Thermal Conduction of Graphene - Nanoscale Energy Transport and Harvesting - page 102

1. G > and G < : they are usually regarded as correlation functions,
and are directly related to observable physical quantities (e.g.,
thermal current, density of states and so on) and dynamic
properties.
2. G r and G a : they have good analytical properties, facilitate
calculations of physical response, and also have simple
relations to observablephysical quantities.

In equilibrium or steady states, the system is time-translation
invariant. The Green's functions depend only on the difference in
time. Then it is more convenient to go from time space to frequency
space through Fourier transformation:

+∞

G ( t ) e i ω t dt .

ω =

G [ ] (3.22)

−∞

In frequency space, there are relations:

G < [ G > [ ] T

− ω = ω

] (3.23)

G < [ ω ] † =− G < [ ω ] (3.24)

] ∗ G r [ G r [

− ω

]

= ω

(3.25)

G r [ ω ] † = G a [ ω ] (3.26)

For steady states, only two Green's functions are linearly inde-
pendent. Thisnumberdecreases to onefor equilibriumstates, since
equilibrium systems satisfy the fluctuation-dissipation theorem,
whichgivesrisetoanadditionalrelationbetweenGreen'sfunctions.

G < [ )( G r [ G a [

ω

]

=

f (

ω ω

]

− ω

]), (3.27)

where f is the Bose-Einstein distribution function. The correlation
function G < contains information of fluctuation, while G r

− G a
describes dissipation of the system. The fluctuation-dissipation
theorem tells us that fluctuation is proportional to dissipation in
equilibriumstates.