Local Thermodynamical Stability (LTS)

• Local thermodynamical stability (LTS) is defined by the negative definiteness of the entropy Hessian, ensuring that small perturbations decrease entropy to maintain local equilibrium.
• It underpins diverse applications from classical statistical mechanics and black hole thermodynamics to advanced hydrodynamics and cosmological models.
• LTS analysis employs both macroscopic and microscopic frameworks, utilizing criteria like entropy concavity, Lyapunov functionals, and the minimization of irreversible entropy production.

Local thermodynamical stability (LTS) is a fundamental property of macroscopic, mesoscopic, and microscopic systems wherein small, localized perturbations to a system near (but generally not in) equilibrium decay over time, ensuring that the system returns to (local) thermodynamic equilibrium. LTS is mathematically characterized by concavity of the entropy or, equivalently, negative definiteness of the entropy’s second variations, and physically underpinned by the Second Law of thermodynamics and the requirement of non-negative entropy production. This principle emerges across diverse domains, from classical statistical mechanics, black hole thermodynamics, and cosmological models, to advanced hydrodynamics and relativistic dissipation theory.

1. Foundations and Mathematical Criteria for LTS

A system possesses local thermodynamical stability if its entropy $S$ is locally maximal for fixed values of its extensive variables. Mathematically, this condition is expressed as negative definiteness of the Hessian matrix of the entropy with respect to its extensive arguments (internal energy $E$, volume $V$, particle number $N$, etc.): $$D^2 S = \left( \begin{array}{cc} \frac{\partial^2 S}{\partial E^2} & \frac{\partial^2 S}{\partial E \partial V} \ \frac{\partial^2 S}{\partial V \partial E} & \frac{\partial^2 S}{\partial V^2} \end{array} \right)\prec 0$$ which, for concrete models, requires

$$\frac{\partial T}{\partial E} > 0,\quad \frac{\partial p}{\partial V} < 0$$

for a simple fluid (Ván, 2023). The concavity of the entropy ensures that small variations in the extensive variables always decrease entropy—that is,

$$\delta^2 S = \sum_{A,B} \frac{\partial^2 S}{\partial X^A \partial X^B} \delta X^A \delta X^B < 0$$

for any allowed perturbation $\delta X$.

In nonequilibrium settings, LTS can be recast as the minimization of time- or path-ensemble-averaged irreversible entropy production over admissible variations, constrained by the system’s dynamics: $[\dot{S}_{\rm bulk}] = \min,$ subject to dynamical constraints such as conservation laws or boundary conditions (Vita, 2017). This universality of the stability criterion persists across both macroscopic and mesoscopic scales.

2. LTS in Classical and Quantum Statistical Frameworks

Sewell (Sewell, 2013) provides a rigorous hierarchical formulation:

• Macroscopic scale: LTE is postulated by the identification of local entropy density $s(q(x,t))$, with $E$0 local densities of conserved quantities, and the conjugate control variables $E$1. LTS materializes as the mathematical uniqueness and convexity of this mapping.
• Mesoscopic scale: Hydrodynamic fluctuations are modeled as Gaussian fields with covariances determined by the local Hessian $E$2; the characteristic function for local fluctuations is

$E$3

again anchoring LTS in the local second derivatives.

• Microscopic/quantum scale: The existence of unique global thermodynamic stability (GTS) states under local conditions is ensured by the KMS (Kubo-Martin-Schwinger) condition

$E$4

with $E$5. In local GTS states, the LTS property is reflected in the local maximization of (quantum) entropy density.

A direct corollary is a local version of the zeroth law: a finite system coupled locally to an infinite system at local temperature $E$6 thermalizes to $E$7, endowing $E$8 with operational significance even in global nonequilibrium (Sewell, 2013).

3. LTS in Relativistic Hydrodynamics and Causality

In relativistic hydrodynamics, the maximization of entropy translates directly into dynamical LTS and, strikingly, implies causality (Gavassino et al., 2021). The requirement that the entropy (or generalized grand potential) attains a local maximum sets the "energy" (or information) current $E$9 to be future-directed and non-spacelike: $V$0 for any spacelike hypersurface normal $V$1. A geometrical argument shows that any localized perturbation is constrained to propagate within the future light-cone; superluminal (acausal) propagation would contradict the entropic stability criterion (Gavassino et al., 2021). The physical rationale is that entropy, as a measure of missing information, cannot be locally decreased by acausal redistribution of information, connecting dynamical stability and relativity at a fundamental level.

