Extended Thermodynamics Overview

Updated 27 June 2026

Extended thermodynamics is a framework that enlarges classical thermodynamics by incorporating extra variables for non-equilibrium processes, including higher-order fluxes and internal stresses.
It employs kinetic, field-theoretic, and variational methods to derive hyperbolic, stable evolution equations applicable to rarefied gases, advanced heat conduction, and gravitational systems.
The approach ensures strict compatibility with the second law, yielding physically consistent models in both relativistic and quantum regimes.

Extended thermodynamics systematically enlarges the theoretical and mathematical framework of conventional thermodynamics to encompass non-equilibrium processes, higher-order fluxes, gravitational systems, and generalized state spaces. The core idea is to include additional thermodynamic variables beyond the classical equilibrium set (such as fluxes, internal variables, or coupling constants), leading to new laws, closure relations, and field equations. This extension is achieved through kinetic-theoretic, field-theoretic, and variational formalisms, and is central to the modeling of complex fluids, rarefied gases, heat conduction beyond Fourier's law, black hole systems with variable couplings, and even cosmological or turbulent systems. Extended thermodynamics is characterized by a strict compatibility with the second law—guaranteeing hyperbolicity, stability, and the emergence of causal and physically consistent evolution equations—even in relativistic and quantum regimes.

1. Foundations and Motivations

Conventional thermodynamics, based on local equilibrium and a small set of state variables (e.g., $(\rho, T, v_i)$ ), fails when describing systems exhibiting finite propagation speeds (e.g., ballistic or "second sound" heat conduction), pronounced non-locality (rarefied gases), or non-equilibrium effects (e.g., dynamic pressure, entropy production beyond classical constitutive laws). Classical constitutive relations—Fourier's law for heat and Navier-Stokes for momentum—are parabolic, predicting infinite signal speeds and giving rise to unphysical behaviors in fast transients, rarefied regimes, or relativistic contexts (Kovács et al., 2018, Gavassino et al., 2021).

Extended thermodynamics remedies these deficiencies by systematically enlarging the set of variables:

In kinetic theory, this means considering higher moments of the one-particle distribution function, yielding rational extended thermodynamics (RET) hierarchies with a finite number $N$ of moments, leading to symmetric hyperbolic PDEs (Ruggeri, 2015, Arima et al., 2021).
In continuum frameworks, it involves adding internal variables (scalar, vector, or tensorial) with physical interpretations as fluxes, stresses, or internal energies (NET-IV) (Kovács et al., 2018).
In gravitational and black hole physics, the set of variables is extended to include coupling constants such as the cosmological constant, scalar field couplings, or brane tension, which act as thermodynamic variables like pressure or chemical potential (Xiao et al., 2023, Frassino et al., 2022, Kumar, 25 Aug 2025).
In turbulence and quantum systems, extended thermodynamics also operates at the level of stochastic, scale-dependent state variables (e.g., spectral entropy in turbulent flows) or accommodates phase spaces permitting negative temperatures (Porporato et al., 2020, Corichi et al., 20 Oct 2025).

2. Mathematical Structure, Closure, and Field Equations

The mathematical structure of extended thermodynamics is inherently more sophisticated than classical theory, requiring full compatibility with the second law and enforcement of hyperbolicity (finite characteristic speeds), convex entropy, and non-negative entropy production.

2.1 Moment Hierarchies and Kinetic Closures

RET constructs a finite $N$ -moment hierarchy by projecting the Boltzmann(-Chernikov) equation onto polynomial basis functions of the velocity and possibly internal (polyatomic) degrees of freedom. The balance laws have the generic form

$\partial_t M^{(n)} + \nabla \cdot M^{(n+1)} = P^{(n)}$

for moments up to order $N$ . Closure is obtained either by the entropy principle via the Liu–Müller method, or by Maximum Entropy Principle (MaxEnt) leading to exponential-family distributions, with the higher-order moments as polynomial functions of the retained fields. These systems are always symmetric hyperbolic, guaranteeing causality and local stability (Kovács et al., 2018, Ruggeri, 2015, Arima et al., 2021).

2.2 NET-IV and Internal Variables

In NET-IV, the state is augmented by internal variables $\xi^A$ ; the entropy density is taken as a quadratic extension,

$s = s_{eq}(e, \rho) - \frac{1}{2} m_{AB} \xi^A \xi^B$

and the entropy flux includes additional couplings. Linear Onsager-type constitutive relations then yield evolution equations for the internal variables, ensuring the entropy production $\sigma \ge 0$ (Kovács et al., 2018).

