Extended Phase Space in Dynamics

• Extended phase space is defined as the enlargement of classical phase space by including extra dynamical variables such as time, control, and thermodynamic parameters to capture complex system behavior.
• Its application in black-hole thermodynamics introduces pressure and volume as dynamic variables, yielding analogues of Van der Waals phase transitions and critical phenomena.
• The framework supports advanced numerical methods and gauge formulations by restoring local symmetries and integrating additional degrees of freedom for accurate simulation and analysis.

The extended phase space is a general concept in Hamiltonian dynamics, control theory, quantum mechanics, and field theory, referring to the enlargement of the standard phase-space manifold (usually positions and momenta) to include additional dynamical variables or parameters. This extension facilitates the modeling of systems where the canonical variables alone are insufficient to capture physical behavior—such as control actions, thermodynamic variables, gauge degrees of freedom, time, or dissipation. Extended phase-space approaches have yielded advances in modeling human-in-the-loop dynamics, black hole thermodynamics, quantum theory, integrable and numerical simulation methods, and gauge theories.

1. Foundational Principles of Extended Phase Space

Classical Hamiltonian mechanics is formulated on a phase space with coordinates $(q^i,p_i)$, or equivalently, a symplectic manifold equipped with the canonical 2-form. The "extended" phase space generalizes this structure by either considering additional variables (such as time, control parameters, thermodynamic quantities, or gauge fields) or duplicating the phase-space degrees of freedom for symplectic integration and computational purposes.

• Bundle Structure: Extended phase space often carries a fiber-bundle structure, with the base manifold representing time or spacetime, and fibers containing the dynamical variables (e.g., positions, momenta, fields, or their conjugates) (Sharan, 2012). For field theories, this becomes a multisymplectic bundle over manifold $M$.
• Covariant Formulation: In field theory, extended phase space allows for time and space to be treated on equal footing, forming bundles $\pi: \mathcal{E} \to M$ where fibers encode field values and their conjugate momenta. Dynamics is then generated by Poincaré–Cartan $n$-forms, with multisymplectic structure $\Omega = -d\Theta$ driving Hamiltonian evolution (Sharan, 2012).
• Thermodynamic Extension: In black-hole thermodynamics, the cosmological constant is reinterpreted as a dynamical pressure $P = -\Lambda/(8\pi)$, and its conjugate thermodynamic volume $V = (\partial M/\partial P)_{S,Q}$ is added to the phase space, so black-hole mass becomes enthalpy, and the first law and Smarr relation accommodate new variables (Pradhan, 2018).

2. Applications in Thermodynamics and Black Hole Physics

Extended phase space has revolutionized black-hole thermodynamics by treating $\Lambda$ as pressure and defining its conjugate volume. This framework, termed "black-hole chemistry," produces direct analogues to conventional thermodynamic systems.

• Equation of State and Van der Waals Analogy: The black-hole equation of state in the extended phase space assumes a form analogous to the Van der Waals equation: $P = T/v - 1/(2\pi v^2) + 2Q^2/(\pi v^4)$, enabling mapping to fluid systems and analysis of first-order and critical phenomena (Pradhan, 2018, Wei et al., 2014).
• Criticality and Phase Transitions: Extended phase-space thermodynamics supports lines of first-order transitions (small/large BH), critical points, swallowtail Gibbs free energy anomalies, and even triple points (for $d=6$ Gauss–Bonnet black holes), closely mirroring liquid-gas systems (Wei et al., 2014). The Ehrenfest equations and Prigogine–Defay ratio $\Pi=1$ are satisfied, confirming second-order transitions (Pradhan, 2018, Poshteh et al., 2016).
• Thermodynamic Geometry: Thermodynamic geometry (Weinhold, Ruppeiner, Quevedo metrics) in extended phase space yields scalar curvatures that diverge at phase transitions. For charged Hořava–Lifshitz black holes, $R^W$ and $R^R$ diverge at second-order transitions, confirming thermodynamic stability (Poshteh et al., 2016, Bairagya et al., 2020).
• Universality of Product Relations: Universal entropy and volume product formulae (independent of mass) for all black-hole horizons emerge naturally in extended phase-space models (Pradhan, 2016). Critical relations such as $P_c^-=P_c^+$ and $T_c^-=-T_c^+$ for inner and outer (Cauchy/Event) horizons mirror those of the Van der Waals gas.

3. Formulation in Gauge Theories and Gravity

The extended phase space formalism enables resolution of integrability issues for Noether charges in gauge-invariant and diffeomorphism-invariant theories. Key developments include:

• Atiyah Algebroid Extension: The extended configuration space in gauge theory employs Atiyah Lie algebroids, combining spacetime diffeomorphisms and internal symmetries within a unified geometric structure. Gauge transformations are interpreted as morphisms between such algebroids (Klinger et al., 2023).
• Edge Modes and Surface Symmetry: In gravitational theories, especially general relativity, extended phase space includes edge modes localized at boundaries or horizons, which restore full gauge invariance and give rise to a universal surface symmetry algebra (Diff$$(\partial\Sigma)\ltimes SL(2,\mathbb{R})^{\partial\Sigma}$$) (Speranza, 2017, Chandrasekaran et al., 2023). These degrees of freedom generate charges that are integrable and capture horizon physics.
• Central Extensions and Poisson Structure: For intersecting null boundaries, half-sided boosts extend the phase space by introducing relative boost angle modes, and the area operator of the black hole becomes canonically conjugate to the boost variable under the Poisson bracket (Chandrasekaran et al., 2023).

