Thermodynamic Diffusion Models

• Thermodynamic diffusion models are frameworks that define transport via thermodynamic potentials, state functions, and entropy production, unifying classical, stochastic, and ML-based methods.
• They leverage key quantities—such as the thermodynamic factor, Onsager reciprocity, and steepest entropy ascent—to predict effective diffusion rates and accommodate multiscale and non-equilibrium effects.
• These models provide practical, energy-based formulations that enable accurate data fitting across dense fluids, porous media, multicomponent systems, and even quantum or anomalous diffusion regimes.

A thermodynamic formulation of diffusion models provides a rigorous basis to describe, predict, and interpret diffusive transport in matter, incorporating the interplay of kinetic processes and thermodynamic driving forces. Across statistical physics, physical chemistry, and statistical learning, thermodynamic diffusion frameworks unify classical, stochastic, and even machine‐learning–based models by expressing transport coefficients, fluxes, and irreversibility directly through thermodynamic potentials, state functions, and entropy production.

1. Thermodynamic Factor and Diffusion in Dense Fluids

A key quantity in the thermodynamic theory of self-diffusion in dense fluids is the thermodynamic factor $\Gamma$, which measures the amplification of collective concentration fluctuations due to particle correlations. For a fluid with temperature $T$, number density $\rho$, and excess chemical potential $\mu_{\mathrm{ex}}(\rho,T)$, the thermodynamic factor is defined as

$$\Gamma(\rho,T) = 1 + \beta \rho \left(\frac{\partial \mu_{\mathrm{ex}}}{\partial\rho}\right)_T$$

where $\beta=1/(k_B T)$. Alternatively, using the compressibility factor $Z=\beta p/\rho$,

$\Gamma = Z + \rho \frac{\partial Z}{\partial\rho}$

In the limit of ideal gases, $\Gamma\to1$, while strong correlations (e.g., near criticality) yield $\Gamma\gg1$.

Sampayo Puelles & Hoyuelos derive a general relation for the self-diffusion coefficient $D$ as a function of $\Gamma$ for dense atomic fluids (Puelles et al., 2023). Motivated by mesoscopic kinetic arguments, they propose

$$\frac{D}{D_0} = \exp\left[a(\Gamma-1) + b(\rho\sigma_{\mathrm{HS}}^3)\right]$$

where $D_0$ is the dilute‐limit diffusion coefficient, $\sigma_{\mathrm{HS}}$ is the (effective) hard‐sphere diameter, and $a<0$, $b<0$ are fitted constants. This model, fit to hard‐sphere simulation data and then transferred without re-fit to Lennard–Jones and real-fluid (xenon) cases, collapses self-diffusion data over a broad density and interaction–strength range. By utilizing only the thermodynamic factor (readily accessible from the equation of state), the model circumvents the need for excess-entropy calculations (as in Rosenfeld scaling), and generalizes more robustly across systems.

2. Non-equilibrium Thermodynamic Models of Coupled Diffusion

Extensions to media where adsorption/desorption or other multi-phase features are non-negligible have been constructed based on non-equilibrium thermodynamic potentials. Santamaría-Holek et al. provide a two-phase model for diffusion in porous materials, with mobile ($c_1$) and adsorbed ($c_2$) species (Santamaria-Holek et al., 2012). The coupled system is governed by:

• Linear Onsager-type force–flux relations coupling chemical potential gradients to fluxes
• Mass-balance equations for both phases, including reversible adsorption/desorption rates
• An explicit reduction to a single effective diffusion equation in the quasi-equilibrium limit:

$$\frac{\partial c}{\partial t} = \frac{\partial}{\partial x}\left[D_{\mathrm{eff}}(c) \frac{\partial c}{\partial x}\right], \qquad D_{\mathrm{eff}}(c) = D_1 \frac{\partial c_1}{\partial c}$$

where $c=c_1+c_2$ is the total local concentration. The effective diffusivity captures the interplay of slow/fast surface processes and particle–particle interactions, leading to regimes of suppression, enhancement, or non-monotonicity in $D_{\mathrm{eff}}$ depending on kinetic and interaction parameters.

3. Quantum and Stochastic Thermodynamic Foundations

The principle of Steepest Entropy Ascent (SEA) offers a foundational dynamical principle for nonequilibrium relaxation—including both heat and mass diffusion—from first thermodynamic principles, without recourse to local equilibrium (Li et al., 2016). SEA dynamics on the state manifold, endowed with the Fisher–Rao metric, yields:

• Conjugate (coordinate-free) fluxes and thermodynamic forces
• Linear Onsager reciprocity and positivity of the dissipation potential, both near and far from equilibrium
• Emergent Fickian and Fourier laws, with explicit expressions for transport coefficients
• Generalized Gibbs relations and Clausius inequalities that hold beyond linear regime

For quantum or stochastic systems, techniques based on stochastic path integrals and martingale structures for thermodynamic functionals are available. These formalisms can be used to define entropy production, characterize its regular and anomalous (coarse-graining–induced) parts, and prove fluctuation theorems for diffusion in singularly perturbed and far-from-equilibrium multiscale systems (Ge et al., 2018).

