Thermodynamic Diffusion Models

• The paper establishes a thermodynamic framework that employs Gibbs-type relations to enforce nonnegative entropy production in diffusion processes.
• It integrates kinetic theory and continuum descriptions, introducing higher-order mass and volume diffusion terms that scale with Kn² for capturing nonlocal transport effects.
• The paper validates its model by demonstrating mechanical compatibility through conservation laws, Onsager reciprocity, and consistency with classical fluid dynamics.

A thermodynamic perspective on diffusion models frames the description and analysis of diffusive processes in terms of energy, entropy, non-equilibrium structures, and the fundamental constraints of thermodynamics. This approach extends beyond classical Fickian frameworks, providing a rigorous foundation that incorporates additional molecular, stochastic, or non-local effects. In kinetic theory, statistical mechanics, and irreversible thermodynamics, the thermodynamic structure of diffusion underpins the conservation laws, drives relaxation to equilibrium, and governs the emergence of anomalous or higher-order transport phenomena. Recent research elucidates how formulations in terms of Gibbs-type relations, entropy production, and generalized fluxes produce models that naturally satisfy the second law, Onsager reciprocity, and mechanical conservation, and account for diffusive effects inaccessible to conventional continuum theories.

1. Thermodynamic Closure: Gibbs-Type Equations and the Second Law

A core principle in thermodynamically consistent diffusion models is the construction of Gibbs-type relations that generate entropy production terms consistent with the second law. In the volume/mass diffusion framework, the entropy per unit mass $s$ is defined via a local generalized Gibbs equation,

$$An\,\frac{Ds}{Dt} = An\,\frac{1}{T'}\left(De^{in} - p' \, D(p^{-1}) + \ldots \right)$$

where the normalization $An$ is related to mass or probability, $T'$ is the effective temperature, and $e^{in}$ and $p'$ are internal energy and kinetic-theoretic pressure, respectively. This generalized Gibbs structure ensures entropy production remains nonnegative after appropriate modeling of dissipative fluxes, such as

$$An\, T' \frac{Ds}{Dt} = \nabla \cdot \left( \text{flux terms} \right)$$

where the fluxes for volume/mass, heat, and momentum may be written using linear irreversible thermodynamics so as to obey Onsager reciprocal relations (Dadzie et al., 2012). Unlike conventional models, these generalized relations allow for extra diffusive processes (e.g., volume diffusion), and their associated entropy production, without violation of the second law.

2. Kinetic and Continuum Descriptions of Volume/Mass Diffusion

Thermodynamic models based on kinetic theory extend the standard Boltzmann approach by introducing either explicit spatial diffusion in the kinetic equation,

$$\frac{\partial f}{\partial t} + \boldsymbol{\xi}\cdot\nabla_X f - k \Delta_X f + F_{\mathrm{ext}} \cdot \nabla_\xi f - I_g(f) = 0$$

or by adding a stochastic variable representing local “microscopic empty volume” $v$,

$$\frac{\partial f}{\partial t} + \boldsymbol{\xi} \cdot \nabla_X f + W \frac{\partial f}{\partial v} + F_{\mathrm{ext}} \cdot \nabla_\xi f - I_g(f) = 0$$

where $W$ is the rate of local volume production. Taking moments of $f$ yields a hierarchy of conservation equations— mass, momentum, energy, and volume—each modified by additional terms encoding stochastic or non-local behavior (Dadzie et al., 2012). These modifications generate diffusive fluxes of mass/volume that cannot be captured by standard continuum closures, highlighting the fundamental role of thermodynamic forcing in mediating higher-order transport.

3. Mechanical and Thermodynamic Compatibility

A significant challenge for thermodynamically generalized diffusion models is preserving the classical conservation laws—mass, momentum, angular momentum—and invariance under Galilean transformations. The distinguishing factor in volume/mass diffusion models is the introduction of both mass-based and volume-based mean velocities. By ensuring that conservation equations respect this distinction and that fluxes are correctly constructed, one demonstrates:

• Conservation of total mass (via a conservative continuity equation)
• Symmetry of the pressure tensor (which guarantees angular momentum conservation)
• Invariance to Galilean transformations and integrability conditions As such, the incorporation of microscopic spatial randomness respects fundamental mechanical principles while providing physically meaningful extra dissipation channels absent in first-order models (Dadzie et al., 2012).

4. Dimensional Analysis and the Nature of Non-Conventional Diffusion

Dimensional analysis reveals that the new volume/mass diffusion terms scale as $\mathrm{Kn}^2$ (where $\mathrm{Kn}$ is the Knudsen number $\lambda/L$), in contrast to the first-order $\mathrm{Kn}$ scaling of Navier-Stokes–Fourier (NSF) terms. These higher-order terms correspond to nonlocal and nonconventional transport: they do not arise in closures that neglect non-equilibrium microstructure, and may manifest as “ghost effects” in certain flow regimes (e.g., high-$\mathrm{Kn}$ microflows or critical cases where entropy production is nonzero despite absence at the NSF level). For typical continuum conditions, these effects are generally negligible, but they become prominent in rarefied or structured fluids (Dadzie et al., 2012).

5. Comparison with Standard Navier–Stokes–Fourier Theory

The classical NSF model is founded on local equilibrium, retaining only linear (first order in $\mathrm{Kn}$) fluxes associated with shear and heat conduction. In the thermodynamic volume/mass diffusion model, however, the equations are enriched by

• explicit evolution for the fluid “volume”
• modification of momentum/energy equations by higher-order fluxes
• replacement of mass-based velocity by a volume-based velocity in the convective terms When the stochastic variable becomes homogeneous (i.e., no fluctuations in $v$), the system collapses to the standard NSF structure. However, the incorporation of volume/mass diffusion is essential for capturing transport in complex microstructured systems (e.g., dispersion, micro- and nanofluidics, high-frequency sound) where nonlocal and fluctuation-induced effects are critical (Dadzie et al., 2012).

Model	Diffusive Terms	Driving Physical Mechanism
Navier–Stokes–Fourier	1st order in Kn	Local equilibrium, molecular collisions
Volume/Mass Diffusion	2nd order in Kn	Spatial molecular randomness, nonlocal

6. Mathematical Structure and Onsager Reciprocity

The flux-gradient relations in the generalized theory are constructed to retain Onsager symmetry. All dissipative fluxes—shear, heat, and volume/mass—are written in the Fick–Fourier–Newton form and constitute a class of bilinear relations to their conjugate thermodynamic forces, ensuring (by construction) the positivity of total entropy production and symmetry under time reversal. This guarantees both thermodynamic consistency and mathematical well-posedness, even when additional dynamical variables (such as microscopic volume) are included in the state description (Dadzie et al., 2012).

7. Broader Significance and Context

The thermodynamic approach to volume/mass diffusion provides a systematic rationale for extending classical fluid models. It unifies kinetic and continuum perspectives, clarifies the precise limitations of closure schemes (i.e., those neglecting non-equilibrium microstructure), and supplies new predictions for flow regimes where spatial stochasticity dominates. This framework also connects with kinetic models previously proposed by Klimontovich and aligns with classical principles under suitable limits, ensuring both quantitative and conceptual agreement with mechanical laws. The higher-order terms and associated entropy production mechanisms in this approach emphasize the rich structure of diffusion beyond the scope of first-order Fickian paradigms, and supply a theoretical basis for studying transport in advanced and microstructured fluidic systems (Dadzie et al., 2012).