Volume/Mass Diffusion & Non-Local Equilibria

• Volume/mass diffusion is a transport phenomenon defined by non-local equilibria arising from extra conservation laws and boundary effects.
• Advanced models reveal that multipole conservation and multicomponent interactions lead to anomalous scaling and highly non-uniform steady states.
• Recent kinetic theory and PDE analyses show that these non-local effects are critical for designing accurate multiscale models in engineering and physics.

Volume/mass diffusion encompasses a broad range of transport phenomena, from classical Fickian diffusion of a conserved scalar (mass, volume, concentration) to more exotic regimes featuring non-local equilibrium structures dictated by additional conservation laws, multi-component interactions, high-order constraints, or nontrivial microscopic dynamics. Advances in both kinetic theory and PDE analysis reveal that such structures—often manifesting as boundary/bulk inhomogeneities, long-range fluctuations, fractal steady states, or anomalous scaling—typically occur where conventional local-equilibrium descriptions fail. This entry surveys the mathematical formulations and physical implications of volume/mass diffusion beyond standard paradigms, with particular focus on non-local equilibria arising under multipole conservation, multicomponent boundary layers, stochastic particle systems, kinetic “ghost effects,” volume-surface coupling, and systems with singular microscopic phase-space structure.

1. Multipole-Conserving Diffusion: Nonlinear Dynamics and Boundary-Localized Equilibria

Imposing center-of-mass (dipole) or higher multipole conservation fundamentally alters the character of diffusive transport. Instead of the classical diffusion equation $\partial_t\rho = D\nabla^2\rho$ (with dynamical exponent $z=2$), the conservation of moments up to order $n$ leads to higher-derivative nonlinear diffusion equations: $$\partial_t\rho = D\,\nabla^{n+1}\cdot\nabla^{n+1}\ln\rho\,,$$ where, e.g., dipole conservation ($n=1$) gives $$\partial_t\rho = D\partial_a\partial_b\partial^a\partial^b\ln\rho$$ (Han et al., 2023). This introduces anomalous scaling of relaxation dynamics: in $d$ dimensions, the characteristic dynamical exponent is $z = 4 + d$ for dipole conservation, generalizing to $z = 2n + 2 + d$ for order-$n$ multipole conservation.

Equilibrium density profiles become boundary-localized and intrinsically non-local: imposing both total mass and center-of-mass fixes the unique zero-current steady state to have exponential localization near boundaries. In 1D, for a region $x>0$ with prescribed center-of-mass, the equilibrium reads

$$\rho_{\rm eq}(x) = \frac{N}{x_{\rm cm}}e^{-x/x_{\rm cm}},$$

with $x_{\rm cm} = Q/N$. In bounded domains, an analogous non-uniform profile is determined by boundary positions and Lagrange multipliers enforcing the global moments. This non-locality arises because the current vanishes only when the entire system—including its spatial boundaries—is accounted for. Physically, the center-of-mass constraint creates an entropic confining potential, resulting in macroscopic transport unfolding predominantly in boundary layers, with the bulk remaining effectively inert (Han et al., 2023).

In the quantum context, the same conservation leads to real-space analogs of Fermi surfaces (for fermions) and Bose-Einstein condensation (for bosons), further highlighting the deep connection between conservation laws and non-local equilibrium phenomena.

2. Multicomponent Diffusion and Non-Local Equilibrium in Boundary Layers

Diffusive transport in multicomponent mixtures, especially under competing mechanisms such as thermophoresis, turbulent mixing, and diffusiophoresis, generically induces inhomogeneous, non-constant composition profiles near boundaries. For inert mixtures with no net wall-normal mass flux, vanishing of the total diffusive flux for each species yields a coupled system of ODEs for the local mass fractions $x_i(y)$: $$\sum_{j=1}^{N} \left( D_{ij}+\frac{v_t}{\mathrm{Sc}_t}\delta_{ij} \right) \frac{d x_j}{d y} = -D_{iT}\frac{d\ln T}{dy} - D_{iP}\frac{d\ln P}{dy} + \sum_{j=1}^{N}D_{ij}\frac{M_{m,j}}{R T}f_{j,y},$$ with $D_{ij}$ the multicomponent (Maxwell–Stefan) diffusion coefficients, and $D_{iT}$, $D_{iP}$ the thermal and barophoretic coefficients (Johnsen, 2018). Numerical simulations in ternary mixtures (e.g., H₂–N₂–CO₂ under a temperature gradient) show pronounced wall-normal gradients in composition, density, viscosity, and thermal conductivity. These output profiles are strongly non-local, determined by the collective action of all transport mechanisms and the global boundary conditions.

