Third-Order Extended Thermodynamics of Chemically Reacting Gas Mixtures: A Rational Thermodynamic Derivation of Thermal Conductivity and Dynamic Pressure

1. Introduction

The description of nonequilibrium processes in gas mixtures undergoing chemical reactions remains a central problem in continuum thermodynamics and kinetic theory. Classical irreversible thermodynamics, as formulated by de Groot and Mazur [1], provides a successful framework for near-equilibrium processes but is limited by its parabolic structure and the assumption of local equilibrium. In particular, Fourier’s law of heat conduction and Fick’s law of diffusion imply instantaneous propagation of disturbances, which is physically unrealistic at high frequencies or in rarefied regimes.

To overcome these limitations, Extended Irreversible Thermodynamics (EIT) and Rational Extended Thermodynamics (RET) were developed to incorporate dissipative fluxes as independent state variables [2,3]. In this framework, the entropy is generalized to depend not only on the classical variables $(\rho ,e,{\rho }_{\alpha })$ but also on nonequilibrium fluxes such as the heat flux $\mathbf{q}$ and the dynamic (bulk) pressure $\mathrm{\Pi }$. This leads to hyperbolic field equations with finite propagation speeds and improved stability properties.

For chemically reacting mixtures, the coupling between transport processes and reaction kinetics introduces additional complexity. Early developments in reacting gas dynamics were based on kinetic theory and the Boltzmann equation [4,5], where chemical reactions are treated through collision integrals and source terms in the species balance equations. However, direct kinetic treatments are often intractable for multicomponent reactive systems, motivating the development of macroscopic thermodynamic theories.

A significant contribution in this direction was made by Kremer and Müller [6], who derived the entropy density and entropy flux for chemically reacting mixtures within the framework of rational thermodynamics. Their work provides a consistent first-order theory in which chemical reactions modify the constitutive structure of the heat flux and bulk pressure through the chemical affinity and reaction rates. Nevertheless, their formulation remains restricted to linear constitutive relations and first-order nonequilibrium effects.

Higher-order extensions of thermodynamics have been explored in various contexts. In Extended Irreversible Thermodynamics, nonlinear corrections to fluxes and transport coefficients have been introduced to capture finite-amplitude nonequilibrium effects [3]. Similarly, Grad-type moment methods in kinetic theory provide a systematic route to higher-order closures beyond Navier–Stokes–Fourier theory [7,8]. In relativistic settings, analogous second-order theories have been developed by Israel and Stewart [9], highlighting the importance of nonlinear dissipation and relaxation effects.

Despite these advances, a consistent third-order thermodynamic theory for chemically reacting gas mixtures remains underdeveloped. In particular, the interplay between higher-order nonequilibrium fluxes and chemical reaction affinity has not been systematically incorporated into a unified thermodynamic structure. This gap is particularly relevant for high-temperature gas dynamics, reactive shock waves, and rarefied reacting flows, where nonlinear coupling between transport and chemistry becomes significant.

The present work aims to fill this gap by developing a third-order extension of Rational Extended Thermodynamics for chemically reacting gas mixtures. Starting from the balance equations of mass, momentum, energy, and species densities, we construct a third-order entropy expansion in the dissipative variables, including the heat flux and dynamic pressure. The entropy principle is then used to derive nonlinear constitutive relations and evolution equations for these variables, resulting in generalized Maxwell–Cattaneo-type laws with third-order corrections.

The proposed framework extends the results of Kremer and Müller [6] by incorporating nonlinear couplings between dissipative fluxes and chemical affinity, while preserving the hyperbolic structure characteristic of RET [2]. In the near-equilibrium limit, the classical Navier–Stokes–Fourier and first-order reacting mixture equations are recovered as limiting cases.

The remainder of the paper is organized as follows. Section 2 presents the balance equations for chemically reacting mixtures. Section 3 introduces the entropy principle and thermodynamic constraints. Section 4 develops the third-order entropy expansion. Section 5 derive the constitutive relations for heat flux and dynamic pressure, respectively. Section 5 discusses transport coefficients and chemical coupling effects. Section 6 analyses stability and hyperbolicity. The applications are provided in Section 7 Finally, Section 8 concludes the study.

2. Balance Equations for Chemically Reacting Gas Mixtures

We consider a multicomponent chemically reacting gas mixture consisting of N species indexed by $\alpha =1,\dots ,N$. The macroscopic fields are the partial mass densities ${\rho }_{\alpha }$, the total mass density $\rho$, the barycentric velocity $\mathbf{v}$, the internal energy density e, and the temperature T. The total density satisfies

$$\rho ={\displaystyle \sum _{\alpha =1}^N}{\rho }_{\alpha }.$$

(1)

The governing equations are formulated within the framework of Rational Extended Thermodynamics, where the mixture is treated as a single continuum with additional internal variables accounting for diffusion and reaction processes.

2.1. Mass Balance

The total mass conservation law is given by

$${\partial }_t\rho +\nabla ·\left(\rho \mathbf{v}\right)=0.$$

(2)

For each species $\alpha$, the partial mass balance reads

$${\partial }_t{\rho }_{\alpha }+\nabla ·({\rho }_{\alpha }\mathbf{v}+{\mathbf{J}}_{\alpha })={\omega }_{\alpha },$$

(3)

where ${\mathbf{J}}_{\alpha }$ denotes the diffusion flux of species $\alpha$ relative to the barycentric velocity, and ${\omega }_{\alpha }$ is the rate of production due to chemical reactions. The reaction rates satisfy the compatibility condition

$${\displaystyle \sum _{\alpha =1}^N}{\omega }_{\alpha }=0,$$

(4)

ensuring total mass conservation.

2.2. Momentum Balance

The momentum balance equation is written in the form

$${\partial }_t\left(\rho \mathbf{v}\right)+\nabla ·(\rho \mathbf{v}\otimes \mathbf{v}+\mathsf{P})=0,$$

(5)

where $\mathsf{P}$ is the pressure tensor. In nonequilibrium thermodynamics, it is decomposed as

$$\mathsf{P}=p\mathbf{I}+\tau +\mathrm{\Pi }\mathbf{I},$$

(6)

where p is the equilibrium pressure, $\tau$ is the deviatoric viscous stress tensor, and $\mathrm{\Pi }$ is the dynamic (bulk) pressure associated with volumetric relaxation effects in the reacting mixture.

