Mori–Zwanzig theory 603to generate the orthogonal dynamics of exp(QiLt) (Darveet al., 2009). Taken as a
phenomenological theory, however, eqn. (15.7.26) implies that if the potential of mean
force is harmonic, and a memory function can be obtained that faithfully represents
the dynamics of the full bath, then the GLE will yield accurate dynamical properties
of a system.
If we wish to use eqn. (15.7.26) to generate dynamical properties ina low-dimensional
subspace of the original system, then several subtleties arise. Consider a one-dimensional
harmonic oscillator with coordinatex, momentump, reduced massμ, frequencyω, and
potential minimum atx 0. For this problem, the force-constant matrix can be shown
to be
iΩ=(
0 1 /μ
−μ ̃ω^20)
(15.7.27)
(see Problem 15.9), where ̃ωis the renormalized frequency, which can be computed
using
ω ̃^2 =kT
μ〈(x−〈x〉)^2 〉. (15.7.28)
More importantly, the memory kernelK(t) is not a simple autocorrelation function.
A closer look at eqn. (15.7.25) makes clear that the required autocorrelation function
is
K(t) =〈
(
eQiLtF)
F†〉〈AA†〉−^1 , (15.7.29)
which requires the orthogonal dynamics generated by exp(QiLt). For this example, it
can be shown that
K(t) =(
0 0
0 ζ(t)/μ)
(15.7.30)
and that
F(t) =(
0
p ̇+μω ̃^2 q)
, (15.7.31)
whereq=x−〈x〉. If we denote the nonzero component ofF(t) asδf, we can express
the exact friction kernel as
ζ(t)
μ
=
〈δfeQiLtδf〉
〈p^2 〉, (15.7.32)
which is nontrivial to evaluate. The standard autocorrelation function
φ(t)
μ=
〈δfeiLtδf〉
〈p^2 〉(15.7.33)
is not equal to the friction kernel. It was shown by Berneet al.(1990) that the Laplace
transforms ofζ(t) andφ(t) are related by
ζ ̃(s)
μ=
[φ ̃(s)/μ]
1 −{
[s/(s^2 + ̃ω^2 )][φ ̃(s)/μ]}, (15.7.34)
from which we can see thatζ ̃(s) andφ ̃(s) are equal only in the limit thats→0. They
are also equal when ̃ω→ ∞, and hence, the standard correlation functionφ(t) is a