Non-Isothermal Maxwell–Stefan System

• Non-Isothermal Maxwell–Stefan system is a thermodynamically consistent framework modeling coupled mass and energy transport in multicomponent mixtures using nonlinear cross-diffusion laws.
• It is derived from the Boltzmann equations through diffusive scaling, leading to conservation laws that capture species interactions, local equilibrium, and entropy production.
• The system underpins applications in reactive flows and industrial processes, and its analytical structure supports robust numerical schemes for simulating coupled heat and mass transfer.

The Non-Isothermal Maxwell–Stefan System extends the classical Maxwell–Stefan formulation for multicomponent diffusion to account for thermal effects, specifically the spatial and temporal evolution of temperature coupled with species transport. Derived from kinetic theory, notably with rigorous hydrodynamic limits from multicomponent Boltzmann models, it serves as a macroscopic, thermodynamically-consistent description of mass and energy transport in gas mixtures, including reactive and non-reactive scenarios. The system is characterized by a set of conservation laws for species and energy, coupled through nonlinear cross-diffusion and temperature-dependent closure relations, and it admits precise mathematical structure allowing for the establishment of existence, uniqueness, and entropy properties.

1. Mathematical and Physical Foundations

The starting point for the non-isothermal Maxwell–Stefan system is a system of multicomponent (n-species) Boltzmann equations, with elastic collision operators for the particle interactions. Under the assumption that solutions remain close to local Maxwellian distributions,

$$f_i^{\varepsilon}(t,x,v) = c_i^{\varepsilon}(t,x) \left(\frac{m_i}{2\pi k T^{\varepsilon}(t,x)}\right)^{3/2} \exp\left\{ -\frac{m_i |v - \varepsilon u_i^{\varepsilon}(t,x)|^2}{2k T^{\varepsilon}(t,x)} \right\}$$

with $c_i^{\varepsilon}$ the local density, $u_i^{\varepsilon}$ the macroscopic velocity (of order $O(\varepsilon)$ — ensuring diffusion is the dominant process), and $T^{\varepsilon}$ the local temperature, one performs a diffusive (parabolic) scaling. The collision kernels are assumed to be Maxwellian (depending only on the deviation angle). The key upshot is that, in the vanishing mean free path limit ($\varepsilon\to0$), all species share a common local temperature, capturing thermodynamic equilibrium in the energetic sense (Hutridurga et al., 2017, Chen et al., 5 Aug 2025).

2. Derivation and Structure of the Non-Isothermal System

Integration over velocity variables yields conservation equations for mass, momentum, and energy:

• Mass Balance for Each Species:

$$\partial_t c_i + \nabla_x \cdot J_i = 0, \quad i=1,\ldots, n,$$

with $J_i$ the macroscopic flux.

• Flux–Gradient (Maxwell–Stefan) Relations:

$$T \nabla_x c_i + c_i \nabla_x T = -\sum_{j \ne i} D_{ij} (c_j J_i - c_i J_j),$$

where $D_{ij} = \dfrac{27\,\|b_{ij}\|_{L^1}}{m_i+m_j}$ (for Maxwellian particles and elastic collisions).

• Energy Equation:

$$\partial_t (C_{\mathrm{tot}} T) + \frac{5}{3} \nabla_x \cdot \left(T \sum_{i=1}^n J_i\right) = 0,$$

with $C_{\mathrm{tot}} = \sum_{i=1}^n c_i$.

• Closure Relation:

In contrast to the isothermal case ($\sum_i J_i=0$), a non-isothermal setting admits

$\sum_{i=1}^n J_i = - a\,\nabla_x C_{\mathrm{tot}},$

where $a>0$ is a proportionality constant linked to the total molar flux. This “decoupling” yields an advection equation for $T$, while $C_{\mathrm{tot}}$ follows a parabolic (Fickian) law (Hutridurga et al., 2017).

A summary of formal structure is given in the following table:

Equation Type	Non-Isothermal Maxwell–Stefan System
Mass balance	$\partial_t c_i + \nabla_x\cdot J_i = 0$
Flux–gradient law	$$T\nabla_x c_i + c_i\nabla_x T = -\sum_{j\ne i} D_{ij}(c_j J_i - c_i J_j)$$
Energy conservation	$$\partial_t (C_{\mathrm{tot}}T) + \frac{5}{3} \nabla_x\cdot (T\sum_{i} J_i) = 0$$
Closure relation	$\sum_{i=1}^n J_i = -a \nabla_x C_{\mathrm{tot}}$

3. Analytical Properties: Well-posedness and Reduction

A rigorous local-in-time existence and uniqueness theory is established by recasting the system as a quasi-linear parabolic system for a reduced set of concentration variables, leveraging:

• Reduction via Spectral Analysis: The original flux–gradient relations have a redundancy due to the closure constraint. Basis transformations (e.g., via Perron–Frobenius theory) yield a reduced $(n-1)$-dimensional system with a modified, invertible cross-diffusion matrix (Hutridurga et al., 2017).
• Normal Ellipticity and Maximal $L^p$-Regularity: The principal coefficient matrix $TB$ is shown to be strictly positive definite, ensuring the system meets the normal ellipticity condition; maximal $L^p$-regularity then yields local strong solutions in standard Sobolev spaces.
• Decoupling of $C_{\mathrm{tot}}$ and $T$: The evolution of total concentration and temperature are shown to decouple: $C_{\mathrm{tot}}$ solves a linear diffusion equation, and $T$ satisfies an advection equation with the transport field determined by gradients of $\log C_{\mathrm{tot}}$.