Such non-uniform, non-local equilibrium structures in the boundary layer have significant feedback effects: wall-derived heat fluxes and friction can be altered by up to $O(10$–$20\%)$, even in idealized systems. The necessity of solving for non-constant $x_i(y)$ to maintain the zero-flux condition means that the boundary region is fundamentally out of local thermodynamic equilibrium, and any attempt to describe multicomponent flows without the full Maxwell–Stefan framework will systematically err in the presence of compositional or thermal gradients. These insights have direct implications for CFD, laboratory rheometry, and applications in chemical engineering and geophysical fluid dynamics (Johnsen, 2018).

3. Kinetic Theories and Emergent Volume/Mass Diffusion beyond Navier–Stokes

Approaches from extended kinetic theory (e.g., Dadzie–Reese–McInnes) introduce an explicit microscopic volume variable $v$ into the single-particle distribution, yielding a fourth macroscopic balance law for the mean volume field beyond the three standard conservation laws (mass, momentum, energy): $$\frac{\partial}{\partial t}(A_n\bar v) + \nabla_X \cdot (A_n\bar v U_m) + \nabla_X \cdot (A_n \mathbf{J}_v) = A_n W,$$ with $A_n$ the molecular count, $\bar v$ the mean microscopic volume, and $\mathbf{J}_v$ the volume/mass diffusion velocity. The corresponding thermodynamic closure delivers a generalized Gibbs-type relation, a positive-definite entropy production rate, and constitutive equations for volume-diffusion fluxes (e.g., $\mathbf{J}_v=-D_m\nabla_X\ln\rho$) (Dadzie et al., 2012).

These models connect to earlier work by Klimontovich, extend the standard Boltzmann equation, and make explicit that volume/mass diffusion effects arise at Burnett or “ghost” order ($O(\mathrm{Kn}^2)$ in the Knudsen number), lying outside the scope of classical Navier–Stokes–Fourier theory. As such, non-local equilibrium structures originate from stochastic volume fluctuations, featuring prominently in flows with curvature, rotation, or other settings sensitive to higher-order deviations from local equilibrium. A key implication is that phenomena such as Sone's “ghost effects” are fundamentally non-classical and cannot be captured by conventional hydrodynamics (Dadzie et al., 2012).

4. Non-Locality from Reaction–Diffusion, Interface Coupling, and Additional Conservation Laws

Volume-surface reaction–diffusion systems—arising, for example, in morphogenesis and cellular signaling—demonstrate that non-local equilibrium structures can be a robust feature of coupled bulk–surface kinetics. Considering models with linear reactions among several bulk and surface species and mass conservation across compartments, the unique steady state is given by the solution of a coupled elliptic system with spatially inhomogeneous profiles (Fellner et al., 2016). The entropy method produces explicit exponential rates of convergence toward these non-local equilibria.

In deterministic models with additional constraints, such as the one-dimensional Galton board under a uniform field, stationary states are not smooth probability densities but rather fractal measures characterized by singular generalized Takagi functions. The fractality of the invariant measure is detected in the persistent scaling of coarse-grained entropy and ensures strictly positive entropy production in line with irreversible thermodynamics, even in volume-preserving, time-reversible settings (0903.3849). Here, the non-locality resides in the singular, system-wide structure of the stationary state.

5. Stochastic Particle Systems: Non-Local Energy Transfer, Crossover, and Hyperuniformity

Volume/mass diffusion in stochastic lattice models with multiple conserved quantities (e.g., the HCVE chain with conserved volume and energy) can yield a hierarchy of transport regimes: local drift–diffusion for the volume field, but non-local fractional (Levy-type) dynamics for the energy current. Systematic expansions around local equilibrium reveal how corrections to the local equilibrium distribution—embodying nonlocal space-time correlations of microscopic currents—drive anomalous transport and the emergence of non-local hydrodynamic kernels (Kundu, 2022). The scaling crossover from ordinary diffusion to anomalous (super-diffusive) transport is set by a system-dependent length scale.