4. Extended Phase Space in Quantum Mechanics

Reformulations of quantum theory in extended phase space address locality and action-reaction principles:

• Complex Amplitude Distribution: Rather than representing quantum states as vectors in Hilbert space, the extended phase space is defined as the Cartesian product of eigenvalues for all possible observables—commuting and noncommuting—and quantum states are specified as complex amplitude distributions $\Psi(\alpha)$ over this set (Lopez, 2015).
• Recovery of Quantum Predictions: Observable statistics for commuting subsets are obtained by marginalizing $\Psi(\alpha)$ and applying Born’s rule. This formalism is immune to Bell’s theorem arguments because the marginals of $\Psi$ are not classical probabilities, sidestepping constraints on local hidden variables.
• Physical Interpretation: This approach models quantum systems as composites of a point-like corpuscle (labeling hidden eigenvalues) and a de Broglie wave (distributed amplitude), restoring local action-reaction symmetry and reconciling quantum measurement with locality (Lopez, 2015).

5. Symplectic Integrators and Numerical Methods

Extended phase space provides powerful machinery for constructing explicit, symplectic integrators for nonseparable or nonautonomous systems—a domain traditionally inaccessible to standard splitting schemes.

• Duplication-of-Variables and Mixing: By doubling the phase-space variables, or including time and energy pairs, systems with inseparable Hamiltonians or implicit velocities can be embedded in a higher-dimensional extended phase space. The extended Hamiltonian is decomposed such that each piece is analytically integrable; explicit leapfrog-type (Strang, Yoshida, Blanes–Moan) splitting methods are then employed (Pihajoki, 2014, Mauger et al., 18 Oct 2025, Pan et al., 2021).
• Projection and Stability: Mixing maps and midpoint permutations are essential for ensuring long-term stability and controlling separation between duplicated copies (Pihajoki, 2014, Pan et al., 2021). Stability criteria and restrain parameters are determined by local Hessians or analysis of the extended system’s linearized flow (Mauger et al., 18 Oct 2025).
• High-Dimensional & Quantum Applications: Extended phase space integrators have been adapted for Kohn-Sham time-dependent density-functional theory (TDDFT) and plasma physics models, handling infinite-dimensional systems and strong time-dependence (Mauger et al., 18 Oct 2025). Cheap accuracy metrics based on the distance between phase-space copies enable adaptive time-stepping and error control.

6. Human-Controlled Systems and Fuzzy Rationality

Extensions of phase space are instrumental in modeling human-in-the-loop control dynamics where decision-making and action are intermittent and threshold-dependent.

• Operator-Motivation Variables: For unstable systems under human control (e.g., stick balancing, vehicle following), the instantaneous control action (e.g., acceleration) is promoted to a dynamic variable, yielding a phase space $(x,v,a)$ where $a$ evolves independently via fuzzy-trap activation functions (Zgonnikov et al., 2012).
• Perceptual Traps and Intermittency: Velocity and action traps encode perceptual thresholds, meaning corrections are applied only when deviations are observable. The extended phase space captures feedback delays, limit cycle oscillations, bifurcation phenomena, and the “on-off” intermittency observed in human control without requiring additive noise (Zgonnikov et al., 2012).
• Generalization: The framework generalizes to any Newtonian control system by promoting the control action(s) $u$ to independent phase-space degrees of freedom whose dynamics activate only above fuzzy thresholds, providing a mechanistic account of intermittent human-in-the-loop corrections.

7. Covariant and Multisymplectic Field Theory

The covariant extended phase-space formalism recasts field theory dynamics and conservation laws in a bundle-theoretic language suited for curved backgrounds.

• Multisymplectic Structure: The covariant Poincaré–Cartan form $\Theta = p_A \wedge d\phi^A - \mathcal{H}(\phi,p)$ leads to the multisymplectic form $\Omega = -d\Theta$. Variational principles require physical sections to annihilate $\Omega$ under vertical variations, ensuring on-shell field equations and Noether identities (Sharan, 2012, Sharan, 2012).
• Torsion and Gravity: For gravity, extended phase space requires treating both the orthonormal frame $e^a$ and connection $\omega^{ab}$ as dynamical variables, leading to explicit coupling of torsion to matter 3-forms and the Einstein tensor. Gravitational dynamics is restricted to the torsion-free surface by algebraic constraints (Sharan, 2012).
• Peierls Bracket and Observables: The Peierls bracket provides a covariant commutator for field observables in extended phase space, yielding an antisymmetric, Jacobi-satisfying algebra on the space of on-shell observables—crucial for quantization on curved backgrounds (Sharan, 2012).

8. Contemporary Impact and Outlook

The extended phase space framework continues to drive developments across diverse domains: high-order simulation algorithms for classical and quantum systems, rigorous quantization schemes for fields on curved spacetime, advanced models of control and cognition, and refined formulations of gravitational thermodynamics and symmetry algebras. A plausible implication is that the further generalization and systematic construction of extended phase-space models will yield unified mechanisms for incorporating observables, conservation laws, and emergent dynamics in complex systems where canonical formulations are insufficient.