Sampling 293A simple remedy for this problem lies at the other extreme, in which we attempt
to only move one particle at a time. Of course, eqn. (7.3.33) includes this possibility
as well. Thus, we begin by choosing a particle at random from among theNparticles
in the system. Suppose the randomly chosen particle has an indexi. Each of the three
components ofriis displaced at random using three uniform random numbersξx,ξy,
andξz, withξα∈[0,1],α=x,y,z. The displacements are then given by
x′i=xi+1
√
3
(ξx− 0 .5) ∆y′i=yi+1
√
3
(ξy− 0 .5) ∆zi′=zi+1
√
3
(ξz− 0 .5) ∆. (7.3.37)All other particle coordinates remain unchanged. The random numbers are shifted to
the interval [− 0. 5 , 0 .5] to ensure that the sphere of radius ∆ is centered onri, and
the
√
3 factor ensures that|r′i−r|<∆/2 and consequently that|r′−r|<∆/2 as
required by eqn. (7.3.33). AMonte Carlo passthrough the system is a collection ofN
such trial moves which, in principle, is a sufficient number to attempt amove on each
of the particles, although in any Monte Carlo pass, attempts will be made on some
particles more than once while others will have no attempted moves.
At this point, several comments are in order. First, for a large majority of systems,
it is not necessary to recompute the potential energyU(r′) in full in order to determine
the acceptance probability when a single particle is moved. We only need to recompute
the terms that involveri. Thus, for a short-ranged pair potential, only the interaction
of particlei with other particles that lie within the cutoff radius ofi need to be
recomputed, which is a relatively inexpensive operation (see Appendix B). Second,
it is natural to ask if maximal efficiency could be achieved via an optimaltarget
for the average number of accepted moves. While this question canbe answered in
the affirmative, it is impossible to give a particular number for the target fraction
of accepted moves. Note that the acceptance probability depends on the choice of
∆, hence the efficiency of the algorithm depends on this critical parameter. A large
value for ∆ generates large displacements for each particle with possiblly significant
increases in the potential energy and consequently, a low acceptance probability. A
small value for ∆ generates small displacements and a correspondingly high acceptance
probability. Thus, choosing ∆ is a compromise between large displacements and a
reasonable number of accepted moves. One occasionally reads in the literature that a
good target is between 20% and 50% acceptance of trial moves, with an optimal value
around 30% (Allen and Tildesley, 1989; Frenkel and Smit, 2002). Thisrange can be
a useful rule of thumb or serve as a starting point for refining the target acceptance
rate. However, the optimal value depends on the system, the thermodynamic control
variablesN,V, andT, and even how the computer program is written. In general, tests
should be performed for each system to determine the optimal acceptance probability
and displacement ∆.