Polymers in Confined Geometry.pdf

44 CHAPTER 4. SIMULATION METHODS
of the problem. As described in [4] the dynamic interpretations of Monte Carlo
averaging can be introduced in terms of a master equation. Besides being able
to use this description to calculate correlations this is the starting point for MC
simulations of dynamic processes.
By this interpretation we write for the probability of having a configuration
x at step i as having it at time t: p(xi) ≡ p(x, t).
With this dynamic interpretation, the direct way to find out after how many
steps the samples are uncorrelated is to measure the time dependence of a suitable
correlation function. This correlation function normally decays as an exponential
in time
φ(t) = exp − |t|
, (4.13)
τ
where the time constant τ gives the time when the correlations are decayed to
1/e of the initial value 4 .
Approximation for the correlation time Besides measuring the, in the
above dynamic interpretation, time-dependent correlation function directly, there
is an alternative way to get an estimate for it by sampling different averages (cf.
[4, 22]). Having a large quantity n ≫ 1 of samples xi of an observable x we use
the statistical error
and take the mean of the square
(δx) 2 = 1
n 2
δx = 1
n
n xi − 〈x〉 2
i=1
= 1 2
x
n
− 〈x〉
2
1 + 2
n
xi − 〈x〉
i=1
+ 2
n 2
i=1
n
n
i=1 j=i+1
〈xixj〉 − 〈x〉 2
(4.14)
n
1 − i
n
〈x0xi〉 − 〈x〉 2
〈x2 〉 − 〈x〉 2
. (4.15)
In the second line the time invariance of the correlation in equilibrium 〈xixj〉 =
〈x0xj−i〉 and a change of the summation index j → j −i has been used. The time
invariance only holds if an initial transient regime, where the system ‘forgets’ its
initial configuration, is omitted.
Now introducing the ‘time’ ti = i δt, where δt is the time unit used, e.g.
δt = 1 for a sample every MC step or δt = N for one MC step per polymer
4For the worm-like chain model described in section 4.4 a good correlation function to
measure is the tangent-tangent correlation function
φ(t) =
〈ti(0) · ti(t)〉 − 〈ti〉 2
,
i
which shows reasonable decaying behavior in the case of an unconfined polymer, but fails to be
usable when confining the molecule, due to correlations by the tube.