Essentials of Computational Chemistry

3.5 ENSEMBLE AND DYNAMICAL PROPERTY EXAMPLES 87
no time delay at all), and this quantity, 〈a 2 〉 (which can be determined from MC simulations
since no time correlation is involved) may be regarded as a normalization constant.
Now let us consider the behavior of C for long time delays. In a system where property
a is not periodic in time, like a typical chemical system subject to effectively random
thermal fluctuations, two measurements separated by a sufficiently long delay time should
be completely uncorrelated. If two properties x and y are uncorrelated, then 〈xy〉 is equal
to 〈x〉〈y〉, so at long times C decays to 〈a〉 2 .
While notationally burdensome, the discussion above makes it somewhat more intuitive
to consider a reduced autocorrelation function defined by
Ĉ(t) = 〈[a(t0) −〈a〉][a(t0 + t) −〈a〉]〉t0
〈[a −〈a〉] 2 〉
(3.46)
which is normalized and, because the arguments in brackets fluctuate about their mean (and
thus have individual expectation values of zero) decays to zero at long delay times. Example
autocorrelation plots are provided in Figure 3.5. The curves can be fit to analytic expressions
to determine characteristic decay times. For example, the characteristic decay time for an
autocorrelation curve that can be fit to exp(−ζt) is ζ −1 time units.
Different properties have different characteristic decay times, and these decay times can
be quite helpful in deciding how long to run a particular MD simulation. Since the point
of a simulation is usually to obtain a statistically meaningful sample, one does not want
to compute an average over a time shorter than several multiples of the characteristic
decay time.
C
Figure 3.5 Two different autocorrelation functions. The solid curve is for a property that shows no
significant statistical noise and appears to be well characterized by a single decay time. The dashed
curve is quite noisy and, at least initially, shows a slower decay behavior. In the absence of a very
long sample, decay times can depend on the total time sampled as well
t