Nonlocal Einstein Relation in Confined Phonon Hydrodynamics

Einstein relations connect carrier mobility to macroscopic transport coefficients and constitute a fundamental link between microscopic dynamics and transport. In phonon hydrodynamics, this connection is conventionally local: the drift-induced thermal conductivity is proportional to the local drift mobility. Here we show that this relation becomes intrinsically nonlocal under spatial confinement. We derive a nonlocal Einstein relation, $κ_H(y) = \int P(y-y')μ(y'),dy'$, in which hydrodynamic thermal conductivity $κ_H(y)$ at one position depends on collective drift mobility $μ(y')$ across the entire cross-section of the channel. In the small-Knudsen-number limit, our nonlocal theory recovers the viscous heat equation and the local Einstein relation. At finite Knudsen number, nonlocality produces intrinsic boundary slip and distinct size-scaling behavior of the drift mobility $μ$ and hydrodynamic thermal conductivity $κ_H(y)$. First-principles calculations for graphite show that nonlocality broadens the channel-size window over which thermal conductivity exhibits superlinear size dependence. Temperature-dependent simulations further show that the nonlocal theory captures the emergence of a hydrodynamic peak in thermal conductivity normalized by ballistic conductance, whereas the local theory fails to accurately predict its magnitude. Our results establish a microscopic connection between collective phonon mobility and heat conduction beyond local hydrodynamics.