1549380323-Statistical Mechanics Theory and Molecular Simulation

Frequency spectra 547

event by a factor of exp(−β ̄hω) in a canonical distribution, for which the probability
of finding the system with a high initial energyEiis smaller than that for finding the
system with a smaller initial energy. Eqn. (14.3.14) is a consequence of the statistical
distribution of initial states; in fact, the individual transition ratesRfi(ω) satisfy the
microscopic reversibility conditionRfi(ω) =Rif(ω). If we followed all of the individual
transitions of an ensemble of systems, they would all obey microscopic reversibility.
However, because we introduce a statistical distribution, we no longer retain such a
detailed microscopic picture, and the ensemble averaged absorption and emission rates,
R(ω) andR(−ω), do not obey the microscopic reversibility condition.
If the analysis leading from eqn. (14.3.3) to eqn. (14.3.7) is carried out on eqn.
(14.3.9), the result is

R(−ω) =

1

̄h^2

|F(ω)|^2

∫∞

−∞

dte−iωt〈Vˆ(t)Vˆ(0)〉, (14.3.15)

and sinceR(−ω) 6 =R(ω), it follows that the correlation functions〈Vˆ(0)Vˆ(t)〉and
〈Vˆ(t)Vˆ(0)〉are not equal. This could also have been gleaned from the fact that the
commutator [Vˆ(0),Vˆ(t)] does not vanish.
We now define the net energy absorption spectrumQ(ω) as the net energy absorbed
per unit time at frequencyω. Since the energy absorbed is just ̄hω, and thenetrate
is the difference between the absorption and emission ratesR(ω)−R(−ω), the energy
spectrumQ(ω) is given by

Q(ω) = [R(ω)−R(−ω)] ̄hω= ̄hωR(ω)

(

1 −e−βhω ̄

)

. (14.3.16)

Note, however, that sinceR(−ω) = exp(−β ̄hω)R(ω), it follows that

R(ω) +R(−ω) =

(

1 + e−β ̄hω

)

R(ω) (14.3.17)

or

R(ω) =

R(ω) +R(−ω)

1 + e−β ̄hω

. (14.3.18)

Using eqn. (14.3.7) and eqn. (14.3.15), we express the sumR(ω) +R(−ω) as

R(ω) +R(−ω) =

1

̄h^2

|F(ω)|^2

∫∞

−∞

dte−iωt〈Vˆ(0)Vˆ(t) +Vˆ(t)Vˆ(0)〉. (14.3.19)

Let us now define a new operator bracket

[A,ˆBˆ]+=AˆBˆ+BˆAˆ (14.3.20)

known as theanticommutatorbetweenAˆandBˆ. It is straightforward to see that the
anticommutator is manifestly Hermitian. Inserting the anticommutator definition into
eqn. (14.3.19), we obtain

R(ω) +R(−ω) =

1

̄h^2

|F(ω)|^2

∫∞

−∞

dte−iωt

〈[

Vˆ(0),Vˆ(t)

]

+

〉

. (14.3.21)