Computational Physics - Department of Physics

13.7 Correlation Functions and Further Analysis of the Ising Model 449

tuating in the same direction, or a negative value if they fluctuate in the opposite direction.
If we then integrate over time, or use the discretized version of Eq. (13.8), the time corre-
lation functionφ(t)should take a non-zero value if the fluctuations are correlated, else it
should gradually go to zero. For times a long way apart the magnetizations are most likely
uncorrelated andφ(t)should be zero. Fig. 13.11 exhibits the time-correlation function for the
magnetization for the same lattice and temperatures discussed in Fig. 13.10.

kBT/J= 2. 4

kBT/J= 1. 5

t

φ(

t)

0 100 200 300 400 500 600 700 800

1.2

1

0.8

0.6

0.4

0.2

0

-0.2

Fig. 13.11Time-autocorrelation function with timetas number of Monte Carlo cycles. It has been normalized
withφ( 0 ). The calculations have been performed for a 100 × 100 lattice atkBT/J= 2. 4 with a disordered state
as starting point and atkBT/J= 1. 5 with an ordered state as starting point.

We notice that the time needed beforeφ(t) reaches zero ist∼ 300 for a temperature
kBT/J= 2. 4. This time is close to the result we found in Fig. 13.10. Similarly, forkBT/J= 1. 5
the correlation function reaches zero quickly, in good agreement again with the results of
Fig. 13.10. The time-scale, if we can define one, for which thecorrelation function falls off
should in principle give us a measure of the correlation timeτof the simulation.
We can derive the correlation time by observing that our Metropolis algorithm is based on
a random walk in the space of all possible spin configurations. We recall from chapter 12 that
our probability distribution functionˆw(t)after a given number of time stepstcould be written
as
ˆw(t) =Wˆtˆw( 0 ),

withˆw( 0 )the distribution att= 0 andWˆ representing the transition probability matrix. We
can always expandˆw( 0 )in terms of the right eigenvectors ofˆvofWˆ as

ˆw( 0 ) =∑

i

αiˆvi,

resulting in
ˆw(t) =Wˆtˆw( 0 ) =Wˆt∑
i

αiˆvi=∑

i

λitαiˆvi,