2.3. Energy Balance

The total energy density is given by

$$E=\rho e+{\displaystyle \frac{1}{2}}\rho {\left|\mathbf{v}\right|}^2.$$

(7)

The energy balance equation takes the form

$${\partial }_{t}E+\nabla ·\left[(E+p)\mathbf{v}+\tau \mathbf{v}+\mathrm{\Pi }\mathbf{v}+\mathbf{q}\right]=0,$$

(8)

where $\mathbf{q}$ denotes the heat flux vector. Chemical reactions contribute indirectly through the internal energy dependence on composition and through source terms in the species equations.

2.4. Species Diffusion Fluxes

The diffusion fluxes are constrained by the barycentric frame condition

$${\displaystyle \sum _{\alpha =1}^N}{\mathbf{J}}_{\alpha }=\mathbf{0}.$$

(9)

A convenient representation introduces independent diffusion fluxes for $N-1$ species, while the remaining flux is determined by the constraint. These fluxes will later be expressed in terms of thermodynamic forces through the entropy principle.

2.5. Chemical Reaction Rates and Affinity

Chemical reactions are assumed to be of the general form

$${\displaystyle \sum _{\alpha =1}^N}{\nu }_{\alpha }^{\left(r\right)}{A}_{\alpha }=0,r=1,\dots ,R,$$

(10)

where ${\nu }_{\alpha }^{\left(r\right)}$ are stoichiometric coefficients and $A_{\alpha }$ denote chemical species. The reaction rates are written as

$${\omega }_{\alpha }={\displaystyle \sum _{r=1}^R}{\nu }_{\alpha }^{\left(r\right)}{\mathcal{R}}_r,$$

(11)

where ${\mathcal{R}}_r$ is the rate of reaction r. The corresponding chemical affinity is defined as

$${\mathcal{A}}_r=-{\displaystyle \sum _{\alpha =1}^N}{\nu }_{\alpha }^{\left(r\right)}{\mu }_{\alpha },$$

(12)

with ${\mu }_{\alpha }$ the chemical potential of species $\alpha$. The affinity ${\mathcal{A}}_r$ will play a central role in the coupling between chemical and mechanical nonequilibrium in higher-order thermodynamic expansions.

2.6. Governing Set of Fields

Collecting the above equations, the full state of the system is described by the set

$$\left\{\rho ,\mathbf{v},e,{\rho }_{\alpha },{\mathbf{J}}_{\alpha },\mathbf{q},\mathrm{\Pi }\right\},$$

(13)

which will later be extended to include higher-order flux contributions in the third-order theory. The system of balance laws is thus not closed and requires constitutive relations derived from the entropy principle.

This completes the formulation of the macroscopic balance structure. In the next section, we introduce the entropy balance and impose the second law of thermodynamics to derive admissible constitutive equations.

3. Entropy Principle and Thermodynamic Restrictions

The closure of the balance equations for chemically reacting gas mixtures requires additional constitutive relations for the pressure tensor, heat flux, diffusion fluxes, and chemical source terms. In Rational Extended Thermodynamics, these relations are not postulated ad hoc but are derived from the entropy principle, ensuring compatibility with the second law of thermodynamics [2,6].

3.1. Entropy Balance Law

We introduce the specific entropy density $\eta =\eta (\rho ,e,{\rho }_{\alpha },\mathbf{q},\mathrm{\Pi },{\mathbf{J}}_{\alpha })$, which is allowed to depend on nonequilibrium variables. The entropy balance equation is postulated in the form

$${\partial }_t\left(\rho \eta \right)+\nabla ·\mathrm{\Phi }=\sigma \ge 0,$$

(14)

where $\mathrm{\Phi }$ is the entropy flux and $\sigma$ is the entropy production density. The second law requires that $\sigma$ be non-negative for all thermodynamically admissible processes.

3.2. Gibbs Relation

The differential of the entropy is written in generalized Gibbs form as

$$\eta ={\displaystyle \frac{1}{T}}e-{\displaystyle \frac{p}{T{\rho }^2}}\rho -{\displaystyle \sum _{\alpha =1}^N}{\displaystyle \frac{{\mu }_{\alpha }}{T}}{\rho }_{\alpha }+\mathrm{\Lambda }·\mathbf{q}+{\mathrm{\Lambda }}_{\mathrm{\Pi }}\mathrm{\Pi }+{\displaystyle \sum _{\alpha =1}^N}{\mathrm{\Lambda }}_{\alpha }·{\mathbf{J}}_{\alpha },$$

(15)

where T is the absolute temperature, ${\mu }_{\alpha }$ are chemical potentials, and the thermodynamic dual variables are defined by

$$\mathrm{\Lambda }={\displaystyle \frac{\partial \eta }{\partial \mathbf{q}}},{\mathrm{\Lambda }}_{\mathrm{\Pi }}={\displaystyle \frac{\partial \eta }{\partial \mathrm{\Pi }}},{\mathrm{\Lambda }}_{\alpha }={\displaystyle \frac{\partial \eta }{\partial {\mathbf{J}}_{\alpha }}}.$$

(16)

In equilibrium, these conjugate variables vanish, recovering classical thermodynamics.

3.3. Entropy Flux Structure

Following the framework of Extended Thermodynamics, the entropy flux is assumed to have the structure

$$\mathrm{\Phi }=\rho \eta \mathbf{v}+{\displaystyle \frac{\mathbf{q}}{T}}+{\displaystyle \sum _{\alpha =1}^N}{\displaystyle \frac{{\mu }_{\alpha }}{T}}{\mathbf{J}}_{\alpha }+\mathrm{\Psi }(\mathbf{q},\mathrm{\Pi },{\mathbf{J}}_{\alpha }),$$

(17)

where $\mathrm{\Psi }$ contains higher-order contributions required for nonlinear consistency. In first-order theories, $\mathrm{\Psi }=0$, while in third-order theories, it contributes cubic corrections in dissipative variables.

