Monte Carlo Methods in Statistical Mechanics: Foundations and ...

The rate ofconvergence to equilibrium from an initial nonequilibrium distribution
can be bounded above interms of R (and hence exp). More precisely, let is a probability
measure on S, andlet us de ne itsdeviation from equilibrium inthel2 sense,
Then, clearly,
d2( ) jj ; 1jjl 2( ) = sup
jjfjj l 2 ( ) 1
And bythe spectral radius formula,
Z fd ; Z fd : (2.13)
d2( P t ) jjP t j n 1 ? jj d2( ) : (2.14)
jjP t j n 1 ? jj R t = exp(;t= exp) (2.15)
asymptotically as t !1,withequality for all t if P is self-adjoint (seebelow).
On the other hand, for a given observable f we de ne the integrated autocorrelation
time
intf = 1
2
= 1
2 +
1X
t=;1 ff(t) (2.16)
1X
t=1
ff(t)
[The factor of 1
2 is purely a matter of convention it is inserted so that intf expf if
ff(t) e ;jtj= with 1.] The integrated autocorrelation time controls the statistical
error in Monte Carlo measurements ofhfi. More precisely, the sample mean
has variance
f
var(f) = 1
n 2
= 1
n
nX
rs=1
n;1 X
1
n
t=;(n;1)
nX
ft
t=1
(2.17)
Cff(r ; s) (2.18)
1 ; jtj
!
Cff(t)
n
(2.19)
1
n (2 intf) Cff(0) for n (2.20)
Thus, the variance of f is a factor 2 intf larger than it would be if the fftg were statistically
independent. Stated di erently, the number of \e ectively independent samples"
in a run of length n is roughly n=2 intf .
It is sometimes convenienttomeasure theintegrated autocorrelation timeinterms of
the equivalent pure exponential decay thatwould produce thesamevalue of P1 t=;1 ff(t),
namely
;1
e intf
log 2 intf;1
2 intf +1
7
: (2.21)