2.3 Variational Principles

Recent developments provide a systematic variational formulation for extended irreversible thermodynamics: the nonequilibrium fluxes (such as heat flux, viscous stresses) are treated as independent variables in the Lagrangian. The Euler–Lagrange equations, derived with appropriate nonholonomic (second law) constraints, yield both the momentum/energy balances and the extended evolution equations for fluxes, e.g., the Maxwell–Cattaneo law for heat conduction and higher-order flux chains (Gay-Balmaz, 24 Feb 2025).

Table: Key Mathematical Structures in Extended Thermodynamics

Approach	Additional Variables	Closure Mechanism
RET	Polynomial-moment fields	Entropy principle/MaxEnt
NET-IV	Internal scalar/tensor variables	Onsager linearity
Gravitational	Couplings (Λ, α, τ, σ, etc.), charges	Covariant variational principle
Variational	Fluxes as dynamical variables	Action extremization + second law constraint

3. Extended Laws of Thermodynamics

Extended thermodynamics consistently modifies both the first and second laws:

Extended First Law: New terms appear, corresponding to the work done by or on the additional variables (e.g., $V\,dP$ for variable cosmological constant, $\Phi_\alpha\,d\alpha$ for scalar field coupling, $N$ 0 or $N$ 1 for entropy splits).
Extended Second Law: The entropy production is generalized, often including contributions from both equilibrium and non-equilibrium components. In the presence of negative temperatures, the law takes the form $N$ 2, restoring a well-defined arrow of time even when $N$ 3 (Corichi et al., 20 Oct 2025).

At the level of field equations,

$N$ 4

where $N$ 5 are extended variables (scalar coupling, brane tension, higher-curvature couplings), and $N$ 6 are their conjugate thermodynamic volumes or potentials, with the corresponding Smarr relations following from homogeneity or Noether identities (Xiao et al., 2023, Ballesteros et al., 2023).

4. Applications: Fluids, Heat Transport, Kinetic Theory

4.1 Rarefied Gases and Polyatomic Fluids

RET and NET-IV have been successfully applied to rarefied gases and polyatomic mixtures, where classical Navier–Stokes–Fourier theory fails (e.g., finite shock thickness, multiple heat conduction modes, second sound). The non-linear ET6 model, directly derived from MaxEnt closure, predicts globally smooth solutions within integrability bounds and incorporates non-equilibrium stress (dynamic pressure) as an independent field (Ruggeri, 2015, Arima et al., 2021).

4.2 Heat Conduction Beyond Fourier

Extended models for heat conduction include not only the classical Fourier law but hyperbolic (Maxwell–Cattaneo–Vernotte), Guyer–Krumhansl, and higher-moment (e.g., 9-field) systems. These generically guarantee dynamic stability of the homogeneous equilibrium state if the entropy is concave and the entropy production is non-negative (i.e., positive-definite Onsager matrix), irrespective of the detailed constitutive model (Somogyfoki et al., 2024).

4.3 Relativistic Dissipative Hydrodynamics

First-order relativistic theories (Eckart, Landau–Lifshitz) are acausal and dynamically unstable. EIT/UEIT (Unified Extended Irreversible Thermodynamics) guarantees hyperbolicity and Lyapunov stability for relativistic fluids by treating dissipative fluxes as independent dynamical variables and deriving their evolution from a first-order field theory manifestly compatible with the second law. All causal relativistic theories for dissipation (Israel–Stewart, divergence-type, Carter's multifluids) are encompassed by this formalism (Gavassino et al., 2021).

4.4 Sub-shocks and Moment Closure Pathologies

In high-order polynomial moment closures, anomalous sub-shocks (internal discontinuities) manifest as phase transitions in the structure of the velocity distribution. These are artifacts of the polynomial basis, rooted in the insufficient degrees of freedom available to represent multimodal distributions. Remediation requires adopting more flexible basis sets (splines, wavelets, neural representations) or including fluctuation-driven corrections (second-order PDEs) (Zheng et al., 2023).

5. Black Hole and Gravitational Extended Thermodynamics

5.1 Extended Black Hole Thermodynamics

The field of black hole chemistry treats quantities such as the cosmological constant ( $N$ 7), higher-curvature couplings ( $N$ 8), brane tension ( $N$ 9), or scalar self-coupling ( $N$ 0) as thermodynamic variables: $N$ 1, $N$ 2, etc. The first law and Smarr relations are derived from variational principles (Iyer–Wald formalism) extended to incorporate these couplings,

$N$ 3

with the conjugate volumes $N$ 4 computed from bulk integrals. These extensions are valid in arbitrary diffeomorphism-invariant gravity theories, not only Einstein gravity (Xiao et al., 2023).