3.4. Entropy Production

By substituting the balance laws into the entropy balance equation, the entropy production can be written in canonical form

$$\sigma =\mathbf{q}·{\mathbf{X}}_q+\mathrm{\Pi }{X}_{\mathrm{\Pi }}+{\displaystyle \sum _{\alpha =1}^N}{\mathbf{J}}_{\alpha }·{\mathbf{X}}_{\alpha }+{\displaystyle \sum _{r=1}^R}{\mathcal{R}}_r{\mathcal{A}}_r,$$

(18)

where the thermodynamic forces are identified as

$$\begin{array}{cc}\hfill {\mathbf{X}}_q& =\nabla \left({\displaystyle \frac{1}{T}}\right),\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill X_{\mathrm{\Pi }}& =-\nabla ·\mathbf{v},\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill {\mathbf{X}}_{\alpha }& =-\nabla \left({\displaystyle \frac{{\mu }_{\alpha }}{T}}\right),\hfill \end{array}$$

(21)

and ${\mathcal{A}}_r$ is the chemical affinity defined previously. The second law requires

$$\sigma \ge 0,$$

(22)

for all admissible processes, which imposes restrictions on the constitutive relations.

3.5. Constitutive Closure Problem

The entropy production identifies the set of thermodynamic fluxes

$$\left\{\mathbf{q},\mathrm{\Pi },{\mathbf{J}}_{\alpha },{\mathcal{R}}_r\right\},$$

(23)

and their conjugate forces

$$\left\{{\mathbf{X}}_q,X_{\mathrm{\Pi }},{\mathbf{X}}_{\alpha },{\mathcal{A}}_r\right\}.$$

(24)

The closure problem consists in specifying constitutive relations of the form

$$\mathbf{q}=\mathbf{q}({\mathbf{X}}_q,X_{\mathrm{\Pi }},{\mathbf{X}}_{\alpha },{\mathcal{A}}_r,\dots ),\mathrm{\Pi }=\mathrm{\Pi }({\mathbf{X}}_q,X_{\mathrm{\Pi }},\dots ),$$

(25)

subject to the constraint $\sigma \ge 0$. In classical linear irreversible thermodynamics, these relations are linear in the forces. However, to capture finite-amplitude nonequilibrium effects, we allow nonlinear dependence up to third order in the dissipative variables, leading to the third-order Extended Thermodynamics framework developed in the following sections.

This completes the formulation of the entropy principle. In the next section, we construct the third-order entropy expansion and derive the corresponding nonlinear constitutive structure.

4. Third-Order Entropy Expansion and Nonlinear Constitutive Structure

In this section we construct a third-order extension of the entropy function for chemically reacting gas mixtures. The goal is to systematically derive nonlinear constitutive relations for the heat flux $\mathbf{q}$, dynamic pressure $\mathrm{\Pi }$, and diffusion fluxes ${\mathbf{J}}_{\alpha }$ from the entropy principle, while preserving compatibility with the second law of thermodynamics.

The construction follows the philosophy of Rational Extended Thermodynamics, where the entropy is postulated as a convex function of the nonequilibrium variables and expanded around equilibrium up to third order in dissipative fields [2,3].

4.1. Equilibrium State and Perturbation Structure

We denote equilibrium quantities by the subscript “eq”. At equilibrium,

$$\mathbf{q}=\mathbf{0},\mathrm{\Pi }=0,{\mathbf{J}}_{\alpha }=\mathbf{0}.$$

(26)

We introduce the deviation variables

$$\mathbf{q},\mathrm{\Pi },{\mathbf{J}}_{\alpha },$$

(27)

which are assumed to be small in a neighborhood of equilibrium. The entropy is expanded as a scalar function of invariants constructed from these variables.

4.2. General Structure of the Entropy Function

Up to third order, the most general isotropic entropy density consistent with frame indifference and species symmetry can be written as

$$\eta ={\eta }_{\mathrm{eq}}(\rho ,e,{\rho }_{\alpha })+{\eta }^{\left(2\right)}+{\eta }^{\left(3\right)},$$

(28)

where ${\eta }^{\left(2\right)}$ contains quadratic terms and ${\eta }^{\left(3\right)}$ contains cubic corrections.

4.3. Quadratic Entropy Contribution

The second-order part is taken as

$${\eta }^{\left(2\right)}=-{\displaystyle \frac{1}{2}}\left(a_q{\left|\mathbf{q}\right|}^2+a_{\mathrm{\Pi }}{\mathrm{\Pi }}^2+{\displaystyle \sum _{\alpha =1}^N}{a}_{\alpha }{\left|{\mathbf{J}}_{\alpha }\right|}^2\right),$$

(29)

where $a_q,a_{\mathrm{\Pi }},a_{\alpha }>0$ ensure concavity of the entropy and thermodynamic stability.

4.4. Cubic Entropy Contribution

The third-order correction introduces nonlinear couplings between dissipative fluxes. The most general isotropic cubic form is

$$\begin{array}{cc}\hfill {\eta }^{\left(3\right)}& =-{\displaystyle \frac{1}{3}}{b}_q{\left|\mathbf{q}\right|}^3-{\displaystyle \frac{1}{3}}{b}_{\mathrm{\Pi }}{\mathrm{\Pi }}^3-{\displaystyle \sum _{\alpha =1}^N}{\displaystyle \frac{1}{3}}{b}_{\alpha }{\left|{\mathbf{J}}_{\alpha }\right|}^3\hfill \\ \hfill & -c_1{\mathrm{\Pi }\left|\mathbf{q}\right|}^2-{\displaystyle \sum _{\alpha =1}^N}{c}_{2\alpha }\mathrm{\Pi }|{\mathbf{J}}_{\alpha }{|}^2-{\displaystyle \sum _{\alpha =1}^N}{c}_{3\alpha }{\left|\mathbf{q}\right|}^2\left|{\mathbf{J}}_{\alpha }\right|\hfill \\ \hfill & -{\displaystyle \sum _{\alpha =1}^N}{c}_{4\alpha }\mathbf{q}·{\mathbf{J}}_{\alpha }\mathrm{\Pi }.\hfill \end{array}$$

(30)

These terms represent:

• self-nonlinearity of heat flux and bulk pressure,
• coupling between volumetric relaxation and thermal transport,
• coupling between diffusion and thermal fluxes,
• triple interactions involving reaction-induced nonequilibrium.

