Reduced Thermodynamic Phase Spaces

• Reduced thermodynamic phase spaces are lower-dimensional state manifolds defined by imposing physical constraints that isolate independent thermodynamic variables.
• They employ geometric formulations, including contact and symplectic reductions, to reformulate thermodynamic laws and highlight signatures of phase transitions and stability.
• Applications in black-hole thermodynamics and isomorph theory demonstrate that reduction techniques yield universal laws, sharper phase diagrams, and consistent curvature diagnostics.

Searching arXiv for recent and foundational papers on reduced thermodynamic phase spaces and closely related formulations. Reduced thermodynamic phase spaces are lower-dimensional state spaces obtained when the physically relevant thermodynamic variables are not all independent, or when one deliberately reformulates thermodynamics in terms of reduced coordinates, constrained submanifolds, or reduced-unit invariants. Across contemporary literature, the term encompasses several distinct but related constructions: submanifolds of a larger black-hole state space obtained by fixing charges or angular momenta; equilibrium Legendre submanifolds inside a contact thermodynamic phase space; reduced descriptions generated by symplectic or contact reduction; phase diagrams compressed by isomorph invariance; and effective phase spaces induced by entanglement or by tracing out auxiliary degrees of freedom. In each case, reduction changes the admissible variations, the geometric structure, or the effective dimensionality of the thermodynamic description, often with direct consequences for the first law, phase transitions, stability criteria, and thermodynamic geometry (Ghosh, 2022, Lim et al., 2022, Huang et al., 2022, Veldhorst et al., 2015, Souza et al., 2017, Błasiak et al., 27 May 2025).

1. Reduced phase spaces as constrained or lower-dimensional state manifolds

A central meaning of reduced thermodynamic phase space is the passage from a full thermodynamic manifold to a submanifold defined by constraints or by fixing part of the state variables. In the Kerr–Newman black hole in a cavity, the canonical phase space at fixed boundary radius $R$ is the $3$-dimensional manifold $M_{\mathrm{KN}}(R):(S,J,Q)$, equipped with the Brown–York quasilocal energy $E(S,J,Q;R)$ and conjugates $T,\Omega,\Phi$. Imposing $J=0$ defines the RN reduced phase space $M_{\mathrm{RN}}(R):(S,Q)$, while imposing $Q=0$ defines the Kerr reduced phase space $M_{\mathrm{Kerr}}(R):(S,J)$; in both cases the dimension decreases from $3$ to $3$0 (Huang et al., 2022).

An analogous reduction appears in five-dimensional Kerr–AdS black holes with equal spins. There, dimensional analysis for a single-characteristic-parameter system characterized by $3$1 motivates reduced variables such as

$3$2

so that the critical point is pinned to $3$3. The reduced formulation rewrites the thermodynamic laws with dimensionless prefactors $3$4, and this paper explicitly states that these reduced laws are “quite different from the conventional thermodynamic laws” (Wei et al., 2019).

A further, more structural, use of reduction occurs when algebraic constraints among black-hole parameters are imposed in addition to the field equations. For regular black holes, such constraints reduce the number of independent thermodynamic variables and make the naive first law inconsistent. The proposed resolution is to define thermodynamic quantities in the full, unconstrained phase space of the underlying singular black hole and only then restrict them to the constraint surface. This yields a reduced thermodynamic phase space understood as a constrained submanifold of the full contact-geometric equilibrium manifold (Ma et al., 13 Jul 2025).

This suggests that reduced phase space is not a single formalism but a family of procedures that share one principle: thermodynamic variables must be varied only along directions that remain independent after all physical constraints are imposed.

2. Contact geometry, symplectic reduction, and equilibrium submanifolds

In geometric thermodynamics, the full thermodynamic phase space is commonly modeled as a contact manifold, and reduction consists in passing from this odd-dimensional space to equilibrium submanifolds. In the Hamilton–Jacobi formulation, the thermodynamic phase space is a contact manifold $3$5 of dimension $3$6, with Darboux coordinates $3$7 and contact form

$3$8

while equilibrium states lie on Legendre submanifolds $3$9 satisfying $M_{\mathrm{KN}}(R):(S,J,Q)$0 (Ghosh, 2022). The paper identifies the reduced space locally with $M_{\mathrm{KN}}(R):(S,J,Q)$1, and formulates thermodynamic transformations directly on $M_{\mathrm{KN}}(R):(S,J,Q)$2, the space of equilibrium states or externally controllable parameters.

