Nonlocal Einstein Relation in Confined Phonon Hydrodynamics
Abstract
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, κH(y) = ∫ P(y−y')μ(y'),dy', in which hydrodynamic thermal conductivity κH(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 κH(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.
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BibTeX
@misc{quan2026nonlocal,
title = {{Nonlocal Einstein Relation in Confined Phonon Hydrodynamics}},
author = {Guanda Quan and Ronggui Yang and Xin Qian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35109},
eprint = {2610.35109},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35109},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Nonlocal Einstein Relation in Confined Phonon Hydrodynamics
AU - Guanda Quan
AU - Ronggui Yang
AU - Xin Qian
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35109
UR - https://arxiv.org/abs/2610.35109
AB - 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) = ∫ 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.
ER -