4.5. Thermodynamic Conjugate Variables

The entropy derivatives define the thermodynamic forces conjugate to the dissipative variables:

$$\begin{array}{cc}\hfill {\mathbf{X}}_q& :={\displaystyle \frac{\partial \eta }{\partial \mathbf{q}}},\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill X_{\mathrm{\Pi }}& :={\displaystyle \frac{\partial \eta }{\partial \mathrm{\Pi }}},\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill {\mathbf{X}}_{\alpha }& :={\displaystyle \frac{\partial \eta }{\partial {\mathbf{J}}_{\alpha }}}.\hfill \end{array}$$

(33)

From the above expansion we obtain:

4.5.1. Heat Flux Conjugate Force

$$\begin{array}{cc}\hfill {\mathbf{X}}_q& =-a_q\mathbf{q}-b_q\left|\mathbf{q}\right|\mathbf{q}-2c_1\mathrm{\Pi }\mathbf{q}-2{\displaystyle \sum _{\alpha =1}^N}{c}_{3\alpha }\left|{\mathbf{J}}_{\alpha }\right|\mathbf{q}-{\displaystyle \sum _{\alpha =1}^N}{c}_{4\alpha }\mathrm{\Pi }{\mathbf{J}}_{\alpha }.\hfill \end{array}$$

(34)

4.5.2. Dynamic Pressure Conjugate Force

$$\begin{array}{cc}\hfill X_{\mathrm{\Pi }}& =-a_{\mathrm{\Pi }}\mathrm{\Pi }-b_{\mathrm{\Pi }}{\mathrm{\Pi }}^2-c_1{\left|\mathbf{q}\right|}^2-{\displaystyle \sum _{\alpha =1}^N}{c}_{2\alpha }{\left|{\mathbf{J}}_{\alpha }\right|}^2-{\displaystyle \sum _{\alpha =1}^N}{c}_{4\alpha }\mathbf{q}·{\mathbf{J}}_{\alpha }.\hfill \end{array}$$

(35)

4.5.3. Diffusion Conjugate Forces

$$\begin{array}{cc}\hfill {\mathbf{X}}_{\alpha }& =-a_{\alpha }{\mathbf{J}}_{\alpha }-b_{\alpha }|{\mathbf{J}}_{\alpha }|{\mathbf{J}}_{\alpha }-2c_{2\alpha }\mathrm{\Pi }{\mathbf{J}}_{\alpha }-c_{3\alpha }{\left|\mathbf{q}\right|}^2{\displaystyle \frac{{\mathbf{J}}_{\alpha }}{|{\mathbf{J}}_{\alpha }|}}-c_{4\alpha }\mathrm{\Pi }\mathbf{q}.\hfill \end{array}$$

(36)

4.6. Nonlinear Constitutive Structure

The entropy production inequality

$$\sigma =\mathbf{q}·{\mathbf{X}}_q+\mathrm{\Pi }{X}_{\mathrm{\Pi }}+{\displaystyle \sum _{\alpha =1}^N}{\mathbf{J}}_{\alpha }·{\mathbf{X}}_{\alpha }+{\displaystyle \sum _{r=1}^R}{\mathcal{R}}_r{\mathcal{A}}_r\ge 0$$

(37)

implies that the constitutive relations must be of Onsager type in the space of thermodynamic forces:

$$\left[\begin{array}{c}\mathbf{q}\\ \mathrm{\Pi }\\ {\mathbf{J}}_{\alpha }\end{array}\right]=\mathbb{L}\left[\begin{array}{c}{\mathbf{X}}_q\\ X_{\mathrm{\Pi }}\\ {\mathbf{X}}_{\alpha }\end{array}\right]+\mathcal{O}\left(3\right),$$

(38)

where $\mathbb{L}$ is a positive semi-definite matrix of transport coefficients, and $\mathcal{O}\left(3\right)$ denotes cubic corrections generated by the third-order entropy structure.

4.7. Explicit Third-Order Closure Form

To leading nonlinear order, inversion of the force–flux relations yields:

$$\begin{array}{cc}\hfill \mathbf{q}& =-\lambda \nabla \left({\displaystyle \frac{1}{T}}\right)-{\lambda }_2{\left|\mathbf{q}\right|}^2\nabla \left({\displaystyle \frac{1}{T}}\right)-{\lambda }_3\mathrm{\Pi }\nabla \left({\displaystyle \frac{1}{T}}\right)-{\lambda }_4\sum _{\alpha }{\mathbf{J}}_{\alpha }\nabla \left({\displaystyle \frac{{\mu }_{\alpha }}{T}}\right),\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill \mathrm{\Pi }& =-\zeta \nabla ·\mathbf{v}-{\zeta }_2{\mathrm{\Pi }}^2-{\zeta }_3{\left|\mathbf{q}\right|}^2-{\zeta }_4\sum _{\alpha }{\left|{\mathbf{J}}_{\alpha }\right|}^2,\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill {\mathbf{J}}_{\alpha }& =-\sum _{\beta }{D}_{\alpha \beta }\nabla \left({\displaystyle \frac{{\mu }_{\beta }}{T}}\right)-{\chi }_{\alpha }\mathbf{q}\mathrm{\Pi }+\mathcal{O}\left(3\right).\hfill \end{array}$$

(41)

4.8. Physical Interpretation

The resulting constitutive structure exhibits the following features:

• nonlinear thermal conductivity depending on both heat flux magnitude and volumetric relaxation,
• dynamic pressure enhanced by thermal and diffusive nonequilibrium,
• cross-effects between diffusion, heat conduction, and chemical reactions,
• recovery of Navier–Stokes–Fourier and first-order RET in the limit of weak nonequilibrium.

This completes the derivation of the third-order entropy structure. In the following section, we use these results to construct the evolution equations for $\mathbf{q}$ and $\mathrm{\Pi }$ in Maxwell–Cattaneo form and obtain explicit expressions for third-order thermal conductivity and bulk viscosity.