In active and driven systems, as in circle-active chiral fluids or binary mixtures under imposed gradients, non-local equilibrium structures are manifest in the form of giant non-equilibrium fluctuations, hyperuniformity at large scales, and microphase clustering at intermediate scales. The preservation of center-of-mass conservation (or similar constraints) at the microscopic dynamics level guarantees long-wavelength suppression of density fluctuations ($S(q)\sim q^2$), while locally breaking homogeneity leads to large, non-Gaussian (high-cumulant) fluctuations at sub-macroscopic scales (Lei et al., 2018, Eyink et al., 2022). Linearized fluctuating hydrodynamics or Kraichnan-type models provide closed-form predictions for structure factors and cumulants, diagnosing the functional form and scale dependence of non-locality in these systems.

6. Mathematical Structures: Entropy Methods, Gradient Flows, and Nonlocal PDEs

The analysis of non-local equilibrium structures associated with volume/mass diffusion employs sophisticated mathematical machinery: entropy dissipation methods, Lyapunov functionals, Poincaré inequalities adapted to nonuniform weights, and Wasserstein gradient-flow formalisms (Fellner et al., 2016, Matthes et al., 2024). For aggregation-diffusion systems with mildly non-convex cross-diffusion perturbations, the exponential convergence to compactly supported steady states persists, controlled by convexity properties of the underlying energy functional. Nonlocal aggregation, cross-diffusion, and small perturbative terms are balanced via $\varepsilon$-dependent splittings, with explicit relationships between convergence rates, nonlocal interaction strengths, and perturbative coupling constants (Matthes et al., 2024).

Recent work also rigorously establishes smoothing and equilibration in nonlocal advection–diffusion models derived from active particle systems, showing (using De Giorgi methods) that instantaneous regularity and eventual convergence to constant steady states is retained provided nonlinear drift terms are sufficiently weak. However, beyond a critical threshold, pattern-forming instabilities may arise, signaling a regime of spontaneously non-uniform equilibria whose mathematical treatment remains open (Alasio et al., 2024).

7. Implications and Outlook: Physical Origins and Model Generality

Non-local equilibrium structures are an intrinsic feature of volume/mass diffusion whenever essential constraints—conservation beyond particle number, multicomponent interactions, stochastic volume dynamics, or coupling across spatial regions—are imposed. In all such cases, global information (boundary positions, total moments, system size, higher conserved quantities) feeds back into the spatial organization and relaxation dynamics, generating equilibrium or stationary states that cannot be written as products of local (Gibbsian) forms. The physical mechanisms encompass entropic barrier formation, competition of diffusive mechanisms, aggregate formation via nonlocal interactions, and global charge or momentum flows.

The generality of these phenomena suggests that any extension of diffusion theory or practical model for real multi-scale, multi-component, or constraint-rich media must incorporate non-local equilibrium effects to accurately predict not only rates and profiles, but also the physical nature of long-time, steady, and fluctuation-driven states. Theoretical advances in high-order kinetic theory, gradient flows in metric spaces, and stochastic dynamics with built-in conservation laws now allow for systematic treatment and quantification of these structures, but challenges remain in the direct experimental observation, the exploration of large-deviation statistics, and the mathematical understanding of pattern formation in strongly nonlinear or singularly perturbed regimes.

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References

• Scaling and localization in multipole-conserving diffusion (Han et al., 2023)
• On Diffusion-Induced Non-Constant Composition Profiles in the Boundary Layer of Inert Multicomponent Mixtures (Johnsen, 2018)
• On the thermodynamics of volume/mass diffusion in fluids (Dadzie et al., 2012)
• Entropy methods and convergence to equilibrium for volume-surface reaction-diffusion systems (Fellner et al., 2016)
• Fractality of the non-equilibrium stationary states of open volume-preserving systems: II. Galton boards (0903.3849)
• The Kraichnan Model and Non-Equilibrium Statistical Physics of Diffusive Mixing (Eyink et al., 2022)
• Non-Equilibrium Strongly Hyperuniform Fluids of Circle Active Particles with Large Local Density Fluctuations (Lei et al., 2018)
• Super-diffusion and crossover from diffusive to anomalous transport in a one-dimensional system (Kundu, 2022)
• Regularity and trend to equilibrium for a non-local advection-diffusion model of active particles (Alasio et al., 2024)
• Convergence to equilibrium for cross diffusion systems with nonlocal interaction (Matthes et al., 2024)
• Steepest-entropy-ascent quantum thermodynamic modeling of heat and mass diffusion in a far-from-equilibrium system based on a single particle ensemble (Li et al., 2016)
• Equilibrium and non-equilibrium concentration fluctuations in a critical binary mixture (Giavazzi et al., 2016)