Diffusion Models & Thermodynamics

Updated 27 October 2025

Diffusion models and thermodynamics are frameworks that combine random molecular motion with energy and entropy constraints, defining transport processes.
They extend classical Fickian dynamics to capture non-local, quantum, and ghost effects, emphasizing Onsager reciprocity and entropy production.
These concepts underpin applications from material science to data-driven generative models, linking micro-level stochasticity to macroscopic conservation laws.

A diffusion model is a mathematical or computational framework describing the transport of quantities—such as mass, energy, charge, or entropy—through continuous media via random molecular motion, typically governed by stochastic processes and conservation laws. Thermodynamics is intimately linked to such models both as a source of driving forces (e.g., chemical potential and entropy gradients) and as a framework establishing constraints (e.g., the second law, Onsager reciprocal relations, and entropy production). The interplay of diffusion and thermodynamic principles underlies a wide range of phenomena, from gas flows and energy transport to pattern formation, quantum transport, and generative modeling. The following sections elucidate core principles, model classifications, thermodynamic constraints, non-classical effects, and paradigmatic applications as found in recent research.

1. Thermodynamic Formulation of Diffusion Models

The thermodynamic underpinnings of diffusion models arise from the connection between microscopic stochasticity and macroscopic irreversible transport. At the microscale, kinetic equations (e.g., Boltzmann or generalized kinetic equations with spatial diffusion terms) capture random molecular behavior. Integrating these equations over velocity and, if applicable, additional microstructural degrees of freedom (like microscopic volume), yields macroscopic conservation laws for mass, momentum, energy, and, in some models, extra structural fields such as “microscopic volume” or “order parameters” (Dadzie et al., 2012).

In the continuum limit, these equations are recast within the framework of nonequilibrium thermodynamics. The entropy production rate (EPR) plays a central role, with irreversible fluxes (mass, heat, charge) linearly related to the gradients of intensive thermodynamic quantities (chemical potential, temperature, electrical potential) via Onsager reciprocal relations. For example, coupled heat and mass transport in polymer solutions is governed by

$\frac{\partial c}{\partial t} = \nabla \cdot \left( D_\mathrm{M} \nabla c + D_T \nabla T \right), \qquad \rho c_p \frac{\partial T}{\partial t} = \nabla \cdot \left( \alpha \nabla T \right) + (\text{coupling terms}),$

where $D_\mathrm{M}$ is the mutual diffusion coefficient and $D_T$ describes thermal diffusion (the Soret effect) (Es-haghi, 2012).

Entropy production decomposes into contributions from each process, and the second law requires that the total EPR is non-negative. Thermodynamics also establishes reciprocity and constraints on cross-effects; for example, thermal and mass diffusion are strictly coupled in nonequilibrium models (Es-haghi, 2012, Santamaria-Holek et al., 2012).

2. Generalization and Extensions Beyond Classical Models

Classical diffusion, as embodied in the Fickian and Navier–Stokes–Fourier (NSF) models, presumes local equilibrium and three macroscopic conservation laws (mass, momentum, energy). This local equilibrium hypothesis constrains the emergent flux-force relations and precludes higher-order or structurally induced effects.

Several advanced models extend this picture:

Volume/Mass Diffusion and Non-Local Equilibrium Structures: Inclusion of explicit spatial randomization at the microscopic level yields an additional evolution equation for “microscopic volume,” necessitating description in terms of mass velocity and volume velocity, whose difference quantifies volume or mass diffusion. These effects are non-conventional, appearing at $O(\mathrm{Kn}^2)$ in the Knudsen number expansion (where $\mathrm{Kn} = \lambda / L$ ) and representing so-called ghost effects—subtle kinetic signatures absent in NSF theory (Dadzie et al., 2012).
Generalized Frameworks for Multicomponent Mixtures: In the case of mixtures with multiple velocities or temperatures, frameworks such as the Maxwell–Stefan and its multiscale generalizations accommodate non-Fickian contributions, cross-diffusion, thermal diffusion (Soret effect), and even electromagnetic driving (Vágner et al., 2022, Bothe et al., 2020). These equations are structurally constrained by thermodynamic projections to ensure mass conservation and positivity of total entropy production.
Quantum and Ballistic Diffusion: Quantum transport in mesoscopic systems is naturally described by a drift–diffusion formalism, but with quantum statistics encoded via “elementary Fermi- or Bose-systems.” This framework allows a unified treatment of particle and entropy transport, reveals quantization of conductances, and predicts quantum interference effects in both electric and heat (entropy) currents (Strunk, 2012).