4. LTS in Dissipative and Nonequilibrium Systems

For systems far from equilibrium or under external driving, LTS is tied to extremal properties of entropy production:

• Invariant entropy production: The rate of Boltzmann-Gibbs entropy production in the bulk is invariant under state-space diffeomorphisms (Vita, 2017), resulting in the universal, constrained minimization criterion for stability:

$V$2

for the path- or time-ensemble-averaged entropy production (Vita, 2017).

• Lyapunov functionals in open systems: For thermodynamically open or spatially inhomogeneous systems, LTS is most naturally encoded in Lyapunov functionals derived from free energy and entropy (rather than in artificial $V$3 norms). For instance, the functional

$V$4

for a temperature perturbation $V$5 around steady state $V$6, is nonnegative and decreases monotonically under the heat equation, providing a physically meaningful characterization of LTS (Bulíček et al., 2017).

• Extended thermodynamics: In advanced theories of heat conduction, LTS is ensured by constructing entropy functions that are concave in all extended state variables (e.g., internal energy $V$7, heat flux $V$8, higher flux tensors $V$9):

$N$0

and by requiring non-negative entropy production. Perturbation analysis reveals that stability is governed by the Routh–Hurwitz criteria applied to the linearized evolution equations, with LTS ultimately rooted in the negative definiteness of the entropy Hessian and the positivity of dissipation (Somogyfoki et al., 2024).

5. LTS in Gravitational and Astrophysical Systems

Black Holes and Thin Shells

Black hole thermodynamics offers a stringent testing ground for LTS:

• Hessian and geometric criteria: For black holes (e.g., Reissner–Nordström surrounded by quintessence), LTS is enforced via positivity of the Hessian of the mass with respect to extensive variables (e.g., entropy $N$1, charge $N$2), which reduces to ensuring specific heat $N$3 and other response function constraints in the canonical ensemble (Azreg-Aïnou et al., 2012). Divergence of the specific heat signals second-order phase transitions.
• Geometrothermodynamics and Poincaré turning-point method: The location of phase transitions is tracked both by divergences in curvature scalars derived from thermodynamic metrics and by tangent bifurcations in sequences of equilibria (Poincaré method), with all three approaches (Hessian, geometric, Poincaré) yielding consistent LTS conditions (Azreg-Aïnou et al., 2012, Zhang, 2018).
• Global versus local formalisms: In $N$4-gravity models, the choice between Hawking (global) and Tolman (local) formalisms influences the description of LTS, especially when additional potentials (e.g., power-law corrections or effective cosmological constants) are present. The key operational criterion remains positivity of the temperature and heat capacity; outside these intervals, systems are locally unstable (Lustosa et al., 2019).

Self-Gravitating Shells and Fluids in Modified Gravity

• Maximum entropy and equivalence with dynamical stability: For self-gravitating thin shells or perfect fluids in $N$5 gravity, the second variation of the total entropy acts as an “energy functional;” the LTS criterion is $N$6, entirely equivalent to the canonical energy or dynamical stability requirement (Fang et al., 2017, Bergliaffa et al., 2020).
• Restrictions on parameter space: Thermodynamical criteria (via entropy's convexity) are often more stringent than dynamical ones, restricting the region of dynamically stable solutions to a proper subset, generally delimited by the point of maximal mass in the equilibrium configuration (Bergliaffa et al., 2020).

6. LTS in Hydrodynamics, Symmetry-Broken Systems, and Magnetic Fields

The most general hydrodynamic systems obeying the first and second laws of thermodynamics exhibit LTS automatically, as the linearized operator governing perturbations has eigenvalues with nonpositive imaginary parts (Goutéraux et al., 2024): $N$7 where the positive definiteness of susceptibilities (from the first law) and dissipative transport coefficients (from the second law) excludes hydrodynamic eigenmodes with exponential growth.

The stability structure extends naturally to systems with:

• Spontaneously or softly broken symmetries: Additional Goldstone or relaxation degrees of freedom enter the susceptibility and damping structure but LTS persists provided the extended susceptibility matrix remains positive definite and the dissipative corrections respect nonnegativity.
• Magnetic fields or parity breaking: Non-dissipative (e.g., Hall) contributions shift real parts of the dispersion but do not affect the positive definiteness of the damping terms, so LTS is preserved unless the magnetic field is unphysically large (Goutéraux et al., 2024).