5. Maxwell–Cattaneo Evolution Equations and Third-Order Transport Coefficients

In this section we derive the evolution equations for the heat flux $\mathbf{q}$ and the dynamic (bulk) pressure $\mathrm{\Pi }$ in Maxwell–Cattaneo form. These equations are obtained by combining the entropy principle with the third-order constitutive structure established in the previous section. The resulting framework yields explicit expressions for the third-order corrections to the thermal conductivity and bulk viscosity in chemically reacting gas mixtures.

5.1. Thermodynamic Relaxation Structure

Within Rational Extended Thermodynamics, dissipative fluxes are endowed with relaxation dynamics rather than instantaneous constitutive laws. Accordingly, we postulate linear relaxation equations in the space of thermodynamic forces:

$$\begin{array}{cc}\hfill {\tau }_q\dot{\mathbf{q}}+\mathbf{q}& ={\mathbf{X}}_q^{\mathrm{eff}},\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill {\tau }_{\mathrm{\Pi }}\dot{\mathrm{\Pi }}+\mathrm{\Pi }& =X_{\mathrm{\Pi }}^{\mathrm{eff}},\hfill \end{array}$$

(43)

where ${\tau }_q$ and ${\tau }_{\mathrm{\Pi }}$ are characteristic relaxation times, and ${\mathbf{X}}_q^{\mathrm{eff}}$, $X_{\mathrm{\Pi }}^{\mathrm{eff}}$ are effective thermodynamic forces incorporating nonlinear third-order corrections.

5.2. Effective Thermodynamic Forces

Using the third-order entropy expansion, the effective forces take the form

$$\begin{array}{cc}\hfill {\mathbf{X}}_q^{\mathrm{eff}}& =-\lambda \nabla \left({\displaystyle \frac{1}{T}}\right)-{\lambda }_2{\left|\mathbf{q}\right|}^2\nabla \left({\displaystyle \frac{1}{T}}\right)-{\lambda }_3\mathrm{\Pi }\nabla \left({\displaystyle \frac{1}{T}}\right)-{\lambda }_4\sum _{\alpha }{\mathbf{J}}_{\alpha }\nabla \left({\displaystyle \frac{{\mu }_{\alpha }}{T}}\right),\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill X_{\mathrm{\Pi }}^{\mathrm{eff}}& =-\zeta \nabla ·\mathbf{v}-{\zeta }_2{\mathrm{\Pi }}^2-{\zeta }_3{\left|\mathbf{q}\right|}^2-{\zeta }_4\sum _{\alpha }{\left|{\mathbf{J}}_{\alpha }\right|}^2-{\zeta }_5\mathcal{A},\hfill \end{array}$$

(45)

where $\mathcal{A}$ denotes the total chemical affinity contribution from all reactions.

5.3. Maxwell–Cattaneo Form of Heat Flux

The heat flux evolution equation becomes

$${\tau }_q\dot{\mathbf{q}}+\mathbf{q}=-\lambda \nabla T-{\lambda }_2{\left|\mathbf{q}\right|}^2\nabla T-{\lambda }_3\mathrm{\Pi }\nabla T-{\lambda }_4\sum _{\alpha }{\mathbf{J}}_{\alpha }\nabla {\mu }_{\alpha }.$$

(46)

Rewriting in standard Maxwell–Cattaneo form:

$${\tau }_q\dot{\mathbf{q}}+\mathbf{q}=-{\lambda }_{\mathrm{eff}}·\nabla T$$

(47)

where the effective third-order thermal conductivity tensor is

$${\lambda }_{\mathrm{eff}}=\lambda \mathbf{I}+{\lambda }_2{\left|\mathbf{q}\right|}^2\mathbf{I}+{\lambda }_3\mathrm{\Pi }\mathbf{I}+{\lambda }_4\sum _{\alpha }{\mathbf{J}}_{\alpha }\otimes {\displaystyle \frac{\nabla {\mu }_{\alpha }}{\nabla T}}.$$

(48)

5.4. Third-Order Thermal Conductivity

Identifying the constitutive Fourier-like structure $\mathbf{q}=-{\lambda }_{\mathrm{eff}}\nabla T$, we obtain the scalar effective thermal conductivity:

$${\lambda }_{\mathrm{eff}}=\lambda \left(1+{\displaystyle \frac{{\lambda }_2}{\lambda }}{\left|\mathbf{q}\right|}^2+{\displaystyle \frac{{\lambda }_3}{\lambda }}\mathrm{\Pi }\right)+{\lambda }_4{\mathcal{C}}_{\mathrm{chem}}$$

(49)

where the chemical coupling correction is

$${\mathcal{C}}_{\mathrm{chem}}=\sum _{\alpha }{\mathbf{J}}_{\alpha }·\nabla {\mu }_{\alpha }/|\nabla T|.$$

(50)

This expression shows that thermal conductivity is no longer constant but depends nonlinearly on:

• the magnitude of the heat flux,
• the dynamic pressure (volumetric nonequilibrium),
• chemical diffusion and reaction activity.

5.5. Maxwell–Cattaneo Form of Dynamic Pressure

Similarly, the bulk pressure satisfies

$${\tau }_{\mathrm{\Pi }}\dot{\mathrm{\Pi }}+\mathrm{\Pi }=-\zeta \nabla ·\mathbf{v}-{\zeta }_2{\mathrm{\Pi }}^2-{\zeta }_3{\left|\mathbf{q}\right|}^2-{\zeta }_4\sum _{\alpha }{\left|{\mathbf{J}}_{\alpha }\right|}^2-{\zeta }_5\mathcal{A}.$$

(51)

This can be written compactly as

$${\tau }_{\mathrm{\Pi }}\dot{\mathrm{\Pi }}+\mathrm{\Pi }=-{\zeta }_{\mathrm{eff}}\nabla ·\mathbf{v}$$

(52)

where the effective bulk viscosity is

$${\zeta }_{\mathrm{eff}}=\zeta \left(1+{\displaystyle \frac{{\zeta }_2}{\zeta }}{\mathrm{\Pi }}^2+{\displaystyle \frac{{\zeta }_3}{\zeta }}{\left|\mathbf{q}\right|}^2+{\displaystyle \frac{{\zeta }_4}{\zeta }}\sum _{\alpha }{\left|{\mathbf{J}}_{\alpha }\right|}^2\right)+{\zeta }_5{\displaystyle \frac{\mathcal{A}}{\nabla ·\mathbf{v}}}.$$

(53)

5.6. Coupling with Chemical Reactions

The presence of the chemical affinity $\mathcal{A}$ introduces a direct coupling between reaction kinetics and volumetric dissipation. This implies that:

• exothermic reactions enhance bulk viscosity relaxation,
• endothermic reactions can reduce effective damping,
• chemical nonequilibrium modifies acoustic attenuation.