The same geometric theme is extended to nonequilibrium thermodynamics through symplecto-contact reduction. Starting from statistical phase space $M_{\mathrm{KN}}(R):(S,J,Q)$3 and kinetic theory phase space $M_{\mathrm{KN}}(R):(S,J,Q)$4, collective observation of observables is treated as a moment map. Marsden–Weinstein reduction then produces a mesoscopic symplectic fibration $M_{\mathrm{KN}}(R):(S,J,Q)$5, and the reduced relative information entropy $M_{\mathrm{KN}}(R):(S,J,Q)$6 serves as a generating function for an equilibrium Legendrian submanifold in

$M_{\mathrm{KN}}(R):(S,J,Q)$7

equipped with contact form $M_{\mathrm{KN}}(R):(S,J,Q)$8 (Lim et al., 2022). In this construction, thermodynamic phase space is not postulated but canonically derived from reduction.

A related but dynamically oriented notion appears in the framework that interprets thermodynamics as pattern recognition between a more detailed model $M_{\mathrm{KN}}(R):(S,J,Q)$9 and a less detailed model $E(S,J,Q;R)$0. There, the reduced thermodynamic phase space is the less-detailed state space $E(S,J,Q;R)$1, endowed with lower entropy $E(S,J,Q;R)$2 when $E(S,J,Q;R)$3, or with lower flux-entropy $E(S,J,Q;R)$4 when $E(S,J,Q;R)$5 (Grmela et al., 2019). Reduction is implemented by gradient or GENERIC evolution, rather than by a purely static quotient.

Taken together, these approaches show that reduced thermodynamic phase spaces can be understood either as equilibrium submanifolds of a contact phase space or as macroscopic quotients derived from microscopic symplectic structures. A plausible implication is that contact geometry and reduction theory provide a common language for both equilibrium and nonequilibrium thermodynamics.

3. Black-hole thermodynamics: reduced variables, restricted spaces, and phase structure

Black-hole thermodynamics provides the most developed set of explicit reduced phase-space constructions. One class fixes background parameters and promotes alternative variables. For the charged AdS black hole in the alternative phase space, the AdS radius $E(S,J,Q;R)$6 is held fixed, no $E(S,J,Q;R)$7–$E(S,J,Q;R)$8 term appears, and $E(S,J,Q;R)$9 is treated as a thermodynamic variable with conjugate

$T,\Omega,\Phi$0

The first law becomes

$T,\Omega,\Phi$1

and the system still exhibits van der Waals-type criticality, with critical values

$T,\Omega,\Phi$2

(Xu, 2020).

A different restricted formulation fixes the AdS radius and excludes the pressure-volume pair altogether. In the restricted phase space used for thermodynamic topology, the cosmological constant is not varied, while one introduces the central charge $T,\Omega,\Phi$3 and its conjugate chemical potential $T,\Omega,\Phi$4. The generalized off-shell Helmholtz free energy

$T,\Omega,\Phi$5

is then used to define thermodynamic topology in the $T,\Omega,\Phi$6-$T,\Omega,\Phi$7 plane. In this framework, AdS RN and AdS EGB black holes have the same total topological number $T,\Omega,\Phi$8, while AdS EPYM has $T,\Omega,\Phi$9, and for EPYM this result coincides across bulk-boundary, restricted, and extended phase space descriptions (Sadeghi et al., 2023).

Reduced spaces also clarify thermodynamic geometry in situations where naive coordinates are singular. For Schwarzschild–AdS, $J=0$0 and $J=0$1 are both functions of the horizon radius, so $J=0$2 makes $J=0$3 a degenerate chart. The proposed remedy is to use reduced independent coordinates $J=0$4 or $J=0$5, with enthalpy $J=0$6 rather than internal energy as the fundamental potential. In these reduced phase spaces, the Ruppeiner curvatures are

$J=0$7

and the paper states that they are equivalent (Xu et al., 2019).