Braneworlds and DGP Model: In DGP braneworlds, varying $N$ 5 induces a new work term $N$ 6 (with $N$ 7) in the first law, and an explicit geometric Smarr relation exists even in asymptotically flat spacetimes (Kumar, 25 Aug 2025).
Higher-dimensional Holographic Origin: In AdS/CFT settings, varying the brane tension $N$ 8 induces a dynamical cosmological constant on the brane, making pressure a physical, variable parameter. The bulk first law,

$N$ 9

maps exactly to the brane first law $\partial_t M^{(n)} + \nabla \cdot M^{(n+1)} = P^{(n)}$ 0, with double-holographic CFT/microstate interpretation (Frassino et al., 2022).

Scalar Charges and Hairy Black Holes: The presence of scalar fields adds new variables (e.g., scalar charge $\partial_t M^{(n)} + \nabla \cdot M^{(n+1)} = P^{(n)}$ 1, scalar-potential "pressure" $\partial_t M^{(n)} + \nabla \cdot M^{(n+1)} = P^{(n)}$ 2) with corresponding volumes $\partial_t M^{(n)} + \nabla \cdot M^{(n+1)} = P^{(n)}$ 3; the extended first law includes these terms and their associated Smarr relation is derived from scaling arguments and generalized Noether charges (Ballesteros et al., 2023).

5.2 Quasi-local and Fractal Extensions

Quasilocal Thermodynamics: Quasilocal energies, surface pressures, and extended phase spaces can be constructed for black holes in finite domains (with boundary at finite radius $\partial_t M^{(n)} + \nabla \cdot M^{(n+1)} = P^{(n)}$ 4), capturing Hawking–Page transitions and new response functions (Fontana et al., 2018).
Quantum/Fractal Corrections: Quasi-fractal (Barrow) corrections to the area law for entropy lead to deformations of the equation of state, critical points, and phase transitions, influencing black hole evaporative lifetimes and microstate degeneracies, while leaving local Tolman/Unruh temperatures unaltered (Ladghami et al., 2024).

6. Generalized State Spaces and Laws: Negative Temperatures, Entropy Splitting

Extended thermodynamics also addresses regimes with negative absolute temperature (e.g., Onsager’s point vortices, quantum cosmological bounces), extending the second law to

$\partial_t M^{(n)} + \nabla \cdot M^{(n+1)} = P^{(n)}$ 5

which ensures the arrow of time is preserved in both positive- $\partial_t M^{(n)} + \nabla \cdot M^{(n+1)} = P^{(n)}$ 6 and negative- $\partial_t M^{(n)} + \nabla \cdot M^{(n+1)} = P^{(n)}$ 7 sectors. The entropy may be split into gravitational and matter parts, or calorimetric and configurational contributions, exposing multiple irreversibility sources and compensation mechanisms among different forms of entropy and energy (Corichi et al., 20 Oct 2025, Suye, 2012).

Table: Law Extensions in Various Contexts

Domain	Extended Law/Variable	Arrow of Time Statement
Kinetic/fluid	Higher moments/internal vars.	$\partial_t M^{(n)} + \nabla \cdot M^{(n+1)} = P^{(n)}$ 8 (local entropy prod.)
Black hole	$\partial_t M^{(n)} + \nabla \cdot M^{(n+1)} = P^{(n)}$ 9, $N$ 0	$N$ 1 includes matter, grav.
Negative $N$ 2	$N$ 3	Time ordering reversed for $N$ 4

7. Open Problems and Future Directions

Moment Expansion Limitations: Sub-shock pathology in high-moment closures motivates active work on richer basis functions and on including fluctuational (dissipative) terms, with neural-net or spline representations an open avenue (Zheng et al., 2023).
Non-uniqueness in Transport Coefficients: In relativistic RET, transport coefficients derived via Maxwellian Iteration depend on the closure order $N$ 5, unlike the $N$ 6-independent coefficients from Chapman–Enskog, indicating a lack of universality that needs clarification (Demontis et al., 2023).
Microscopic/Quantum Gravitational Effects: The full implications of fractal microstructure for black hole evaporation, horizon microstate counting, and observer-dependent thermodynamics remain incompletely understood (Ladghami et al., 2024).
Numerical and Variational Methods: Variational formulations of extended irreversible thermodynamics enable structure-preserving, thermodynamically consistent discretizations, crucial for faithful simulations of nonequilibrium systems (Gay-Balmaz, 24 Feb 2025).