4. Multicomponent and Nonlinear Diffusion: Thermodynamic Structure

Systematic construction of multicomponent, nonlinear diffusion models is enabled by quasichemical–mechanism representations and thermodynamic constraints (Gorban et al., 2010, Bothe et al., 2020). The central elements are:

• State variables: Concentrations $\mathbf{c}(x,t)$ of $N$ species, with total Helmholtz free energy $F[\mathbf{c}]$ and chemical potentials $\mu_i = \partial f/\partial c_i$
• Driving forces: Gradients of chemical potentials $\nabla \mu_i$ (including activity and non-ideality effects)
• Constitutive relations: Fluxes are given by

$\mathbf{J} = -L(\mathbf{c}) \nabla (\mu/T)$

with $L$ a symmetric, positive semidefinite Onsager matrix (Fick–Onsager form), or in the Maxwell–Stefan formalism as force–balance among interspecies frictions.

A recent advancement is the operator-level closure that is explicit, modular, and unifies the Fick–Onsager and Maxwell–Stefan approaches (Bothe et al., 2020). In all cases, the entropy production (dissipation) density

$$\zeta = -\sum_{i} \mathbf{J}_i \cdot \nabla (\mu_i/T)$$

is manifestly non-negative provided $L$ is positive on the constraint subspace (mass conservation).

The mean-field approach in binary mixtures yields a detailed dependence on the so-called thermodynamic factor $\Phi = 1 + \frac{d\ln \gamma_A}{d\ln x_A}$, directly linking non-ideality to diffusion kinetics, and allows explicit fitting and prediction of intrinsic and tracer diffusion coefficients in alloys (Martínez et al., 2019).

5. Thermodynamic Constraints, Bounds, and Uncertainties

Stochastic thermodynamics provides universal bounds and relations for fluctuation and irreversibility in diffusion. The Thermodynamic Uncertainty Relation (TUR) gives a lower bound on the variance of currents (such as particle displacement) in terms of entropy production rate (dissipated power) (Hartich et al., 2021). For a current $J_t$ in a NESS,

$$\mathrm{Var}(J_t) \geq \frac{2 \langle J_t\rangle^2}{\sigma t}$$

This result imposes sharp limits on the possible extent and duration of anomalous diffusive regimes (e.g., subdiffusion or superdiffusion) in finite systems, expressed as a bound on the time scale

$$t^* = \left(\frac{K_\alpha}{C}\right)^{1/(1-\alpha)}$$

where $K_\alpha$ parameterizes the anomalous MSD scaling.

In reaction–diffusion systems, geometric thermodynamic decompositions of the entropy production rate facilitate identification of dissipation trade-offs and optimal transport paths for pattern formation (Nagayama et al., 2023). In such settings, the Pythagorean decomposition $$\sigma = \sigma^{\mathrm{ex}} + \sigma^{\mathrm{hk}}$$ (excess + housekeeping EP) as well as inverse inequalities (speed limits) derived from Wasserstein distance metrics, impose tight speed-dissipation constraints and underpin energetic analyses of pattern dynamics.

6. Thermodynamic Formulation in Modern Diffusion Models and Machine Learning

Recent developments in statistical learning, particularly score-based diffusion models for generative modeling, have exposed deep thermodynamic analogies and limitations (Kodama et al., 7 Oct 2025, Yu et al., 2024, Máté et al., 2024). In these models:

• The forward SDE is taken as a time-dependent Langevin (diffusion) process, while generative sampling involves the reverse SDE driven by a learned score (gradient of log probability).
• A path-integral or Fokker–Planck formulation recasts training, sampling, and performance metrics in terms of entropy production, detailed fluctuation theorems, and the total heat/work budget along sample trajectories (Yu et al., 2024).
• A fundamental link is established between the negative log-likelihood of the generative model and the integrated entropy rate along the diffusion path (Kodama et al., 7 Oct 2025):

$$\mathrm{NLL} \geq \frac{S_0+S_1}{2} - \frac{1}{2} \int_0^1 [\dot{S}^i(t) + \dot{S}^e(t)] dt$$

where the integral represents the irreversibility cost and entropy export of the generative process.

"Neural Thermodynamic Integration" views the entire denoising diffusion process as a sequence of thermodynamically-consistent intermediate Boltzmann distributions, enabling free-energy differences to be framed as trajectory integrals, and bridges classical thermodynamic integration techniques with energy-based diffusion sampling (Máté et al., 2024).

7. Beyond Classical Models: Volume/Mass Diffusion, Multiscale, and Mesoscopic Effects

In certain multi-scale or rarefied-gas applications, additional "non-conventional" diffusion processes (volume or mass diffusion) emerge, invisible to Chapman–Enskog–type expansions truncated at Navier–Stokes–Fourier (NSF) order (Dadzie et al., 2012). By introducing kinetic models with extra microstructure variables (e.g., per-molecule volume), fully consistent frameworks with four coupled balances (mass, momentum, energy, and volume) can be constructed, with explicit entropy balances and Gibbs-like relations that maintain mechanical and thermodynamic consistency where simpler models fail.

Similarly, phase-field models of non-equilibrium diffuse interfaces make use of coupled Allen–Cahn and Cahn–Hilliard equations, with energy-dissipation and Onsager reciprocity, to partition driving forces into interface friction and solute drag, and recover atomic-scale features such as "partial drag" self-consistently from mesoscopic thermodynamics (Li et al., 2023).

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In summary, thermodynamic formulations of diffusion models offer a physically and mathematically grounded approach to transport phenomena, capable of encompassing dense-fluid kinetics, heterogeneous media, non-ideal mixtures, quantum–stochastic systems, multicomponent/multiphase regimes, and energy-based machine learning paradigms. A unifying feature in all cases is the encoding of dissipative kinetics and structure–property relationships in a transparent, variational, and entropy-centered language—facilitating direct connection to both fundamental physical limits and practical computation or modeling.