Unified Statistical Theory of Heat Conduction in Nonuniform Media

TL;DR

This paper develops a unified microscopic theory of heat conduction in nonuniform media using the Zwanzig formalism, capturing nonlocality, memory effects, and heterogeneity, and connecting microscopic dynamics to continuum models.

Contribution

It introduces a causal spatiotemporal kernel for heat conduction that unifies diffusive, ballistic, and hydrodynamic regimes, including interfacial transfer and atomistic evaluation.

Findings

Kernel recovers classical diffusion, nonlocal transport, and hydrodynamic models as limits.

Explicit kernel forms derived for crystalline and disordered solids.

Application to silicon shows nonlocality and memory effects influence heat transport.

Abstract

Using the Zwanzig projection-operator formalism, we derive a causal two-point spatiotemporal kernel for heat conduction, defined microscopically as a space-resolved equilibrium heat-flux time-correlation function, that encodes temporal memory, spatial nonlocality, and material heterogeneity on equal footing. Classical diffusion, nonlocal transport, and hydrodynamic models emerge as controlled asymptotic limits of this kernel, providing a unified constitutive description across diffusive, quasi-ballistic, and hydrodynamic regimes. Interfacial heat transfer is incorporated through a spatially resolved kernel formulation, in which the conventional Kapitza resistance arises as a coarse-grained limit. The kernel admits a spatiotemporal Green--Kubo representation and can, in principle, be evaluated from atomistic simulations for bulk media, providing a direct connection between microscopic dynamics and continuum transport without empirical closure. For crystalline solids, we derive explicit kernel forms in the hydrodynamic and attenuated-streaming limits and introduce a hybrid reduction that captures the coexistence of collective and quasi-ballistic transport. For disordered harmonic solids, the framework recovers a spatial diffusion kernel consistent with the Allen--Feldman limit. To illustrate the theory, we construct the kernel for silicon at room temperature within the relaxation-time approximation and apply it to transient thermal grating configurations. Spatial nonlocality associated with the phonon mean-free-path distribution is the primary source of deviation from Fourier transport under these conditions, while temporal memory mainly influences short-time dynamics. These findings identify the spatiotemporal kernel as a unifying constitutive descriptor whose coarse-grained limits recover conventional transport coefficients.