Polymers in Confined Geometry.pdf

42 CHAPTER 4. SIMULATION METHODS
This means in terms of energy: If the energy is lowered the system evolves to the
preferred lower energy state. If the energy is increased, the system moves only
with a Boltzmann factor into that less preferred region. But one has to allow for
this movement “up the energy-hill” since the system is fluctuating around the
equilibrium configuration.
In an computer algorithmic form this consists of either accepting the move
right away since ∆E is lowered. Or a random number of a uniform deviate has
to be drawn which must lie below the transition probability exp(−β∆E).
So the new state is:
Cr accepted with probability: p
Ck+1 =
. (4.9)
Ck rejected with probability: 1 − p
Instead of the usual non-differential acceptance probability, eq. (4.8), other
schemes are possible—like the Glauber dynamics—but we will not go into the
details of this but stick we the equations derived above.
The condition of detailed balance is a sufficient but not necessary condition
for the Monte-Carlo scheme to work properly. It can be shown that e.g. sequential
updating of the particles—after a particle is moved, in the next trials this particle
is skipped, so detailed balance is obviously not satisfied—gives a working MCscheme
as well (see [13]).
4.3.3 Correlations and error calculation
When using the Metropolis algorithm one has to assure that the simulation is
exploring the configuration space. There are several problems which have to be
taken into account. Normally two configurations which are only separated by
some small number of MC steps are “quite close” together in configuration space
(in particular when being close to a steep energy minimum, a large number of MC
steps are rejected, so that one has to be careful to sample the fluctuation around
the minimum efficiently). When taking a sample configuration too frequently (in
the worst case every step), one has an ensemble of samples which are correlated.
This asks for careful treatment of error estimation and for a smart way of choosing
trial steps.
Choosing a trial Monte-Carlo step must be done in the following way:
• Make as large steps as possible to explore all configuration space without
having too many steps rejected, since large changes in energy ∆E are not
very probable to be accepted.
• On the other hand the steps should be small enough to have an overall
high enough acceptance rate. But choosing the step size to small—which
makes the acceptance rate large—will not drive the simulation far through
configuration space.