The two-dimensional integral in Eq. (11.11) may be reduced to two one-dimensional integrals by a change of variables. This is illustrated in Fig. G.0.1, which shows a Cartesian coordinate system with unit axes (i,j) and associated coordinates τ1 and τ2.
We now rotate the (i,j) system by 45° to the (i',j') system. The orthonormal transformation is given by
so a vector R
represented in the two coordinate systems is
Fig. G.0.1 A sketch of the coordinate transformation for the evaluation of the double integral in Eq. (11.11).
Then the relation between the ‘old’ and ‘new’ coordinates is
or the inverse
and, since the Jacobian determinant for the transformation is unity, the area element in the new coordinate system is identical to the one in the old coordinate system, that is,
. The integral spans the ‘hatched’ square with side t
in Fig. G.0.1
. To cover the same area in the new coordinate system, the integral is divided into two integrals, one for the triangle below the line l
and one for the triangle above this line. This division is naturally introduced in order to set the limits for
when we integrate over
. For the lower triangle we see from Eqs (G.3) and (G.4) that the region spanned by
for a given value of
and for the upper triangle
The integral in Eq. (11.11) then becomes
where we have used Eq. (11.7
) for the correlation function of the random force. We may now introduce the substitution
into Eq. (G.7) and find
where we have used
Our assumption that ϕ(θ) is sharply peaked around θ = 0 and drops to zero for θ > τ Cimplies that its integral f(z) reaches a constant value f = f(∞) for τ ∼ τC. Therefore,
when we are interested in much longer times, we may replace f(ψ) and f(2t − ψ) by their constant asymptotic value f in the right-hand side of Eq. (G.9). Hence we may write, for t ≫ τc,
Substitution of this result for the double integral in Eq. (11.11) finally gives us Eq. (11.12).