292 Monte Carlo
A simple yet fairly standard way to apply the M(RT)^2 algorithm is based on
choosing the trial probabilityT(x|y) to be uniform for x in some domain of radius ∆
about y. For this choice,T(x|y) takes the form
T(x|y) ={
1 /∆ |x−y|<∆/ 2
0 otherwise. (7.3.33)
ThisT(x|y) clearly satisfies eqn. (7.3.24) and also has the property thatT(x|y) =
T(y|x). Consequently, the acceptance probability becomes simply
A(x|y) = min[
1 ,
f(x)
f(y)]
. (7.3.34)
Sampling the canonical distribution
Eqns. (7.3.33) and (7.3.34) can be straightforwardly applied to the problem of cal-
culating the canonical configurational partition function for a system of monatomic
particles such as a Lennard-Jones liquid (see Section 3.14.2). Recallthat the partition
function for a system ofNparticles with coordinatesr 1 ,...,rN and potential energy
U(r 1 ,...,rN) is given by
Q(N,V,T) =
1
N!λ^3 N∫
dr 1 ···drNe−βU(r^1 ,...,rN), (7.3.35)whereλ=
√
βh^2 / 2 πmand the integral is the configurational partition function. In-
troducing the usual notationr≡r 1 ,...,rN as the complete set of coordinates, we
wish to devise a trial move fromrtor′and determine the corresponding acceptance
probability. If the move is based on eqn. (7.3.33), then, sincef(r)∝exp[−βU(r)], the
acceptance probability is simply
A(r′|r) = min[
1 ,e−β[U(r′)−U(r)]]. (7.3.36)
In other words, the acceptance probability is determined solely by the change in the
potential energy that results from the move. If the potential energy decreases in a trial
move, the move will be accepted with probability 1; if the energy increases, the move
will be accepted with a probability that decreases exponentially with the change in
energy.
An immediate problem arises when attempting to apply eqn. (7.3.33) aswritten.
As a general rule, it is not possible to moveallof the particles simultaneously! Re-
member that the potential energyUis an extensive quantity, meaning thatU∼N.
Thus, if we attempt to move all of the particles at once, the changein the potential
energy can be quite large unless the particle positions are varied onlyminimally. This
problem becomes increasingly severe as the number of particles grows. According to
eqn. (7.3.36), a large change in the potential energy leads to a verysmall probability
that the move is accepted, and the M(RT)^2 algorithm becomes inefficient as a means
of generating canonical configurations.