3. Non-Classical and Emergent Thermodynamic Phenomena

Higher-order and structurally induced diffusion phenomena are critical in several advanced contexts:

Ghost Effects and Knudsen-Number Scaling: In rarefied flows or microfluidics, higher-order terms (e.g., volume/mass diffusion proportional to $\mathrm{Kn}^2$ ) become non-negligible and manifest as ghost effects—nonzero orthogonal fluxes or nonlocal entropy production that elude conventional continuum mechanics (Dadzie et al., 2012).
Pattern Formation and Thermodynamic Trade-Offs: In reaction-diffusion systems, entropy production can be systematically decomposed—via geometric thermodynamics—into “excess” dissipation driving genuine pattern evolution and “housekeeping” dissipation. The minimal dissipation required for a given pattern transition is governed by optimal transport geometry (e.g., the 2-Wasserstein metric) and appears in thermodynamic speed limits:

$W_2^2([c](0), [c](\tau)) \leq \tau \int_0^\tau \sigma^\mathrm{ex} dt,$

where $W_2$ is the Wasserstein-2 distance between patterns and $\sigma^\mathrm{ex}$ the excess EPR (Nagayama et al., 2023, Ikeda et al., 5 Jul 2024).

Nonequilibrium Steady States and Kinetic Potential Minimization: In open, non-ideal reaction-diffusion systems driven by chemical networks, the steady state is not a minimum of the free energy, but rather of a shifted “kinetic potential” dictated by the network topology and the driving (e.g., chemostats). Dissipation is spatially distributed among reactions and diffusion, and energetic costs for maintaining structures such as coacervates and biomolecular condensates can be quantified (Avanzini et al., 12 Jul 2024).
Entropy Rates and Thermodynamic Bounds in Generative Diffusion Models: In score-based generative models, the negative log-likelihood (NLL) of the data is fundamentally bounded from below by entropy rates associated with the stochastic process:

$\mathrm{NLL} \geq \frac{S_0 + S_1}{2} - \frac{1}{2}\int_0^1 \dot{S}_\theta(t) dt,$

where $S_0$ is the data entropy, $S_1$ the equilibrium entropy, and $\dot{S}_\theta$ the system entropy rate along the model path. Achieving lower NLL thus directly relates to representing and “removing” entropy induced by the forward noisy process—an explicit thermodynamic operation analogous to Maxwell’s demon (Kodama et al., 7 Oct 2025).

4. Thermodynamic Constraints, Irreversibility, and Entropy Production

Thermodynamic consistency imposes stringent restrictions on diffusion models:

Second Law Compliance: All physically plausible models must guarantee non-negative entropy production in every process. This requirement often manifests as positivity constraints on the full Onsager tensor or the transport operator, not necessarily on individual scalar diffusivities, especially in the case of multicomponent systems (Bothe et al., 2020).
Reciprocity and Cross-Coupling: Onsager relations enforce symmetry constraints on cross-coupling coefficients (e.g., $l_{12}=l_{21}$ in Onsager matrices), and dictate the structure of kinetic coefficients. These constraints are essential for correct phenomenology in coupled heat-mass and adsorption-diffusion systems (Es-haghi, 2012, Santamaria-Holek et al., 2012).
Irreversible Cost of Pattern Evolution and Data Generation: The excess EPR in pattern-forming systems or the entropy-rate correction to the NLL in generative models quantifies the fundamental dissipative cost required by thermodynamics for a desired dynamical transformation (Nagayama et al., 2023, Ikeda et al., 5 Jul 2024, Kodama et al., 7 Oct 2025).
Optimal Protocols via Thermodynamic Geometry: Minimizing dissipation for a given pattern change or for high-fidelity data generation corresponds to traversing geodesics in the Wasserstein (optimal transport) metric. This links physical irreversibility directly to the geometric structure of the transport and generation process (Ikeda et al., 5 Jul 2024, Nagayama et al., 2023).

