🔬 mesoscale physics

Emergence of Nonlocal Hydrodynamics in Many-Body Radiative Heat Transport

This paper develops a nonlocal hydrodynamic theory for many-body radiative heat transport by coarse-graining microscopic electrodynamic interactions, revealing that long-range exchanges can sustain non-Fourier transport when the interaction kernel's second spatial moment diverges, thereby bridging discrete fluctuational electrodynamics with continuous macroscopic descriptions.

Original authors: Mauricio Gomez Viloria, Florian Herz, Riccardo Messina, Philippe Ben-Abdallah

Published 2026-10-14

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CC BY 4.0

Original authors: Mauricio Gomez Viloria, Florian Herz, Riccardo Messina, Philippe Ben-Abdallah

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Heat is a traveler that usually follows a very predictable path. In the solid world we touch every day, thermal energy moves from hot spots to cold ones by bumping into neighbors, passing its energy along a chain of atoms in a steady, local handoff. This familiar process, known as Fourier's law, dictates that the speed of heat flow at any specific point depends only on how steep the temperature change is right there. It is a rule that has governed our understanding of engineering and physics for centuries, assuming that what happens far away has no immediate effect on what happens right here. But when we shrink our view down to the scale of tiny particles and look at how they exchange heat through light rather than touch, this simple local rule begins to break down. In these microscopic systems, particles can send energy directly to one another across empty space, skipping over the neighbors in between. This long-range whispering of heat creates a complex web of connections that traditional physics struggles to describe, leaving scientists to wonder if the old rules of heat flow still apply or if a new, more complicated set of laws is needed to explain how energy moves through a crowd of interacting particles.

A team of researchers in France has now built a new mathematical framework to answer this question, bridging the gap between the chaotic dance of individual particles and the smooth flow of heat we see in the macroscopic world. They started with a system of many tiny thermal emitters, each radiating energy to every other particle in the group. In this microscopic view, the heat exchange is not a simple neighbor-to-neighbor transfer but a vast network where every particle talks to every other particle, with the strength of the conversation depending on the distance and the material properties. The researchers took this intricate network of interactions and smoothed it out, averaging the behavior over small volumes to create a continuous description of heat flow. What they found was that the resulting description is inherently nonlocal. This means that the heat current at any given point is not determined solely by the temperature gradient at that exact spot, but rather by the temperature differences across a wide surrounding region. The heat flowing through a specific point is influenced by conditions far away, a phenomenon that persists even when we look at the system from a distance.

The study reveals that the famous Fourier law, which assumes heat flow depends only on local conditions, is actually just a special case that emerges under very specific circumstances. It appears only when the influence of distant particles fades away quickly enough as they get farther apart. The researchers showed that if the strength of the connection between particles drops off slowly with distance, the long-range interactions survive the smoothing process and remain a dominant feature of the heat flow. In such cases, the heat current at a point is a weighted sum of temperature gradients from a large area around it, creating a transport kernel that links distant points together. This kernel acts like a map of the underlying radiative connections, carrying information about the strength and geometry of the channels through which energy travels. The team demonstrated that for certain materials and arrangements, such as chains of gold or silicon carbide nanoparticles, these long-range channels are so persistent that the heat flow never settles into a simple local pattern, no matter how large the system becomes.

To prove their theory, the researchers applied their new equations to simulated chains of nanoparticles and compared the results with the exact microscopic calculations. They found that while the temperature profiles in these systems often look almost linear, just as they would in a standard conductor, the underlying physics is fundamentally different. The deviations from the standard local prediction, though small, are significant and reveal the hidden influence of the distant particles. In materials where the radiative coupling is strong, like silicon carbide, these nonlocal effects are much more pronounced, causing the heat flow to behave in ways that a local model cannot predict. The researchers also established a clear mathematical criterion for when the old local rules apply and when the new nonlocal rules take over. They found that if the number of particles and the way their connections decay with distance meet a certain threshold, the system will behave locally; if not, the long-range nature of the radiation ensures that the heat transport remains nonlocal.

This work provides a systematic way to describe radiative heat transport in complex, many-body systems without getting lost in the details of every single particle interaction. By deriving a continuous description that retains the memory of the microscopic network, the researchers have created a tool that can handle large and geometrically complex systems where traditional methods fail. This framework not only explains why heat sometimes behaves in unexpected ways in nanoscale systems but also opens the door to designing new materials where heat can be steered, concentrated, or redirected in ways that were previously impossible to imagine. The findings suggest that the universe of heat transport is richer and more interconnected than the local laws of the past century implied, revealing that even in the quiet flow of thermal energy, the distant past and the far-off future are always in conversation.

