Non-Hermitian Exceptional Dynamics in First-Order Heat Transport

TL;DR

This paper introduces a unified non-Hermitian operator framework for heat transport, revealing exceptional point physics as fundamental to thermal dynamics, bridging ballistic, diffusive, and wave-like regimes.

Contribution

It develops a minimal first-order dynamical model that naturally incorporates Fourier and Cattaneo laws within a non-Hermitian spectral structure.

Findings

Spectral structure governed by an exceptional point separates diffusion and wave-like propagation.

Nonanalytic spectral transitions and nonmodal transient dynamics are induced by the exceptional point.

The framework generalizes to anisotropic media, enabling intrinsic steering of heat flow.

Abstract

Heat transport exhibits distinct regimes ranging from ballistic propagation to diffusive relaxation, traditionally described by disparate theoretical frameworks. Here, we introduce a unified first-order operator formulation in which temperature and heat flux are treated as a coupled state vector, yielding a minimal dynamical closure of heat transport. The resulting generator is intrinsically non-Hermitian and gives rise to a spectral structure governed by an exceptional point that separates overdamped diffusion from underdamped wave-like propagation. In this framework, Fourier law emerges as a singular limit of a hyperbolic dissipative system, while the Cattaneo equation arises naturally as the minimal hydrodynamic closure of kinetic theory. We show that the exceptional point induces nonanalytic spectral transitions, nonmodal transient dynamics, and a breakdown of conventional modal decomposition. The theory further generalizes to anisotropic media, where direction-dependent exceptional surfaces enable intrinsic steering of heat flow. Our results establish a unified non-Hermitian dynamical framework for heat transport and reveal exceptional-point physics as a fundamental organizing principle underlying thermal dynamics across scales.