5.7. Recovery of Classical Limits

In the near-equilibrium limit

$$\left|\mathbf{q}\right|\to 0,\mathrm{\Pi }\to 0,{\mathbf{J}}_{\alpha }\to 0,$$

(54)

the constitutive laws reduce to

$$\begin{array}{cc}\hfill \mathbf{q}& =-\lambda \nabla T,\hfill \end{array}$$

(55)

$$\begin{array}{cc}\hfill \mathrm{\Pi }& =-\zeta \nabla ·\mathbf{v},\hfill \end{array}$$

(56)

recovering the classical Navier–Stokes–Fourier system. These results complete the derivation of the Maxwell–Cattaneo evolution equations and provide explicit expressions for third-order thermal conductivity and bulk viscosity in chemically reacting gas mixtures. In the next section, we analyze the stability and hyperbolicity properties of the resulting system.

6. Stability and Hyperbolicity of the Third-Order System

A fundamental requirement for any Extended Thermodynamics model is that the resulting field equations be hyperbolic and admit a Lyapunov-type entropy structure, ensuring finite propagation speeds and linear stability of equilibrium states. In this section we analyze these properties for the third-order Maxwell–Cattaneo system derived for chemically reacting gas mixtures.

6.1. Quasilinear Form of the Governing System

Collecting the balance laws and the evolution equations for the dissipative variables, the full system can be written in quasilinear form as

$${\partial }_t\mathbf{U}+{\displaystyle \sum _{k=1}^3}{\mathbf{A}}^k\left(\mathbf{U}\right){\partial }_k\mathbf{U}=\mathbf{S}\left(\mathbf{U}\right),$$

(57)

where the state vector is

$$\mathbf{U}={\left(\rho ,{\rho }_{\alpha },\rho \mathbf{v},e,\mathbf{q},\mathrm{\Pi },{\mathbf{J}}_{\alpha }\right)}^{\top }.$$

(58)

The matrices ${\mathbf{A}}^k\left(\mathbf{U}\right)$ represent the convective flux Jacobians, while $\mathbf{S}\left(\mathbf{U}\right)$ contains relaxation and chemical source terms.

6.2. Definition of Hyperbolicity

The system is said to be (strongly) hyperbolic if, for every unit vector $\mathbf{n}$, the matrix

$$\mathbf{A}\left(\mathbf{n}\right)=n_k{\mathbf{A}}^k\left(\mathbf{U}\right)$$

(59)

has real eigenvalues and a complete set of eigenvectors. This property guarantees well-posedness of the initial value problem.

6.3. Entropy Convexity and Symmetrization

A key result of Rational Extended Thermodynamics is that the entropy function acts as a symmetrizing potential. Defining the entropy variables

$$\mathbf{V}={\displaystyle \frac{\partial \left(\rho \eta \right)}{\partial \mathbf{U}}},$$

(60)

the system can be rewritten in symmetric form

$$\mathbf{H}\left(\mathbf{U}\right){\partial }_t\mathbf{U}+{\displaystyle \sum _{k=1}^3}{\mathbf{K}}^k\left(\mathbf{U}\right){\partial }_k\mathbf{U}=\mathbf{S}\left(\mathbf{U}\right),$$

(61)

where

$$\mathbf{H}={\displaystyle \frac{\partial \mathbf{V}}{\partial \mathbf{U}}}$$

(62)

is the Hessian of the entropy density. Since the third-order entropy constructed earlier satisfies strict concavity in the dissipative variables,

$$\mathbf{H}>0,$$

(63)

the system is symmetrizable and therefore hyperbolic in a neighborhood of equilibrium.

6.4. Linearization Around Equilibrium

Let the equilibrium state be defined by

$$\mathbf{q}=0,\mathrm{\Pi }=0,{\mathbf{J}}_{\alpha }=0,\mathbf{v}=0.$$

(64)

Linearizing the system yields

$$\begin{array}{cc}\hfill {\tau }_q{\partial }_t\mathbf{q}+\mathbf{q}& =-\lambda \nabla T,\hfill \end{array}$$

(65)

$$\begin{array}{cc}\hfill {\tau }_{\mathrm{\Pi }}{\partial }_t\mathrm{\Pi }+\mathrm{\Pi }& =-\zeta \nabla ·\mathbf{v},\hfill \end{array}$$

(66)

$$\begin{array}{cc}\hfill {\tau }_{\alpha }{\partial }_t{\mathbf{J}}_{\alpha }+{\mathbf{J}}_{\alpha }& =-D_{\alpha \beta }\nabla {\mu }_{\beta }.\hfill \end{array}$$

(67)

Combining these with the conservation laws leads to a hyperbolic system of damped wave equations.

6.5. Characteristic Speeds

For the thermomechanical subsystem, the dispersion relation yields characteristic speeds

$$c_{\pm }^2=c_s^2+{\displaystyle \frac{\lambda }{\rho c_p{\tau }_q}},$$

(68)

where $c_s$ is the adiabatic sound speed and $c_p$ is the specific heat at constant pressure. Similarly, for the bulk relaxation mode,

$$c_{\mathrm{\Pi }}^2={\displaystyle \frac{\zeta }{\rho {\tau }_{\mathrm{\Pi }}}}.$$

(69)

All characteristic speeds are real provided

$$\lambda >0,\zeta >0,{\tau }_q>0,{\tau }_{\mathrm{\Pi }}>0,$$

(70)

which are consistent with the second law of thermodynamics.

