Computational Physics - Department of Physics

450 13 Monte Carlo Methods in Statistical Physics

withλitheitheigenvalue corresponding to the eigenvectorˆvi. If we assume thatλ 0 is the
largest eigenvector we see that in the limitt→∞,ˆw(t)becomes proportional to the corre-
sponding eigenvectorˆv 0. This is our steady state or final distribution.
We can relate this property to an observable like the mean magnetization. With the prob-
abiltyˆw(t)(which in our case is the Boltzmann distribution) we can write the mean magneti-
zation as
〈M(t)〉=∑
μ

ˆw(t)μMμ,

or as the scalar of a vector product

〈M(t)〉=ˆw(t)m,

withmbeing the vector whose elements are the values ofMμin its various microstatesμ.
We rewrite this relation as
〈M(t)〉=ˆw(t)m=∑
i

λitαiˆvimi.

If we definemi=ˆvimias the expectation value ofMin theitheigenstate we can rewrite the
last equation as
〈M(t)〉=∑
i

λitαimi.

Since we have that in the limitt→∞the mean magnetization is dominated by the the largest
eigenvalueλ 0 , we can rewrite the last equation as

〈M(t)〉=〈M(∞)〉+∑

i 6 = 0

λitαimi.

We define the quantity

τi=−

1

logλi

,

and rewrite the last expectation value as

〈M(t)〉=〈M(∞)〉+∑

i 6 = 0

αimie−t/τi. (13.9)

The quantitiesτi are the correlation times for the system. They control also the auto-
correlation function discussed above. The longest correlation time is obviously given by the
second largest eigenvalueτ 1 , which normally defines the correlation time discussed above.
For large times, this is the only correlation time that survives. If higher eigenvalues of the
transition matrix are well separated fromλ 1 and we simulate long enough,τ 1 may well define
the correlation time. In other cases we may not be able to extract a reliable result forτ 1.
Coming back to the time correlation functionφ(t)we can present a more general definition in
terms of the mean magnetizations〈M(t)〉. Recalling that the mean value is equal to〈M(∞)〉
we arrive at the expectation values

φ(t) =〈M( 0 )−M(∞)〉〈M(t)−M(∞)〉,

and using Eq. (13.9) we arrive at

φ(t) =∑

i,j 6 = 0

miαimjαje−t/τi,

which is appropriate for all times.