass as a cohomology generator in non-relativistic kinematics.

Historical & philosophical context Master

Souriau introduced the Gibbs state on a symplectic $G$-space in Structure des Systèmes Dynamiques (Dunod, 1970), Chapter IV §16 ^{[Souriau 1970]}. The construction was part of his broader programme to recast classical mechanics and statistical mechanics in symplectic-geometric language, replacing local-coordinate Hamiltonian formulations with coordinate-free symplectic-manifold statements. The key insight — that the scalar inverse-temperature $\beta$ of the canonical ensemble should be promoted to a Lie-algebra element conjugate to the moment map — has parallels in earlier work by Iglesias-Zemmour (his doctoral thesis under Souriau, published in 1973 in the Annales de l'Institut Henri Poincaré ^{[Iglesias-Zemmour 1973]}) and was independently developed by Marsden and Weinstein in their 1974 work on symplectic reduction, though Souriau's framing predates and is more general.

The recovery of the Jüttner distribution from the Poincaré-group specialisation was historically significant. Ferencz Jüttner derived the relativistic Maxwell-Boltzmann distribution in 1911 (Annalen der Physik 339, 856-882) ^{[Jüttner 1911]} by direct relativistic-kinematic arguments. The status of Jüttner's distribution was disputed in the early twentieth century — Einstein and Planck offered conflicting derivations of the temperature transformation under Lorentz boosts — and was not fully clarified until the Souriau framing showed that the Jüttner distribution is the canonical Gibbs state for the proper orthochronous Poincaré group, with the rest-frame inverse-temperature four-vector ${\beta }^{\mu }$ as the relativistic analogue of the scalar $\beta$. The temperature-transformation question is then settled in the standard way: ${\beta }^{\mu }$ is a future-directed timelike four-vector, and its rest-frame component is $1/(k_B{T}_{\mathrm{rest}})$.

Marle's 2016 review (Journal of Geometry and Symmetry in Physics 40, 33-99) ^{[Marle 2016]} consolidated the Souriau framework as the canonical modern reference, with explicit recovery of the Maxwell-Boltzmann, Jüttner, and rotating-ensemble cases. Barbaresco's 2020 work (Entropy 22, 642) ^{[Barbaresco 2020]} developed the information-geometric reformulation, identifying the Souriau-Fisher metric on the convergence cone $\Omega (M,\mu )$ as a Hessian Riemannian structure that places the Souriau framework inside the modern information-geometry programme of Amari and Chentsov. The Souriau cocycle and its cohomology class in $H^1(G,{\mathfrak{g}}^{\ast })$ have found application in the geometric-quantisation literature, where the cocycle obstruction is the algebraic input to the Kirillov character formula and to the orbit-method classification of irreducible unitary representations of Lie groups.

Bibliography Master

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