Environmental Engineering Reference
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Figure 3.3 A representative thermal-transport system composed of a
center part (C)as well as thermal leads on the left (L) and right (R).
hand is very di cult for the present theory, on the other hand
is unnecessary because we are mainly interested in scatterings
happened in the center part. Therefore, for simplicity we assume
that all the phonon scatterings occur in the center part only.
Since phonons are the thermal-transport carriers here, the
Hamiltonian ( H ) of thermal-transport systems is in fact the
Hamiltonian of phonon systems.
=
H α +
+
H
V
H n .
(3.12)
α = L,C,R
H L H C ,and H R are the harmonic Hamiltonian of the decoupled
left, center and right parts of the transport system.
1
1
2 ( u α ) T u α +
2 u T
D αα u α ( α = L, C, R), (3.13)
where u α = ( u 1 , u 2 , u 3 , ··· ) T ; u i = M i x i ; M i and x i respectively
represent atomic mass and displacement of the i th degree of
freedom in the α part; u α is the conjugated momentum of u α ; D is
the mass-reduced force constantmatrix.
H α =
α
D LL D LC 0
D CL D CC D CR
0 D RC D RR
.
D =
(3.14)
Here, by default the coupling between the left and the right
thermal leads are zero. In principle, this condition can be satisfied
always by enlarging the size of the center part so as to decouple the
 
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