Computational Physics

7.4 Determination of averages in simulations 193

We can estimate the standard deviation as a time average:

σ^2 =A^2 −A

2

. (7.71)

For a long enough simulation this reduces to the ensemble average, and the
expectation value of this estimate becomes independent of the simulation time.
Equation(7.71)estimates the standard deviation irrespective of time correlations
between subsequent samples generated by the simulation. However, thestandard
deviation of the mean valueofAcalculated overMsamples generated by the sim-
ulation, i.e. the statistical error, depends on the number ofindependentsamples
generated in the simulation, and this is the total number of samples divided by the
correlation ‘time’τ, measured in simulation steps.
In order to study the standard deviation of the mean (the statistical error), we first
analyse the time correlations. These manifest themselves in the time correlation
function:

cAA(k)=

〈

(An−〈An〉)(An+k−〈An+k〉)

〉

=〈AnAn+k〉−〈An〉^2. (7.72)

Note that the right hand side of this expression does not depend onnbecause of time
translation symmetry. Fork=0 this function is equal toσ^2 , and time correlations
manifest themselves in this function assuming nonzero values fork=0. The time
correlation function can be used to determine theintegrated correlation timeτ,
defined as

τ=

1

2

∑∞

n=−∞

cAA(n)

cAA( 0 )

(7.73)

where the factor 1/2 in front of the sum is chosen to guarantee that for a correlation
function of the form exp(−|t|/τ )withτ 1, the correlation time is equal toτ. Note
that this definition of the time correlation is different from that given inEq. (7.59).
The current one is more useful as it can be determined throughout the simulation,
and not only at the beginning when the quantityAdecays to its equilibrium value.
A third definition is theexponential correlation timeτexp:

τexp=−t/ln

∣

∣∣

∣

cAA(t)

cAA( 0 )

∣

∣∣

∣, larget. (7.74)

This quantity is the slowest decay time with which the system relaxes towards
equilibrium (such as happens at the start of a simulation when the system is not yet
in equilibrium), and it is in general not equal to the integrated correlation time.
Now let us return to the standard deviation of the mean value ofAas determined
in a simulation generatingMconfigurations (with time correlations). It is easy to
see that the standard deviation in the mean,ε,isgivenby

ε^2 =

〈

1

M^2

∑M

n,m= 1

AnAm

〉

−

(〈

1

M

∑M

n= 1

An

〉) 2

=

1

M^2

∑M

n,m= 1

cAA(n−m). (7.75)