Generalized Fourier's law in mesoscopic systems

Abstract: Fourier's law fails when the mean free path of the energy carriers becomes comparable to the length and time scales over which the temperature field varies. We derive a thermodynamically consistent generalization in which the conductivity is promoted to a nonlocal memory operator $\bmκ<em>{\mathrm{eff}}(\mathbf{k},ω)$, obtained by combining mesoscopic nonequilibrium thermodynamics with the Mori--Kubo--Zwanzig projection-operator formalism. The Onsager kernel decomposes exactly into a tensor sum over vibrational normal modes weighted by their Bose heat capacities and relaxation functions, and satisfies the second law by construction. Two consequences follow. First, because the modal weights carry the directional group velocity, the kernel is anisotropic, so a nominally isotropic crystal exhibits direction-dependent apparent conductivities $Λ_z\neqΛ_r$. Second, in a pump--probe experiment the modulation frequency does not introduce temporal memory but sets the probed wavevector through the thermal penetration depth, so the suppression of the apparent conductivity measured by time-domain thermoreflectance on Si, Ge and Si$</em>{1-x}$Ge$_x$ is a spatial-nonlocality effect set by a sub-micron carrier mean free path. Fitting the data of Wilson and Cahill yields nonlocality lengths of $0.25$--$0.4~μ$m consistent with the mean-free-path spectra of these crystals. The framework supplies a thermodynamic foundation for the two-channel ballistic/diffusive picture of nondiffusive heat transport.

• The paper establishes a thermodynamically consistent generalization of Fourier's law by promoting conductivity to a nonlocal, tensorial memory operator.
• It employs mesoscopic nonequilibrium thermodynamics and modal decomposition to connect microscopic carrier dynamics with anisotropic heat transport phenomena observed in experiments.
• Quantitative fits to TDTR data reveal a nonlocality length scale that captures size-dependent thermal conductivity in materials like Si, Ge, and SiGe alloys.

Generalized Fourier's Law and Nonlocal Heat Transport in Mesoscopic Systems

Motivation and Context

Classical heat conduction theory, encapsulated by Fourier's law, relies on the assumption that the carrier mean free path (MFP) is much shorter than the characteristic system length and timescale. This local diffusive framework fails in systems where MFP approaches or exceeds these scales, such as nanostructures, colloid suspensions, and biological organelles. The breakdown manifests through size-dependent thermal conductivity, direction-dependent apparent conductivities in nominally isotropic crystals, and non-Gaussian temperature profiles. Existing kinetic models (e.g., phonon Boltzmann transport equation) reproduce these phenomena but are computationally intensive, while phenomenological extensions (e.g., Cattaneo-Vernotte) lack a rigorous thermodynamic basis.

Theoretical Framework

The paper introduces a thermodynamically consistent generalization of Fourier's law using mesoscopic nonequilibrium thermodynamics (MNET) and the Mori-Kubo-Zwanzig projection-operator formalism. The conductivity is elevated to a nonlocal memory operator $K_{\rm eff}(k, \omega)$, whose Onsager kernel decomposes exactly into a tensor sum over vibrational normal modes weighted by Bose heat capacities and relaxation functions. This formulation rigorously respects the second law of thermodynamics by construction, with the symmetric part of the nonlocal kernel positive semi-definite in all regimes.

The generalized constitutive relation is given by:

$$J(\mathbf{r}, t) = \int d\mathbf{r}' \int dt'~ L_{qq}(\mathbf{r}, \mathbf{r}', t - t') \nabla T(\mathbf{r}', t')$$

and, in reciprocal space, as:

$$J_a(\mathbf{k}, \omega) = -i K_{\rm eff}(\mathbf{k}, \omega) \mathbf{k} \delta T(\mathbf{k}, \omega)$$

The modal decomposition, derived via the GLE, gives:

$$L_{A} (\mathbf{r}, \mathbf{r}',t) = \sum_{\mu\nu} v_\mu(\mathbf{r}) v_\nu(\mathbf{r}') R_{\mu\nu}(t) C_{\mathrm{ph}}(\omega_\nu, T)$$

where $v_\mu$ is the group velocity for mode $\mu$, $R_{\mu\nu}(t)$ encapsulates relaxation dynamics, and $C_{\mathrm{ph}}$ is the quantum Bose heat capacity for mode $\nu$.

Directional Anisotropy and Spatial Nonlocality

A key result is that the modal kernel is inherently anisotropic due to the directional group velocities of modes. Even in a nominally isotropic crystal, time-domain thermoreflectance (TDTR) experiments exhibit direction-dependent apparent conductivities (azimuthal $A_z \neq A_r$). This arises because TDTR samples the kernel at different spatial wavevectors depending on modulation frequency and spot size, thereby probing effective thermal length scales ($$J(\mathbf{r}, t) = \int d\mathbf{r}' \int dt'~ L_{qq}(\mathbf{r}, \mathbf{r}', t - t') \nabla T(\mathbf{r}', t')$$0 for through-plane, $$J(\mathbf{r}, t) = \int d\mathbf{r}' \int dt'~ L_{qq}(\mathbf{r}, \mathbf{r}', t - t') \nabla T(\mathbf{r}', t')$$1 for in-plane).

