Environmental Engineering Reference
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essential steps. There is also a Kubo formula for electrons [53],
which relates electrical conductivity to the electrical current auto-
correlation function. For the electrical case, there is a mechanical
disturbance term [60] in the Hamiltonian, which describes the
energy perturbation due to the applied external electrical field.
The derivation of electrical Kubo formula follows the standard
perturbationtheory.Forthethermalcase,however,thederivationof
GKFislessstraightforwardbutreliesonthestatisticalhypothesisof
localthermalequilibrium,whichcanbedescribedbythelocalspace-
dependent temperature T ( r )
( r )] 1 .
Consider a system with Hamiltonian H and volume V at
temperature T ( r ) subjected to a small thermal disturbance
=
[ k B β
δ
T ( r ).
The local thermal equilibrium holds and can be described by the
local equilibriumdensity matrix:
ρ = exp
d 3 r β ( r ) h ( r )
/ Z ,
(1.37)
where Z is the partition function, and h ( r ) is the Hamiltonian
density operator related to the Hamiltonian as H = d 3 r h ( r ).
Similarly, the heat current density operator s ( r )isdefinedas
d 3 rs ( r ),
S =
(1.38)
where S is the total heat current operator. Due to the energy
conservation, these two density operators are related through
h ( r )
t +∇· s ( r ) = 0.
(1.39)
Without δ T ( r ), the system is in thermal equilibrium so that
T ( r ) = T 0 =
[ k B β 0 ] 1 is a constant, and there is no net heat current
( J
0). Afterapplying thesmallthermal disturbance, the
density matrix becomes
ρ = exp β ( H + H ) / Z , (1.40)
where H isthesmallperturbationtotheHamiltoniancausedbythe
thermal disturbance. There is a fundamental equality in quantum
mechanics for any two operators a and b [60] so that
=
Tr[
ρ 0 S ]
=
= e β a 1 +
d λ e λ a be λ ( a + b ) ,
β
e β ( a + b )
(1.41)
0
 
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