An even sharper issue arises when the black hole is defined by a constraint among parameters, as in regular black holes. The constraint $J=0$8 means that not all variations are independent. If one solves the constraint too early and differentiates on the reduced space as though all remaining variables were independent, the resulting “temperature” can differ from the geometric Hawking temperature. The consistent restricted first law is instead obtained by defining all conjugates in the full unconstrained phase space and projecting onto the tangent space of the constraint surface, yielding effective coefficients such as

$J=0$9

(Ma et al., 13 Jul 2025).

These examples show that reduced black-hole phase spaces are not merely calculational conveniences. They determine which thermodynamic variables are admissible, which free energies are meaningful, and whether the first law and curvature diagnostics remain well defined.

4. Thermodynamic geometry and critical behavior on reduced spaces

Reduced thermodynamic phase spaces are especially useful for thermodynamic geometry because curvature formulas often require an independent coordinate chart. In the Kerr–Newman black hole in a cavity, the Ruppeiner framework is applied separately to the RN and Kerr reduced phase spaces, using ensemble-dependent correspondences $M_{\mathrm{RN}}(R):(S,Q)$0 or $M_{\mathrm{RN}}(R):(S,Q)$1. The resulting scalar curvature $M_{\mathrm{RN}}(R):(S,Q)$2 exhibits strong differences between the reduced spaces despite similar macroscopic phase diagrams: RN is generically attraction-dominated in the small and large black-hole regions, whereas Kerr can have either sign and diverges as $M_{\mathrm{RN}}(R):(S,Q)$3 (Huang et al., 2022).

In the alternative $M_{\mathrm{RN}}(R):(S,Q)$4 phase space for RN–AdS, the Ruppeiner metric can be written equivalently in $M_{\mathrm{RN}}(R):(S,Q)$5 or $M_{\mathrm{RN}}(R):(S,Q)$6 coordinates, and the scalar curvatures

$M_{\mathrm{RN}}(R):(S,Q)$7

are stated to be equivalent and positive in the physical region (Xu, 2020). Based on the empirical interpretation adopted in that paper, this suggests repulsive effective interactions when $M_{\mathrm{RN}}(R):(S,Q)$8 is the extensive variable.

Reduced variables also sharpen universal critical behavior. In $M_{\mathrm{RN}}(R):(S,Q)$9-dimensional charged AdS black holes, photon-sphere observables become order parameters in reduced parameter space. The reduced photon-sphere radius and reduced minimum impact parameter exhibit non-monotonic behavior for pressures below critical, and their jumps across the coexistence line scale as

$Q=0$0

with universal exponent $Q=0$1 for any spacetime dimension $Q=0$2 (Wei et al., 2017).

For five-dimensional equal-spin Kerr–AdS, reduced thermodynamic laws yield coexistence curves and critical exponents directly in dimensionless form. The differences

$Q=0$3

along coexistence all scale with exponent $Q=0$4, while the reduced Clapeyron equation and latent heat are formulated entirely in reduced variables (Wei et al., 2019).

A common misconception is that reduction necessarily removes physical information. The examples above indicate the opposite: in many cases reduction is what makes curvature, criticality, and universality visible, because it removes redundant coordinates and isolates the actual control parameters of the system.

5. Effective reduction through invariants, entanglement, and thermal tracing

Not all reduced thermodynamic phase spaces are defined by explicit constraints among thermodynamic variables. In several contexts, reduction arises because a large phase space is effectively compressed by invariants, entanglement, or partial tracing.

In isomorph theory for the Yukawa system, hidden scale invariance implies that along curves of constant excess entropy one has

$Q=0$5

and many structural and dynamical observables are invariant in reduced units. The paper states that this “effectively reduces the dimensionality of the phase diagram” because many observables become functions of a single isomorph index rather than independent functions of density and temperature. For the Yukawa fluid, strong virial–potential energy correlations with $Q=0$6 across the studied range support this reduction (Veldhorst et al., 2015).

