Calculating time correlation functions 515as an independent “sampling” of the correlation function because over each segment,
the correlation function decays to zero.
Suppose a molecular dynamics trajectory is long compared to the correlation time
and consists ofMpoints, xm∆t,m= 1,...,M. We can break the trajectory up into seg-
ments each havingKpoints, whereK≪MandK∆tis comparable to the correlation
time, and apply the procedure of eqn. (13.4.1) to each segment. For example, we could
choose x 1 ,...,xK∆tto be the first segment, x(K+1)∆t,...,x 2 K∆tto be the next, etc., and
apply eqn. (13.4.1) on these segments. Note, however, that we could just as well start
with x∆tand assume x∆t,...,x(K+1)∆tis the first segment, x(K+2)∆t,...,x(2K+1)∆tis
the next, etc. By segmenting the trajectory in as many different ways as possible, each
point of the trajectory serves both as an independent sampling off 0 (H(x)) and as a
time point in the correlation function. In fact, when eqn. (13.4.2) is written in discrete
form for a finite-time trajectory
CAB(n∆t) =1
M−nM∑−nm=1a(xm∆t)b(x(m+n)∆t), n= 0,...,K, (13.4.3)it is clear that eqn. (13.4.3) automatically exploits the idea of dividing the trajectory
in all possible ways. Each point of the trajectory, xm∆t, serves as an “initial condition”
from which the point x(m+n)∆ta timen∆tlater is determined, i.e., as an independent
sampling off 0 (H(x)), and as a point generated by an “initial condition” at some
earlier time in the trajectory. This approach is illustrated in Fig. 13.5(b). Indeed, as
nincreases, the number of time intervals that fit into the trajectory decreases, and
hence, the statistics degrade. This is why it is imperative to ensure that the trajectory
is long compared to the correlation time when using eqn. (13.4.3).
13.4.3 The fast Fourier transform method
For correlation functions with very long decay times, eqn. (13.4.3) requires a very
long trajectory and could potentially need to be evaluated at a largenumber of
points. Consequently, the computational overhead of eqn. (13.4.3) could be quite high,
increasing roughly asM^2 for a trajectory ofMtime steps. The third method we will
discuss is a highly efficient Fourier transform-based method that can take advantage
of fast Fourier transform algorithms (Futrelle and McGinty, 1971).
In order to derive the method, we start by noting that Hamilton’s equations are
invariant with respect to a change in the origin of time, so that we canshift the
time origin in eqn. (13.4.2) fromt= 0 tot=−T/2. Such a shift gives an equivalent
definition of the time correlation function:
CAB(τ) = lim
T→∞1
T
∫T/ 2
−T/ 2dt a(xt)b(xt+τ). (13.4.4)SinceTformally is taken to be infinite, we may writea(xt) andb(xt) in terms of their
Fourier transforms:
̃a(ω) =1
√
2 π∫∞
−∞dte−iωta(xt)