5. Application Domains: Classical, Quantum, and Data-Driven Diffusion

The intersection of diffusion models and thermodynamics spans a diversity of domains:

Classical Gaseous and Liquid Systems: High-order volume/mass diffusion models resolve rarefaction, microstructure, and ghost effects in gases—critical for micro/nano flows and for capturing transport phenomena at finite Knudsen number (Dadzie et al., 2012, Puelles et al., 2023).
Polymeric and Porous Media: Coupled heat and mass diffusion in polymer evaporation, effective medium approaches for adsorption-diffusion in nanopores, and modeled interplay of adsorption kinetics and mobility are central to material processing, separation, and catalysis (Es-haghi, 2012, Santamaria-Holek et al., 2012).
Quantum and Mesoscopic Transport: Unified drift–diffusion models incorporating quantum statistics, conductance quantization, and entropy current interference underpin theoretical descriptions of quantum wires, thermoelectric effects, and ballistic–diffusive crossovers (Strunk, 2012, Langley et al., 2018).
Pattern Formation and Active Matter: Spatial self-organization—chemical waves, Turing patterns, biomolecular condensates—is governed by a balance between kinetic (reaction network) nonlinearities, diffusion, non-ideal free energies, and the allocation of dissipation among processes (Avanzini et al., 12 Jul 2024, Nagayama et al., 2023).
Machine Learning and Generative Models: Diffusion probabilistic models for data synthesis, notably in vision, are fundamentally inspired and constrained by nonequilibrium and stochastic thermodynamics. Recent work demonstrates that model efficiencies, accuracy bounds, and computational costs are quantitatively governed by entropy production rates, speed–accuracy trade-offs, and thermodynamic geometry (Ho et al., 2020, Ulhaq et al., 2022, Kodama et al., 7 Oct 2025, Ikeda et al., 5 Jul 2024).

6. Emerging Concepts and Future Prospects

Recent developments have foregrounded several cross-cutting concepts:

Speed–Accuracy Trade-offs: Tight inequalities relate data generation error (e.g., in Wasserstein distance) to total entropy production and the “speed” of the forward diffusion, providing practical metrics for protocol optimization (Ikeda et al., 5 Jul 2024).
Entropy-Driven Performance Bounds: The negative log-likelihood in score-based models is physically limited by entropy rate integrals, implying a direct link between generative model performance and fundamental thermodynamic principles (Kodama et al., 7 Oct 2025).
Geometric and Optimal Transport Perspectives: The language of optimal transport (Wasserstein metrics, geodesics) offers a natural setting for expressing efficiency limits, controlling dissipation, and designing optimal evolution protocols in both physical and data-generative diffusion (Nagayama et al., 2023, Ikeda et al., 5 Jul 2024).
Thermodynamically Consistent Closure and Numerical Modeling: Symmetric and computationally attractive closure schemes for multicomponent diffusion have been shown to be equivalent to classical forms under natural positivity conditions, facilitating robust numerical simulations and parameterizations (Bothe et al., 2020).
Quantum-Classical Crossovers: Quantum noise–induced decoherence in models such as the Haken–Strobl–Reineker SQLE reveals how temperature-dependent dephasing mediates the transition from quantum-coherent to classical diffusive regimes, with diffusion coefficients analytically scaling with thermodynamic quantities (Barford, 18 Jun 2024).
Energetic Costs and Biological Functionality: Quantitative estimation of the free energy required to sustain out-of-equilibrium structures in living cells and engineered systems—such as coacervates and biomolecular condensates—is enabled by integrating nonideal free energy terms with reaction-diffusion thermodynamics (Avanzini et al., 12 Jul 2024).

This synthesis reflects the breadth and depth of current research at the interface of diffusion modeling and thermodynamic theory, highlighting the essential role of entropy, non-classical effects, and optimality principles in modern applications spanning physical, chemical, quantum, and computational domains.