Environmental Engineering Reference
In-Depth Information
Time-to-time the list is reconstructed: it is important to do this before any
unlisted pairs have crossed the safety zone and come within interaction range. It
is possible to trigger the list reconstruction automatically, if a record is kept of the
distance traveled by each atom since the last update. The choice of list cutoff dis-
tance is a compromise: larger lists will need to be reconstructed less frequently,
but will not give as much of a saving on CPU time as smaller lists. This choice can
easily be made by experimentation. For larger systems (or so, depending on the
potential range) another technique becomes preferable. The cubic simulation box
(extension to non-cubic cases is possible) is divided into a regular lattice of cells.
The first part of the method involves sorting all the atoms into their appropriate
cells. This sorting is rapid, andmay be performed every step. Then, within the
force routine, pointers are used to scan through the contents of cells, and calculate
pair forces. This approach is very efficient for large systems with short-range
forces. A certain amount of unnecessary work is done because the search region
is cubic, not (as for the Verlet list) spherical. Molecular dynamics evolves a nite-
sized molecular configuration forward in time, in a step-by-step fashion. There
are limits on the typical time scales and length scales that can be investigated and
the consequences must be considered in analyzing the results. Simulation runs
are typically short: typicallyMD steps, corresponding to perhaps a few nanosec-
onds of real time, and in special cases extending to the microsecond regime. This
means that we need to test whether or not a simulation has reached equilibrium
before we can trust the averages calculated in it. Moreover, there is a clear need
to subject the simulation averages to a statistical analysis, to make a realistic esti-
mate of the errors. How long should we run? This depends on the system and the
physical properties of interest as the variables and become uncorrelated; this de-
cay occurs over a characteristic time . Formally we may define a correlation time:
( ) ( )
2
∞
τ=
∫
dta
0
a t
(113)
a
a
0
It is almost essential for simulation box sizes to be large and for simulation run
lengthsto be large compared with , for all properties of interest Only then can we
guarantee that reliablysampled statistical properties are obtained. Roughly speak-
ing, the statistical error in a property calculated as an average over a simulation
run of length is proportional to the time average is essentially a sum of inde-
pendent quantities, each an average over time. Near critical points, special care
must be taken, in that these inequalities will almost certainly not be satisfied, and
indeed one may see the onset of non-exponential decay of the correlation func-
tions. In these circumstances a quantitative investigation of nite size effects and
correlation times, with some consideration of the appropriate scaling laws, must
be undertaken. Phase diagrams of soft-matter systems often include continuous
phase transitions, or weakly first-order transitions exhibiting significant pretran-
sitional actuations. One of the most encouraging developments of recent years
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