7. Cosmological and Universal Aspects of LTS

In the expanding Universe, LTE—and therefore LTS of the primordial plasma—is set by comparing the microscopic reaction time $N$8 to the Hubble time $N$9, through the relation $$D^2 S = \left( \begin{array}{cc} \frac{\partial^2 S}{\partial E^2} & \frac{\partial^2 S}{\partial E \partial V} \ \frac{\partial^2 S}{\partial V \partial E} & \frac{\partial^2 S}{\partial V^2} \end{array} \right)\prec 0$$0 (Ignatyev, 2011). At superhigh energies where quantum field theory predicts a “scaling” regime for cross-sections ($$D^2 S = \left( \begin{array}{cc} \frac{\partial^2 S}{\partial E^2} & \frac{\partial^2 S}{\partial E \partial V} \ \frac{\partial^2 S}{\partial V \partial E} & \frac{\partial^2 S}{\partial V^2} \end{array} \right)\prec 0$$1), reactions are too slow to maintain LTE, so LTS is absent initially and recovers only at later epochs when expansion dilutes the redshifted energies and increases interaction rates. This overturns the standard cosmological assumption of initial LTE and implies non-equilibrium initial conditions—fundamentally altering relic abundance computations and baryogenesis scenarios (Ignatyev, 2011).

The universal aspects of LTS are synthesized in thermodynamic formalisms where entropy acts as a Lyapunov function for evolution, with local maxima guaranteeing attractor behavior. Stability is a precondition for persistence in natural and engineered systems, underlying natural selection at the level of physical law and ensuring the robustness of observed steady states (Ván, 2023).

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Summary Table: Key LTS Criteria Across Domains

Context	LTS Criterion	Mathematical Formulation
Classical/statistical	Max concavity of entropy	$$D^2 S = \left( \begin{array}{cc} \frac{\partial^2 S}{\partial E^2} & \frac{\partial^2 S}{\partial E \partial V} \ \frac{\partial^2 S}{\partial V \partial E} & \frac{\partial^2 S}{\partial V^2} \end{array} \right)\prec 0$$2
Hydrodynamics	Positive susceptibilities, dissipation	$$D^2 S = \left( \begin{array}{cc} \frac{\partial^2 S}{\partial E^2} & \frac{\partial^2 S}{\partial E \partial V} \ \frac{\partial^2 S}{\partial V \partial E} & \frac{\partial^2 S}{\partial V^2} \end{array} \right)\prec 0$$3, $$D^2 S = \left( \begin{array}{cc} \frac{\partial^2 S}{\partial E^2} & \frac{\partial^2 S}{\partial E \partial V} \ \frac{\partial^2 S}{\partial V \partial E} & \frac{\partial^2 S}{\partial V^2} \end{array} \right)\prec 0$$4
Black hole thermodynamics	Positive specific heat, Hessian positivity	$$D^2 S = \left( \begin{array}{cc} \frac{\partial^2 S}{\partial E^2} & \frac{\partial^2 S}{\partial E \partial V} \ \frac{\partial^2 S}{\partial V \partial E} & \frac{\partial^2 S}{\partial V^2} \end{array} \right)\prec 0$$5, Hessian$$D^2 S = \left( \begin{array}{cc} \frac{\partial^2 S}{\partial E^2} & \frac{\partial^2 S}{\partial E \partial V} \ \frac{\partial^2 S}{\partial V \partial E} & \frac{\partial^2 S}{\partial V^2} \end{array} \right)\prec 0$$6
Nonequilibrium/dissipative	Constrained minimum of entropy production	$$D^2 S = \left( \begin{array}{cc} \frac{\partial^2 S}{\partial E^2} & \frac{\partial^2 S}{\partial E \partial V} \ \frac{\partial^2 S}{\partial V \partial E} & \frac{\partial^2 S}{\partial V^2} \end{array} \right)\prec 0$$7
Cosmological plasma	Reaction rate vs. expansion rate	$$D^2 S = \left( \begin{array}{cc} \frac{\partial^2 S}{\partial E^2} & \frac{\partial^2 S}{\partial E \partial V} \ \frac{\partial^2 S}{\partial V \partial E} & \frac{\partial^2 S}{\partial V^2} \end{array} \right)\prec 0$$8, i.e., $$D^2 S = \left( \begin{array}{cc} \frac{\partial^2 S}{\partial E^2} & \frac{\partial^2 S}{\partial E \partial V} \ \frac{\partial^2 S}{\partial V \partial E} & \frac{\partial^2 S}{\partial V^2} \end{array} \right)\prec 0$$9

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• (Ignatyev, 2011, Azreg-Aïnou et al., 2012, Sewell, 2013, Vita, 2017, Fang et al., 2017, Bulíček et al., 2017, Zhang, 2018, Lustosa et al., 2019, Bergliaffa et al., 2020, Gavassino et al., 2021, Salazar et al., 2022, Ván, 2023, Somogyfoki et al., 2024, Goutéraux et al., 2024)