1549380323-Statistical Mechanics Theory and Molecular Simulation

Numerical evaluation 485

we obtain

∑j

k=0

(xl+k−xl+k+1)^2 =

∑j

k=1

k+ 1

k

u^2 l+k+

1

j+ 1

(xl+j+1−xl)^2 (12.6.28)

(see, also, eqn. (4.5.44)). Since thejstaging variables have a distribution proportional
to exp(−βmkωP^2 u^2 l+k/2), wheremk= (k+ 1)m/k, we sample eachul+krandomly
from its corresponding Gaussian distribution and then use eqn. (12.6.27) to generate
the proposed move to new primitive variablesx′l+k,k= 1,...,j. Let x denote the
original coordinatesxl+k, withk= 1,...,j, and let x′denote the coordinatesx′l+kof
the proposed move. The change in potential energy will be

∆U(x,x′) =

1

P

∑j

k=1

[

U(x′l+k)−U(xl+k)

]

, (12.6.29)

and the acceptance probability is then

A(x′|x) = min

[

1 ,e−β∆U(x,x

′)]

. (12.6.30)

Thus, if the move lowers the potential energy, it will be accepted with probability 1;
otherwise, it is accepted with probability exp[−β∆U(x,x′)]. On the average,P/jsuch
moves will displace all the beads of the cyclic polymer chain, and hencethe set ofP/j
staging moves is called aMonte Carlo pass.
The algorithm works equally well with a normal-mode transformation between
fixed endpointsxlandxl+j+1, as in eqn. (1.7.12). In this case, the normal-mode
variables are sampled from independent Gaussian distributions, andeqn. (12.6.30) is
used to determine whether the move is accepted or rejected.
The staging and normal-mode schemes described above move only the internal
modes of the cyclic chain. Therefore, in both algorithms, one additional move is needed,
which is a displacement of the chain as a whole. This can be achieved by an attempted
displacement of the uncoupled mode variableu 1 (recall thatu 1 is the centroid variable
in the normal-mode scheme) according to

u′ 1 ,α=u 1 ,α+

1

√

d

(ζα− 0 .5) ∆. (12.6.31)

Here,αruns over the spatial dimensions and ∆ is a displacement length (see,also,
eqn. (7.3.33)). Since the positions of all the beads change under such a trial move, the
potential energy changes as

∆U(x,x′) =

1

P

∑P

k=1

[U(x′k)−U(xk)]. (12.6.32)

Again, eqn. (12.6.30) determines whether the move is accepted or rejected. A complete
Monte Carlo pass, therefore, requiresP/jstaging or normal-mode moves plus one
move of the centroid. Typically, the parameterjis chosen such that the average