Technical Summary: Emergence of Nonlocal Hydrodynamics in Many-Body Radiative Heat Transport

Problem Statement
Fluctuational electrodynamics provides the microscopic framework for radiative heat exchange in many-body systems, describing energy transfer via electromagnetic transmission coefficients between thermal emitters. While this microscopic description reveals collective, long-range transport regimes characterized by anomalous (non-Fourier) thermal spreading, a systematic derivation of the corresponding continuum transport kernel remains an open question. Specifically, it is unclear under what conditions the underlying radiative interactions yield a scale-independent, local thermal conductivity (Fourier's law) and what continuum description is required when long-range transfer channels prevent such a local limit. Conventional heat conduction coarse-grains to a local constitutive relation, but radiative transport is intrinsically nonlocal because energy is exchanged directly between spatially separated constituents.

Methodology
The authors develop a nonlocal hydrodynamic theory by coarse-graining the microscopic electrodynamic interactions of a system of $N$ reciprocal thermal emitters. The methodology proceeds through the following steps:

Microscopic Formulation: Starting from the linearized energy balance equation for discrete particles, the system is modeled as a weighted spatial network where radiative thermal conductance $G_{ij}$ defines transfer rates $\Gamma_{ij}$ between emitters $i$ and $j$ .
Coarse-Graining Construction: The discrete temperature field is smoothed over a representative elementary volume (REV) of characteristic size $\ell$ using a smoothing kernel $W_\ell$ . This defines a continuum temperature field $T_\ell(\mathbf{r}, t)$ under the scale-separation condition $d_0 \ll \ell \ll L_T$ (where $d_0$ is inter-emitter spacing and $L_T$ is the macroscopic temperature variation scale).
Current Construction: Following Irving-Kirkwood constructions, the continuum heat current is derived by distributing the power exchanged along microscopic bonds $(i, j)$ over the connecting segments. The thermodynamic driving force is sampled continuously along these bonds.
Derivation of Constitutive Relation: By summing contributions from all unordered pairs, the authors derive a nonlocal constitutive relation where the heat current $\mathbf{J}_\ell(\mathbf{r}, t)$ depends on temperature gradients over an extended region via a two-point transport kernel $\mathbf{K}_\ell(\mathbf{r}, \mathbf{r}')$ .
Analysis of Limits: The paper investigates the conditions under which this nonlocal kernel reduces to a local conductivity. This involves analyzing the convergence of the spatial moments of the microscopic transfer network, specifically the second spatial moment $M_2$ .

Key Contributions and Results

Nonlocal Hydrodynamic Framework: The central result is the derivation of a nonlocal hydrodynamic equation (Eq. 18) directly from the microscopic radiative network. The transport kernel $\mathbf{K}_\ell$ explicitly encodes the strength, range, and geometry of radiative channels. Energy conservation retains its local conservative form, but the constitutive relation is intrinsically nonlocal.
Emergence of Fourier's Law: Fourier's law is shown to emerge as a controlled long-wavelength limit only when the second spatial moment of the microscopic transfer network converges. In this limit, the kernel becomes effectively local ( $\mathbf{K} \to \kappa_{rad} \delta(\mathbf{r}-\mathbf{r}')$ ), and the radiative diffusivity is determined by the converged second moment.
Persistence of Nonlocality: When the second spatial moment fails to converge (e.g., when the transfer rate decays as $\Gamma(r) \sim r^{-\alpha}$ with $\alpha \leq d+2$ in $d$ dimensions), spatial nonlocality survives coarse-graining. In such cases, increasingly distant radiative channels continue to renormalize the effective transport coefficient, preventing the reduction to a local conductivity.
Numerical Validation: The theory is applied to one-dimensional chains of Au and SiC nanoparticles.
- For these systems, the pairwise conductance scales as $G(r) \sim r^{-2}$ . In 1D, this leads to a second moment $M_2(\ell)$ that grows linearly with the coarse-graining length $\ell$ rather than saturating.
- This linear growth confirms that no scale-independent radiative diffusivity exists for these chains over the considered length scales.
- The resulting nonlocal kernel exhibits significant off-diagonal weight, particularly in SiC due to surface-phonon-polariton enhancement, indicating strong long-range coupling.
Stationary Profiles: The nonlocal theory predicts that a uniform heat current does not necessarily imply a uniform temperature gradient. While the temperature profiles in the studied chains remain nearly linear, the nonlocal theory captures finite deviations from the Fourier solution, which are more pronounced in SiC than in Au.

Significance
The paper establishes a systematic bridge between discrete many-body fluctuational electrodynamics and a continuous description of radiative heat transport. It clarifies that the emergence of a local thermal conductivity is not guaranteed but is a specific limit dependent on the convergence of spatial moments of the interaction network. When this condition is not met, the continuum description must remain nonlocal. This framework enables the treatment of large and geometrically complex many-body systems where long-range radiative interactions dominate, and it provides a natural route for combining local conductive mechanisms (phononic/electronic) with nonlocal radiative exchange in a unified continuum description.

Technical Summary: Emergence of Nonlocal Hydrodynamics in Many-Body Radiative Heat Transport

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