In the one-dimensional transverse-field Ising model at criticality, entanglement produces a compact occupancy of phase space for a block of length $Q=0$7. The entanglement spectrum has only $Q=0$8 thermodynamically active fermionic modes, with numerical fit

$Q=0$9

and the effective volume becomes

$M_{\mathrm{Kerr}}(R):(S,J)$0

Standard Boltzmann–Gibbs thermodynamics is then extensive in $M_{\mathrm{Kerr}}(R):(S,J)$1, not in $M_{\mathrm{Kerr}}(R):(S,J)$2, while the full subsystem Hilbert-space dimension $M_{\mathrm{Kerr}}(R):(S,J)$3 is effectively replaced by $M_{\mathrm{Kerr}}(R):(S,J)$4 (Souza et al., 2017).

A different type of reduction occurs in thermofield dynamics. There, doubling the Hilbert space introduces tilde modes, and physically relevant phase-space distributions are obtained only after back-transforming the density operator and tracing out the tilde degrees of freedom. In the inverse Bogoliubov transformation variant, the physical one-particle density matrix is

$M_{\mathrm{Kerr}}(R):(S,J)$5

so the reduced thermodynamic phase space of the physical mode is obtained only after a nontrivial back-transformation and partial trace (Błasiak et al., 27 May 2025).

These cases broaden the notion of reduction beyond classical thermodynamic manifolds. This suggests that reduced thermodynamic phase space can also mean an effective macroscopic description obtained after eliminating inactive, auxiliary, or hidden degrees of freedom while retaining the thermodynamically relevant observables.

6. Conceptual consequences, limitations, and open directions

Several recurring consequences follow from the literature. First, reduction changes the meaning of thermodynamic conjugacy. When variables are constrained, the naive partial derivative on the reduced surface need not equal the physical conjugate variable; regular black holes provide the clearest example (Ma et al., 13 Jul 2025). Second, reduction often changes the natural thermodynamic potential. For Schwarzschild–AdS, enthalpy rather than internal energy is the correct fundamental characteristic function in reduced Ruppeiner geometry (Xu et al., 2019). Third, reduced laws can display universal structure more transparently than unreduced ones, as in equal-spin Kerr–AdS or in black-hole topology classified by total winding number (Wei et al., 2019, Sadeghi et al., 2023).

At the same time, reduction introduces limitations. In the Kerr–Newman cavity system, the mapping $M_{\mathrm{Kerr}}(R):(S,J)$6 or $M_{\mathrm{Kerr}}(R):(S,J)$7 used for Ruppeiner geometry is explicitly described as heuristic, and the inferred microstructure is strongly ensemble- and state-dependent (Huang et al., 2022). In the nonequilibrium symplecto-contact construction, the reduction is formulated formally in an infinite-dimensional setting, and functional-analytic subtleties are not the focus (Lim et al., 2022). In isomorph theory, a single constant nearest-neighbor scaling factor $M_{\mathrm{Kerr}}(R):(S,J)$8 cannot perfectly describe invariance over very large density spans (Veldhorst et al., 2015).

Open questions are correspondingly diverse. For constrained black holes, systematic treatment of multiple constraints and projected Smarr relations remains important (Ma et al., 13 Jul 2025). For contact and symplectic constructions, the relation between different reduced thermodynamic representations and nonequilibrium closures remains active (Ghosh, 2022, Grmela et al., 2019). For thermal quantum dynamics, higher-dimensional reduced Wigner functions remain computationally costly even when exact mapping formulas are known (Błasiak et al., 27 May 2025).

Reduced thermodynamic phase spaces therefore unify a broad set of methods rather than a single doctrine. Their common role is to identify the true thermodynamic degrees of freedom—whether by geometric projection, scaling invariance, dynamical coarse graining, partial tracing, or explicit constraint—and to reformulate thermodynamics on that smaller space so that phase behavior, geometry, and response become internally consistent.