Molecular modelling of entangled polymer fluids under flow The ...

2.II. RESCALING A GAUSSIAN WALK 43
correlations decay rapidly in time.
F p (X i ) remains essentially fixed then
For a time interval, t ′ = t − ∆t...t, over which
∫ t
X p (t) = X p (t − ∆t) + ∆tF p (X i (t − ∆t)) + g p (t ′ )dt ′ . (2.46)
t−∆t
This allows the following useful correlation to be calculated
∫ t
〈X p (t)g q (t)〉 = X p (t − ∆t) 〈g q (t)〉 + ∆tF p (X i (t − ∆t)) 〈g q (t)〉 +
=
∫ t
t−∆t
= 1 2 Γδ pq.
Γδ(t − t ′ )δ pq dt ′
t−∆t
〈
gp (t ′ )g q (t) 〉 dt ′
The factor of 1/2 arises from the integration over half of the delta function.
(2.47)
This
result hinges crucially on the existence of a time interval, ∆t, which is simultaneously
small enough that the F p (X i ) do not change significantly and large enough for the
correlations of the Brownian forces to decay. The separation of time-scales between
these two processes is often wide enough that such a ∆t can be found. If the F p (X i )
are all linear functions then a more rigorous proof along the same lines as above can
be made.
Appendix 2.II
Rescaling a Gaussian walk
This problem is described in Doi and Edwards (1986) [section 2.1] and is recast in my
notation in this section. Figure 2.14 demonstrates the division of the molecule from
steps n = 0..N of length b to steps s = 0..Z of length a. Where each segment a contains
b
a
Figure 2.14: Sketch of a random walk rescaled from N steps of length b to Z steps of
length a.