The spatial nonlocality effect produces apparent suppression in the measured conductivity at high frequencies, which is not due to temporal memory but the spatially probed wavevector. The paper demonstrates that the frequency-dependent suppression of apparent conductivity observed in Si, Ge, and Si$$J(\mathbf{r}, t) = \int d\mathbf{r}' \int dt'~ L_{qq}(\mathbf{r}, \mathbf{r}', t - t') \nabla T(\mathbf{r}', t')$$2Ge$$J(\mathbf{r}, t) = \int d\mathbf{r}' \int dt'~ L_{qq}(\mathbf{r}, \mathbf{r}', t - t') \nabla T(\mathbf{r}', t')$$3 alloys is quantitatively accounted for by a simple Lorentzian wavevector dependence:

$$J(\mathbf{r}, t) = \int d\mathbf{r}' \int dt'~ L_{qq}(\mathbf{r}, \mathbf{r}', t - t') \nabla T(\mathbf{r}', t')$$4

with $$J(\mathbf{r}, t) = \int d\mathbf{r}' \int dt'~ L_{qq}(\mathbf{r}, \mathbf{r}', t - t') \nabla T(\mathbf{r}', t')$$5 a nonlocality length scale consistent with dominant carrier MFPs.

Quantitative Results and Experimental Validation

Fits to TDTR data (Wilson and Cahill) for Si, Ge, and SiGe alloys yield nonlocality lengths ranging from 0.25–0.41 $$J(\mathbf{r}, t) = \int d\mathbf{r}' \int dt'~ L_{qq}(\mathbf{r}, \mathbf{r}', t - t') \nabla T(\mathbf{r}', t')$$6m, matching the experimentally determined MFP spectra. Pure Si and heavily doped Si:B remain flat across the frequency band, reflecting unresolved $$J(\mathbf{r}, t) = \int d\mathbf{r}' \int dt'~ L_{qq}(\mathbf{r}, \mathbf{r}', t - t') \nabla T(\mathbf{r}', t')$$7 due to high bulk conductivity, while alloys demonstrate roll-off at MHz frequencies with well-constrained values for $$J(\mathbf{r}, t) = \int d\mathbf{r}' \int dt'~ L_{qq}(\mathbf{r}, \mathbf{r}', t - t') \nabla T(\mathbf{r}', t')$$8. The framework also accounts for the deepening suppression with increased alloying, as enhanced mass-disorder scattering shortens MFP, strikingly consistent with measured conductivity accumulation functions.

Further, the modal kernel clarifies the origin of interfacial nonequilibrium resistance observed in TDTR interfacial measurements. The truncation in mode overlap near boundaries leaves off-diagonal correlations that fail to relax, demonstrating residual excess resistance within a layer of thickness $$J(\mathbf{r}, t) = \int d\mathbf{r}' \int dt'~ L_{qq}(\mathbf{r}, \mathbf{r}', t - t') \nabla T(\mathbf{r}', t')$$9. This links the diffusive reservoir and the ballistic channel of two-fluid descriptions on a common entropy-producing foundation.

Transport Regime Classification

The memory kernel's low-frequency spectral structure classifies transport regimes, recovering classical diffusion (Ohmic), sub/superdiffusive regimes, ballistic saturation, and the Cattaneo-Vernotte equation as limiting cases. The framework thus unifies disparate nonlocal transport phenomena—including anomalous diffusion, second-sound, and nondiffusive heat transport—within a single entropy-consistent formalism.

Implications and Outlook

The generalization of Fourier's law to a nonlocal, tensorial memory kernel establishes a rigorous link between microscopic carrier dynamics and macroscopic observables, without recourse to ad hoc extensions or fitted relaxation times. The identification of a single length scale $$J_a(\mathbf{k}, \omega) = -i K_{\rm eff}(\mathbf{k}, \omega) \mathbf{k} \delta T(\mathbf{k}, \omega)$$0 governing spatial nonlocality has direct implications for nanowire, superlattice, and quantum dot thermal management, and for the temperature dependence of nondiffusive transport through quantum modal weighting. The formulation is directly extensible to electronic transport, thermoelectric devices, and active quantum systems. The rigorous thermodynamic underpinning facilitates predictive modeling of heat transport in regimes where classical diffusion is inapplicable, and enables falsifiable experimental tests by connecting nonlocality lengths to the accumulation function of measured conductivity.

Conclusion

The paper provides a thermodynamically consistent, microscopically grounded extension of Fourier's law for mesoscopic systems, promoting conductivity to a nonlocal memory operator with modal decomposition and tensorial structure. Direction-dependent sampling yields apparent anisotropy in nominally isotropic crystals, and spatial nonlocality underlies frequency-resolved suppression of apparent conductivity in TDTR experiments. The nonlocality length $$J_a(\mathbf{k}, \omega) = -i K_{\rm eff}(\mathbf{k}, \omega) \mathbf{k} \delta T(\mathbf{k}, \omega)$$1 coincides with dominant carrier MFPs, and modal overlap truncation at interfaces explains interfacial resistance. The framework supplies a robust foundation for hydrodynamic and ballistic-diffusive theories of heat transport, and is broadly applicable to modern nanostructured and quantum thermal devices (2606.23909).