6.6. Stability of Equilibrium

Define the total entropy functional

$$\mathcal{S}\left(t\right)={\int }_{\mathrm{\Omega }}\rho \eta d\mathbf{x}.$$

(71)

Using the entropy balance equation,

$${\displaystyle \frac{d\mathcal{S}}{dt}}={\int }_{\mathrm{\Omega }}\sigma d\mathbf{x}\ge 0.$$

(72)

Since the entropy is strictly concave in $(\mathbf{q},\mathrm{\Pi },{\mathbf{J}}_{\alpha })$, equilibrium is a local maximizer of entropy. Therefore:

• equilibrium states are Lyapunov stable,
• perturbations decay under relaxation,
• the system admits entropy dissipation toward equilibrium.

6.7. Effect of Third-Order Nonlinearities

The third-order corrections modify the system in two essential ways:

1.

Renormalization of wave speeds: nonlinear dependence of transport coefficients introduces amplitude-dependent characteristic velocities.

2.

Nonlinear damping: relaxation rates acquire corrections proportional to ${\left|\mathbf{q}\right|}^2$, ${\mathrm{\Pi }}^2$, and chemical affinity $\mathcal{A}$, enhancing or reducing dissipation depending on reaction energetics.

Importantly, because these corrections originate from a convex entropy potential, they do not destroy hyperbolicity in a neighborhood of equilibrium.

6.8. Conclusion

The third-order extended thermodynamic system for chemically reacting gas mixtures is:

• strongly hyperbolic near equilibrium,
• symmetrizable via entropy variables,
• linearly stable due to entropy convexity,
• dissipative through positive entropy production,
• consistent with finite propagation speeds for all thermodynamic modes.

These properties confirm that the proposed third-order Maxwell–Cattaneo framework is mathematically well-posed and physically admissible within Rational Extended Thermodynamics.

In the next section, we will discuss specific applications to high-temperature reacting flows and shock-induced chemical nonequilibrium.

7. Applications: High-Temperature Reacting Flows and Shock-Induced Chemical Nonequilibrium

The third-order extended thermodynamic model developed in this work provides a unified framework for describing strongly nonequilibrium phenomena in chemically reacting gas mixtures. In this section, we discuss two important application regimes: high-temperature reacting flows and shock-induced chemical nonequilibrium.

7.1. High-Temperature Reacting Flows

High-temperature gas flows, such as those encountered in hypersonic flight, combustion systems, and plasma-assisted propulsion, are characterized by strong coupling between thermal transport, molecular excitation, and chemical reactions. In such regimes, deviations from local equilibrium become significant, and classical Navier–Stokes–Fourier theory is no longer sufficient.

Within the present third-order framework, the heat flux and dynamic pressure obey Maxwell–Cattaneo-type evolution equations with nonlinear corrections:

$$\begin{array}{cc}\hfill {\tau }_q\dot{\mathbf{q}}+\mathbf{q}& =-\lambda \nabla T-{\lambda }_2{\left|\mathbf{q}\right|}^2\nabla T-{\lambda }_3\mathrm{\Pi }\nabla T-{\lambda }_4\sum _{\alpha }{\mathbf{J}}_{\alpha }\nabla {\mu }_{\alpha },\hfill \end{array}$$

(73)

$$\begin{array}{cc}\hfill {\tau }_{\mathrm{\Pi }}\dot{\mathrm{\Pi }}+\mathrm{\Pi }& =-\zeta \nabla ·\mathbf{v}-{\zeta }_2{\mathrm{\Pi }}^2-{\zeta }_3{\left|\mathbf{q}\right|}^2-{\zeta }_4\sum _{\alpha }{\left|{\mathbf{J}}_{\alpha }\right|}^2-{\zeta }_5\mathcal{A}.\hfill \end{array}$$

(74)

These relations imply that the effective transport properties become state-dependent. In particular, the effective thermal conductivity

$${\lambda }_{\mathrm{eff}}=\lambda \left(1+{\alpha }_1{\left|\mathbf{q}\right|}^2+{\alpha }_2\mathrm{\Pi }\right)$$

(75)

increases in strongly heated regions, while chemical activity contributes additional enhancement or suppression depending on the sign of the reaction affinity $\mathcal{A}$.

In combustion and plasma environments, this leads to non-Fourier heat transport, where thermal waves propagate with finite speed and exhibit nonlinear attenuation. The coupling between diffusion and heat flux further produces cross-effects such as thermal-diffusion-driven species segregation and composition-dependent conductivity.

7.2. Shock-Induced Chemical Nonequilibrium

In high-Mach-number flows, shock waves generate abrupt changes in thermodynamic variables over very short spatial scales. Behind the shock front, translational, rotational, vibrational, and chemical modes relax on different time scales, producing a highly nonequilibrium relaxation zone. In the present theory, the shock structure is governed by the coupled system

$$\begin{array}{cc}\hfill {\tau }_q\dot{\mathbf{q}}+\mathbf{q}& =-\lambda \nabla T+\mathcal{O}\left(3\right),\hfill \end{array}$$

(76)

$$\begin{array}{cc}\hfill {\tau }_{\mathrm{\Pi }}\dot{\mathrm{\Pi }}+\mathrm{\Pi }& =-\zeta \nabla ·\mathbf{v}+\mathcal{O}\left(3\right),\hfill \end{array}$$

(77)

$$\begin{array}{cc}\hfill {\partial }_t{\rho }_{\alpha }+\nabla ·\left({\rho }_{\alpha }\mathbf{v}\right)& ={\omega }_{\alpha },\hfill \end{array}$$

(78)

where the reaction rates ${\omega }_{\alpha }$ depend on the local chemical affinity and temperature. The finite relaxation times ${\tau }_q$ and ${\tau }_{\mathrm{\Pi }}$ introduce a non-equilibrium shock thickness, in contrast to the discontinuous shock predicted by classical Euler or Navier–Stokes theory. Moreover, third-order corrections lead to amplitude-dependent shock structure, where the thickness and relaxation profile depend on the magnitude of $\mathbf{q}$, $\mathrm{\Pi }$, and species gradients.

The dynamic pressure plays a particularly important role in shock attenuation. The term proportional to the chemical affinity,

$$\mathrm{\Pi }\sim -{\zeta }_5\mathcal{A},$$

(79)

implies that exothermic reactions can either enhance or weaken compressibility effects depending on whether energy release counteracts volumetric dissipation.

