Heat Conduction Operators

• Heat Conduction Operators (HCOs) are mathematical operators that generalize Fourier’s law by integrating memory, spatial nonlocality, and dissipative effects.
• They derive from statistical-mechanical projection methods, enabling rigorous operator-based descriptions of heat transport across ultrafast and nanoscale regimes.
• HCOs support practical applications from modeling nanoscale thermal phenomena to enhancing computational methods in imaging and composite media homogenization.

A Heat Conduction Operator (HCO) is a mathematical entity—often an integro-differential, spatiotemporal, or spectral operator—that quantitatively encodes the constitutive law relating heat flux to the temperature gradient, including memory, nonlocal, and dissipative effects. HCOs encompass and generalize Fourier’s law, yielding the operator framework for describing heat transport across ultrafast, nanoscale, inhomogeneous, and non-Fourier regimes. They arise naturally from both statistical-mechanical projection methods (e.g., Zwanzig-Mori), thermodynamic formalisms, and homogenization in composite or structured media, providing exact or systematically improvable descriptions of heat transfer mechanisms beyond classical phenomenology.

1. Microscopic Origins and Projection-Operator Formalism

The modern HCO formalism is rooted in Zwanzig’s statistical theory of irreversible processes, with the Mori–Zwanzig projection yielding a generalized Langevin equation (GLE) for heat-flux operators. The heat-flux correlation (memory) kernel $Z(t)$ is given by

$$Z(t) = \frac{1}{k_B T^2 V} \langle \widehat{\mathbf{J}}_q(0) \cdot \widehat{\mathbf{J}}_q(t) \rangle_{eq},$$

where $\widehat{\mathbf{J}}_q(t)$ is the microscopic heat-flux operator, $k_B$ Boltzmann’s constant, $T$ the equilibrium temperature, and $V$ the system volume (Crawford et al., 31 Dec 2024). The constitutive law in the time domain becomes the generalized Fourier law: $$\mathbf{q}(t) = - \int_0^t Z(t - \tau) \nabla T(\tau) d\tau.$$ This form naturally incorporates intrinsic memory and applies at all temporal and spatial scales. In nonuniform or structurally complex media, a fully spatiotemporal operator kernel $K_{ij}(\mathbf{r}, \mathbf{r}', \tau)$ replaces $Z(t)$: $$q_i(\mathbf{r}, t) = - \int_0^\infty d\tau \int_V d^3 r'\, K_{ij}(\mathbf{r}, \mathbf{r}', \tau) \, \partial_j T(\mathbf{r}', t - \tau),$$ with $K_{ij}$ corresponding to space-time correlations of local heat fluxes, thus encoding both temporal memory and spatial nonlocality (Zeng et al., 8 Jul 2025).

2. Operator Structure, Spectral Representations, and Reduction to Classical Laws

The convolutional HCO kernel $Z(t)$ or $K_{ij}(\mathbf{r}, \mathbf{r}', \tau)$ is fundamentally more general than classical local or first-order time-local laws. Taking Laplace transforms yields frequency-domain operators: $\widehat{Z}(s) = \int_0^\infty e^{-s t} Z(t) dt,$ so the flux in Laplace space is

$Q(s) = -\widehat{Z}(s) \nabla T(s).$

$\operatorname{Re} \widehat{Z}(i \omega)$ is the dissipative component; $\operatorname{Im} \widehat{Z}(i \omega)$ characterizes wave-like (second-sound) regimes.

In the long-memory, slow-gradient (“steady-state”) or long-wavelength/low-frequency limit, the operator reduces to classical conductivity: $$\mathbf{q}(t) \approx -\left( \int_0^\infty Z(\tau) d\tau \right) \nabla T(t) = -\kappa \nabla T(t),$$ where $\kappa$ is the Green–Kubo thermal conductivity. Higher-order Taylor expansions of spatially nonlocal HCOs generate the Guyer–Krumhansl, Maxwell–Cattaneo–Vernotte, and hierarchy of high-order tensorial conductivity corrections (Zeng et al., 8 Jul 2025).

3. Hierarchies, Generalizations, and Memory Kernels

HCOs can be systematically generalized to include multiple time and length scales:

• Hierarchies of HCOs $HCO_n[T]$ (with $n$ iterated memory relaxations) yield finite combinations of exponentials as memory kernels, unifying Maxwell–Cattaneo, Gurtin–Pipkin, and Moore–Gibson–Thompson laws through Volterra-type or high-order differential equations (Dell'Oro et al., 2022). The general PDE form is:

$$HCO_n[T](x, t) = \partial_t T(x, t) + \int_0^t G_n(t - s) \Delta T(x, s) ds = 0,$$

where $G_n(t)$ is a sum of exponentials encoding multiple relaxation modes.

