588 Langevin and generalized Langevin equations
vibrational and energy relaxation phenomena as an application of the GLE in greater
detail in Section 15.4.) According to eqn. (13.3.39), the velocity autocorrelation func-
tion of a harmonic oscillator in isolation is proportional to cosωt, whereωis the bare
frequency of the oscillator. In this case, the correlation functiondoes not decay because
the system retains infinite memory of its initial condition. However, when coupled to a
bath, the oscillator exchanges energy with the bath particles via collision events and,
as a result, loses memory of its initial state on a time scaleT 2. If there is a very large
disparity of frequencies between the oscillator and the bath, the coupling between
them will be weak andT 2 will be long, whereas if the oscillator frequency lies near or
within the spectral density of the bath, vibrational energy exchange will occur readily
andT 2 will be short. Thus,T 2 is an indicator of the strength of the coupling between
the oscillator and the bath. As the frequency of the oscillator is increased, the coupling
between the oscillator and the bath becomes weaker, andγ′( ̃ω) decreases. According
to eqn. (15.3.34), this means that the correlation functionsCqq(t) andCvv(t) decay
more slowly, and the number of oscillations that can cycle through onthe time scale
T 2 grows. Two examples of the velocity autocorrelation functionCvv(t) are shown in
Fig. 15.2. In this example, the values of ̃ω,γ′( ̃ω), andγ′′( ̃ω) correspond to a harmonic
0 5 10 15 20
t /T-1
0
1
C
vv(t)
0 20 40 60 80
t /T-1
0
1
C
vv(t)
w = 60 w = 90Fig. 15.2Velocity autocorrelation function of the bond length of a harmonic diatomic cou-
pled to a Lennard-Jones bath as described in the text. The bond frequenciesω= 60 and
ω= 90 are expressed in units of
√
ǫ/(mσ^2 ).diatomic molecule of atomic type A coupled to a bath of A atoms interacting with each
other and with the molecule via a Lennard-Jones potential at reduced temperature
Tˆ=T/ǫ= 2.5 and reduced density ˆρ=ρσ^3 = 1.05. The frequenciesω= 60 and
ω= 90 are expressed in Lennard-Jones reduced frequency units
√
ǫ/(mσ^2 ). A method
for calculating the friction kernel for a high-frequency oscillator weakly coupled to a
bath will be discussed in Section 15.7.
15.4 Vibrational dephasing and energy relaxation in simple fluids
An application of the GLE that is of particular interest in chemical physics is the study
of vibrational and energy relaxation phenomena. As we noted in Section 15.3.3, quan-
tifying energy exchange between the system and the bath provides direct information
about the strength of the system–bath coupling. For a harmonic oscillator coupled