4. Kinetic Consistency, Entropy, and Thermodynamics

The system is not merely phenomenological but is tightly constrained by thermodynamics:

• Microscopic Consistency: The diffusion coefficients $D_{ij}$ are derived explicitly from Boltzmann collision integrals. Entropy production (second-law compliance) is built-in via these coefficients and the structure of the flux–gradient equations (Anwasia et al., 2021, Georgiadis et al., 2023).
• Entropy Structure:

The entropy density for the system can be expressed (in the Maxwell–Stefan formulation) as

$H(\{c_i\}) = \sum_{i=1}^{n} c_i ( \log c_i - 1).$

The system admits a rigorous entropy balance (with no anomalous dissipation): all entropy dissipation arises directly from the modeled frictional terms between species, with no hidden or “anomalous” loss (Berselli et al., 14 Jul 2024).

• Weak–Strong and Renormalized Uniqueness: Using relative entropy methods, uniqueness of weak solutions is established as long as a strong solution exists with the same initial data, even in the presence of non-isothermal coupling (Georgiadis et al., 2023, Georgiadis et al., 2023).
• Methodological Advances: The framework employs the Bott–Duffin inverse to handle the singularity inherent in the Maxwell–Stefan (cross-diffusion) matrix, thus permitting the derivation of closed parabolic systems on the physically relevant subspace (Huo et al., 2021, Georgiadis et al., 2023).

5. Extensions: Reactions, Porous Media, Multiscale Thermodynamics

The model admits several notable generalizations:

• Reactive Mixtures: For polyatomic, chemically reacting systems, production terms (reflecting reaction rates) explicitly couple to the species and energy balances. The corresponding Maxwell–Stefan diffusion coefficients and chemical source terms are derived from detailed kinetic models, incorporating internal energy modes (Anwasia et al., 2019).
• Porous Media and Compressible Flows: In the context of compressible multicomponent flows (e.g., in porous media), the dissipative Maxwell–Stefan operator appears in hyperbolic balance laws, ensuring global classical solutions near equilibrium and characterizing the large time, high-diffusion limit as a parabolic (diffusive) system (Ostrowski et al., 2019).
• GENERIC and Multiscale Extensions: The non-isothermal framework is enriched by multiscale methods where individual constituents may possess distinct temperatures before equilibration; reduction via the Maximum Entropy principle recovers the classical non-isothermal Maxwell–Stefan or dusty gas equations, with temperature differences contributing directly to Soret (thermodiffusion) coefficients (Vágner et al., 2022).

6. Numerical Methods and Practical Implementation

The development of robust, structure-preserving numerical methods is critical for simulation:

• Finite element frameworks have been developed around augmented variational formulations that enforce mass-average constraints and maintain symmetry in the presence of non-idealities and eventual anisothermal (non-isothermal) extensions. These systems are solved using Picard linearization, and appropriate test spaces ensure inf-sup stability and preservation of the underlying physical structure (Aznaran et al., 2022).
• Asymptotic and sharp-interface analysis connects diffuse interface models for phase transitions with non-isothermal Maxwell–Stefan systems, supporting simulation of moving interfaces and latent heat effects (Liu et al., 19 Aug 2024).
• Decoupling between Fickian mass diffusion (parabolic PDE) and advective temperature evolution is a feature that aids both in analytical estimates and in the design of operator-splitting numerical schemes (Hutridurga et al., 2017, Hutridurga et al., 2017).

7. Applications and Significance

Non-isothermal Maxwell–Stefan systems are foundational in the description of:

• Multicomponent gas separation, reactive flows, and combustion processes where heat and mass transfer are coupled.
• Industrial processes such as distillation, absorption, drying, evaporation, and condensation, where spatially varying temperature and cross-diffusion are both prevalent and pivotal for process efficiency.
• Physically rigorous and computationally tractable upscaling from kinetic (Boltzmann-level) models to macroscopic transport phenomena, enabling integration with experimental and engineering design data.

The theoretical completeness of the model, extending from kinetic theory through rigorous existence and entropy arguments to numerical implementation frameworks, makes it central to the modern theory of multicomponent, reactive, and thermally complex transport phenomena.