• In rate-type thermodynamic models, the HCO is a rational operator polynomial:

$$\mathcal{H}(\partial_t) = \frac{k_0 + k_1 \partial_t + \cdots}{\tau_0 + \tau_1 \partial_t + \cdots},$$

covering all classical and higher-order non-Fourier laws as special cases (Giorgi et al., 1 Feb 2024).

In inhomogeneous or composite systems, homogenization/cluster-expansion approaches can rigorously define effective HCOs involving spatially dependent coefficients or integral operators, with quantitative error bounds derived from asymptotic analysis (Sini et al., 2019). In fractal or hierarchical materials, the Laplacian and hence the HCO is generalized to noninteger dimensions, impacting decay rates and transport behavior (Lin et al., 2017).

4. Numerical Implementation and Atomistic Foundations

Practical HCO deployment relies on accurately estimating $Z(t)$ or $K_{ij}(\mathbf{r}, \mathbf{r}', \tau)$. The Green–Kubo formalism enables direct computation from equilibrium atomistic simulations: $$Z(t) = \frac{1}{k_B T^2 V} \langle \widehat{\mathbf{J}}_q(0) \cdot \widehat{\mathbf{J}}_q(t) \rangle_{eq},$$ or, for nonuniform media,

$$K_{ij}(\mathbf{r}, \mathbf{r}', \tau) = \frac{1}{k_B T_0^2} \langle j_i(\mathbf{r}, 0) j_j(\mathbf{r}', \tau) \rangle_{eq}.$$

In practice, $Z(t)$ is often modeled as a Prony series,

$$Z(t) \sim \sum_{\lambda=1}^M \zeta_{\lambda 0} e^{-\gamma_\lambda t},$$

allowing the convolution to be rewritten as $M+1$ coupled PDEs for $T$ and auxiliary mode-flux fields. This approach preserves causality and reduces the computational cost of evaluating memory integrals to updating local variables (Crawford et al., 31 Dec 2024).

Spatially-discretized or spectral HCOs can be implemented in classical, neural, or hybrid architectures (e.g., DCT-diagonalized diffusion in U-Mamba for deep medical image segmentation). In such architectures, the HCO acts as a global, learnable low-pass filter, rapidly propagating context with $O(N\log N)$ complexity and provable eigen-spectral stability (Wu et al., 5 Nov 2025).

5. Comparison with Phenomenological Theories and Physical Interpretation

Unlike traditional models (e.g., Fourier, Maxwell–Cattaneo, Guyer–Krumhansl), the HCO framework is rigorously derived from first principles, with parameters computed from microscopic physics rather than fitted phenomenologically. The dependence of memory, nonlocal effects, and interfacial discontinuities on atomic-scale mechanisms is explicit via the equilibrium time-correlation functions of fluxes. HCOs avoid ad hoc constructs such as “phonon drift velocity” and provide an unambiguous, operator-level relation between temperature gradients and heat flux valid at all scales and in all regimes (diffusive, wave-like, ballistic, or anomalous).

HCOs also seamlessly recover well-established boundary conditions such as Kapitza conductance at sharp interfaces, by integrating the operator kernel against a temperature jump across the interface (Zeng et al., 8 Jul 2025).

6. Extensions: Stochastic and Gradient-Flow Structures

In stochastic lattice models (e.g., the Brownian Energy Process, Kipnis–Marchioro–Presutti process) and associated large-deviation theory, HCOs emerge as Onsager operators driving hydrodynamic limits; the HCO then determines the metric and dissipation structure underlying gradient flows for energy profiles. Unlike mass-diffusion cases (which yield Wasserstein metrics and linear mobilities), heat conduction HCOs exhibit quadratic mobility in the fluctuating variable and a logarithmic “entropy” function, endowing the gradient flow with fundamentally different geometric properties (Peletier et al., 2014).

7. Applications and Physical Ramifications

HCO-based frameworks have proven essential in:

• Predictive modeling of ultrafast, nanoscale, or nonlocal heat flow in solids, nanostructures, and heterogeneous materials
• Design/interpretation of thermal grating, pump-probe, and pulsed heat transport experiments in EUV/X-ray regimes (Crawford et al., 31 Dec 2024, Zeng et al., 8 Jul 2025)
• Medical image segmentation, where HCOs provide stable, physically meaningful context propagation operators in frequency space (Wu et al., 5 Nov 2025)
• Optimization and homogenization of effective conductivity in media with structured inclusions, fractal networks, or random sinks (Sini et al., 2019, Lin et al., 2017)
• Unification of non-Fourier, hyperbolic, and memory/cross-over heat transfer models in a single thermodynamically consistent framework (Giorgi et al., 1 Feb 2024, Ván et al., 2011, Kovács et al., 2014).

The HCO concept, by embedding full spatiotemporal memory and nonlocality directly anchored in microscopic dynamics, thus provides the mathematically and physically rigorous foundation for both advanced modeling and next-generation experimental and computational methodologies in heat transport science.