Polymers in Confined Geometry.pdf

48 CHAPTER 4. SIMULATION METHODS
where the prefactor k is related to the persistence length lp by eqs. (A.8) and
(A.10). As input parameter for the simulation the flexibility ɛ is taken so that
everything is measured in units of total length L (cf. eq. (3.2)). All output is
normalized by L as well.
This Hamiltonian is used to perform an off-lattice simulation where the polymer
can orient without any restriction (cf. [25]).
In a straightforward algorithm the polymer is represented by just its N segments,
where the positions of the joints can be ignored as long as the polymer is
unconfined. The basic steps are:
1. perform a random movement on a randomly chosen segment
2. calculate the change in configuration energy ∆e
3. accept/reject the move corresponding to the Metropolis algorithm
(4. take samples after a specific number of steps)
This three (four) basic steps are the essential part of the MC scheme. They will
be discussed now in more detail.
Step 1: The trial movement in the n-th MC-step is chosen by drawing a
random number m to specify a segment to change ˆt (n)
m → ˜t (n)
m . The orientation
is changed by a multiplication with a rotational matrix R l k
, which is chosen out
of a set that has been calculated in the initial phase of the simulation. The
superscript l numbers one of the d spacial dimensions and k the specific matrix
chosen from the set. It turned out to be optimal having 20 d equidistant discrete
matrices (20 for each dimension) out of a maximal rotation angle ±δα/2. This
angels must be tuned to give an overall acceptance rate of 25–50%.
Detailed balance is obviously guaranteed: choosing a rotation matrix by first
choosing one direction at random and than again randomly a specific rotation 6 .
Thus this step is subsumed by—with random numbers m = 1, . . . , N, l = 1, . . . , d
and k = 1, . . . , 20—
˜t (n)
i
=
Rl k · ˆt (n)
m for i = m
ˆt (n)
. (4.26)
i for i = m
One segment m is changed, where the orientation of all others is kept fixed as
shown in figure 4.1.
An alternative approach to the simulation in two dimension is by noting that
the polymer configuration is fully specified by knowing the angles θi between each
segment and a fixed global direction (see figure 4.1). The simulation can than
be build on trial moves changing these angles, i.e. taking a random change on
6 E.g. it would be wrong to use in each step a rotation in all d-direction, when applying the
rotation always in the same order. This is due to the non-commutativity of rotations, which
breaks detailed balance in this case. But choosing the order of application randomly, again
obeys detailed balance, but has an unnecessary calculation overhead.