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Landauer formula for phonon heat conduction: relation between energy transmittance and transmission coefficient
The heat current across a quantum harmonic system connected to reservoirs at
different temperatures is given by the Landauer formula, in terms of an
integral over phonon frequencies \omega, of the energy transmittance T(\omega).
There are several different ways to derive this formula, for example using the
Keldysh approach or the Langevin equation approach. The energy transmittance
T({\omega}) is usually expressed in terms of nonequilibrium phonon Green's
function and it is expected that it is related to the transmission coefficient
{\tau}({\omega}) of plane waves across the system. In this paper, for a
one-dimensional set-up of a finite harmonic chain connected to reservoirs which
are also semi-infinite harmonic chains, we present a simple and direct
demonstration of the relation between T({\omega}) and {\tau}({\omega}). Our
approach is easily extendable to the case where both system and reservoirs are
in higher dimensions and have arbitrary geometries, in which case the meaning
of {\tau} and its relation to T are more non-trivial.Comment: 17 pages, 1 figur
Heat conduction in the \alpha-\beta -Fermi-Pasta-Ulam chain
Recent simulation results on heat conduction in a one-dimensional chain with
an asymmetric inter-particle interaction potential and no onsite potential
found non-anomalous heat transport in accordance to Fourier's law. This is a
surprising result since it was long believed that heat conduction in
one-dimensional systems is in general anomalous in the sense that the thermal
conductivity diverges as the system size goes to infinity. In this paper we
report on detailed numerical simulations of this problem to investigate the
possibility of a finite temperature phase transition in this system. Our
results indicate that the unexpected results for asymmetric potentials is a
result of insufficient chain length, and does not represent the asymptotic
behavior.Comment: 14 pages, 6 figure
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