Computational Physics

194 Classical equilibrium statistical mechanics

If we definel=n−m, then this can be rewritten as

ε^2 =

1

M^2

∑M

n= 1

n∑−M

l=n− 1

cAA(l). (7.76)

The lowest and highest values taken on bylare−(M− 1 )andM−1 respectively,
and some fixed value oflbetween these two boundaries occursM−|l|times. This
leads to the expression

ε^2 =

1

M

M∑− 1

l=−(M− 1 )

(

1 −

|l|

M

)

cAA(l)

largeM

−−−−→ 2

τ

M

cAA( 0 )= 2

τ

M

σ^2. (7.77)

We see that time correlations cause the errorεto be multiplied by a factor of

√

2 τ
with respect to the uncorrelated case. The obvious procedure for determining the
statistical error is first to estimate the standard deviation and the correlation time,
using (7.71) and (7.73) respectively, and then calculate the error using (7.77).
In practice, however, a simpler method is preferred. The values of the physical
quantities are recorded in a file. Then the data sequence is chopped into a number
of blocks of equal size which is larger than the correlation time. We calculate the
averages ofAwithin each block. For blocks of sizem, thejth block average is then
given as

Aj=

1

m

m∑(j+ 1 )

k=jm+ 1

Ak. (7.78)

The averages of the physical quantities in different blocks are uncorrelated and
the error can be determined as the standard deviation of the uncorrelated block
averages. This method should yield errors which are independent of the block size
provided the latter is larger than the correlation time and sufficiently small to have
enough blocks to calculate the standard deviation reliably. This method is called
data-blocking.

Exercises

7.1 In this problem we analyse the relation between the differential scattering cross
section for elastic X-ray scattering by a collection of particles and the structure factor
in more detail. Consider an incoming X-ray with wave vectork 0 , which is scattered
intok 1 by particle numberjatrjat timet′. When the wave ‘hits’ particlejat timet′,
its phase factor is given by
eik^0 rj−iωt′.
(a) Give the phase of the scattered wave when it arrives at the detector located atrat
timet.