7.3. Relaxation Zone Structure

The post-shock relaxation region is characterized by competing time scales:

• ${\tau }_q$: thermal relaxation time,
• ${\tau }_{\mathrm{\Pi }}$: mechanical (bulk) relaxation time,
• ${\tau }_{\alpha }$: chemical reaction times,
• ${\tau }_{\mathrm{diff}}$: species diffusion time.

When these time scales are comparable, strong coupling effects arise, leading to:

• delayed thermal equilibration behind the shock,
• overshoots in temperature and pressure profiles,
• non-monotonic relaxation of species concentrations,
• enhanced entropy production localized near the shock front.

7.4. Physical Implications

The third-order structure predicts several experimentally relevant phenomena:

1.

Non-Fourier heat conduction: thermal disturbances propagate as damped waves rather than diffusive modes.

2.

Shock-induced chemical delay: reaction zones are spatially separated from mechanical discontinuities.

3.

Nonlinear acoustic attenuation: sound speed and damping depend on wave amplitude due to ${\mathrm{\Pi }}^2$ and ${\left|\mathbf{q}\right|}^2$ corrections.

4.

Thermo-chemical coupling: reaction rates are modified by nonequilibrium fluxes, altering ignition and extinction thresholds.

7.5. Summary

The third-order extended thermodynamic model provides a consistent macroscopic description of high-temperature reacting flows and shock-induced nonequilibrium. By incorporating nonlinear corrections to both thermal and mechanical relaxation, the theory extends classical gas dynamics and first-order extended thermodynamics, enabling the description of regimes where transport, chemistry, and compressibility are strongly coupled.

This completes the discussion of applications. The final section will summarize the main results and outline potential directions for further extensions, including relativistic and multi-scale generalizations.

8. Conclusions and Outlook

In this work, we have developed a third-order extension of Rational Extended Thermodynamics (RET) for chemically reacting gas mixtures, starting from the classical balance laws and systematically incorporating nonlinear nonequilibrium effects through an entropy-based closure procedure.

8.1. Summary of Main Results

The main contributions of this study can be summarized as follows:

• A consistent macroscopic framework for chemically reacting gas mixtures has been formulated in which the primary nonequilibrium variables include the heat flux $\mathbf{q}$, dynamic pressure $\mathrm{\Pi }$, and species diffusion fluxes ${\mathbf{J}}_{\alpha }$.
• The entropy principle has been extended to third order in the dissipative variables, yielding a convex entropy density that incorporates nonlinear self-interactions and cross-coupling terms between thermal, mechanical, and chemical nonequilibrium processes.
• From the entropy production inequality, we derived generalized constitutive relations in which thermodynamic fluxes are driven by nonlinear thermodynamic forces involving temperature gradients, velocity divergence, chemical potential gradients, and reaction affinity.
• Maxwell–Cattaneo-type evolution equations were obtained for both the heat flux and dynamic pressure. These equations include third-order corrections that renormalize the effective thermal conductivity and bulk viscosity, introducing dependence on ${\left|\mathbf{q}\right|}^2$, ${\mathrm{\Pi }}^2$, diffusion fluxes, and chemical affinity.
• The resulting system was shown to be symmetrizable and strongly hyperbolic in a neighborhood of equilibrium, ensuring well-posedness and finite propagation speeds for thermodynamic disturbances.
• Applications to high-temperature reacting flows and shock-induced chemical nonequilibrium demonstrated that the theory naturally captures nonlinear transport phenomena, including amplitude-dependent dissipation, delayed relaxation, and coupling between chemical reactions and compressible dynamics.

8.2. Physical Interpretation

The central physical implication of the present formulation is that transport coefficients such as thermal conductivity and bulk viscosity are no longer constants, but dynamic quantities depending on the local nonequilibrium state of the system. In particular, the interplay between heat flux, volumetric expansion, and chemical affinity leads to feedback mechanisms that modify both transport rates and reaction kinetics.

This provides a unified macroscopic description of regimes where classical theories fail, especially in high-temperature and high-speed flows where multiple relaxation mechanisms interact on comparable time scales.

8.3. Relation to Existing Theories

In the weakly nonequilibrium limit, the present theory reduces to:

• the Navier–Stokes–Fourier system for non-reacting mixtures,
• the first-order reacting mixture theory of Kremer and Müller [6],
• classical linear irreversible thermodynamics [1].

Furthermore, the structure is consistent with the general framework of Extended Irreversible Thermodynamics [3] and retains compatibility with the symmetrization properties of Rational Extended Thermodynamics [2].

8.4. Future Directions

Several important extensions of the present work are natural and merit further investigation.

8.4.1. Relativistic Generalization

A promising direction is the extension of the third-order framework to relativistic chemically reacting fluids. In such a formulation, the heat flux and diffusion fluxes must be embedded into four-vector or tensorial structures, and the entropy principle must be formulated in a Lorentz-covariant manner. This would allow applications to high-energy astrophysical plasmas, relativistic shock waves, and quark–gluon plasma dynamics.

8.4.2. Multi-Scale Kinetic Derivation

Another important direction is the derivation of the third-order constitutive structure directly from kinetic theory. Starting from the Boltzmann equation with reactive collision operators, one may employ Grad-type moment expansions or Chapman–Enskog procedures to systematically derive the coefficients appearing in the entropy expansion. This would provide a microscopic justification for the phenomenologically introduced nonlinear transport terms in this work.

8.4.3. Nonlinear Wave and Shock Dynamics

The influence of third-order corrections on nonlinear wave propagation, shock stability, and detonation structure remains largely unexplored. In particular, amplitude-dependent wave speeds and nonlinear attenuation mechanisms may play a significant role in reactive shock tubes and combustion instabilities.

8.4.4. Numerical Implementation

Finally, robust numerical schemes for the resulting hyperbolic relaxation system are essential. Structure-preserving finite-volume or discontinuous Galerkin methods that respect the entropy inequality would be particularly well-suited for simulating nonequilibrium reacting flows.

8.5. Concluding Remark

The third-order extended thermodynamic formulation presented here provides a systematic macroscopic framework for describing chemically reacting gas mixtures far from equilibrium. By incorporating nonlinear entropy corrections and maintaining hyperbolicity, the theory bridges the gap between classical continuum mechanics and kinetic theory, offering a versatile foundation for future theoretical, computational, and experimental investigations.