284 Monte Carlo
Eqn. (7.2.18) is a function ofyalone, and we can now analyze itsydependence in
greater detail. First, the extrema ofF( ̃σ(y),y) are given by the solution of
0 =dF
dy=
∂F
∂y+
∂F
∂σ ̃∂σ ̃
∂y
=−i ̃σ(y). (7.2.19)Since∂F/∂ ̃σ(y) = 0 by the definition of ̃σ(y), the extrema ofFoccur where ̃σ(y) = 0.
According to eqn. (7.2.16), this implies
y=∫
dxφ(x)f(x)
∫
dxf(x)=〈φ〉f. (7.2.20)Because this solution is unique, we can expandF( ̃σ(y),y) to second order abouty=
〈φ〉f. For this, we need
d^2 F
dy^2
∣
∣
∣
∣
̃σ=0=−i
d ̃σ
dy∣
∣
∣
∣
̃σ=0. (7.2.21)
Differentiating eqn. (7.2.16) atσ= ̃σ(y) with respect toy, we obtain
1 =−ig′′( ̃σ)d ̃σ
dy, (7.2.22)
so that
d ̃σ
dy
∣
∣
∣
∣
̃σ=0=
i
g′′(0). (7.2.23)
Note, however, that
g′′(0) =i[∫
dxφ^2 (x)f(x)
∫
dxf(x)−
(∫
dxφ(x)f(x)) 2
(∫
dxf(x)) 2
]
=i[
〈φ^2 〉f−〈φ〉^2 f]
, (7.2.24)
which (apart from the factor ofi) is just the square of the fluctuationδφinφ(x) with
respect to the distributionf(x). From this analysis, we see thatP(y) has a single
maximum aty=〈φ〉fand decreases monotonically in either direction from this point.
In the limit thatM becomes very large, all higher-order contributions, which are
simply higher-order moments offwith respect toP(y), vanish, so thatP(y) becomes
just a Gaussian normal distribution
P(y)−→√
M
2 πδφ^2exp[
−
M(y−〈φ〉f)^2
2 δφ^2]
. (7.2.25)
We conclude, finally, that for largeM, eqn. (7.2.5) can be approximated via eqn.
(7.2.3) with a variance consistent with a normal distribution in the limit of largeM,
i.e.,
∫
dxφ(x)f(x) =1
M
∑M
i=1φ(xi)±1
√
M
[
〈φ^2 〉f−〈φ〉^2 f] 1 / 2
=
∑M
i=1φ(xi)±δφ, (7.2.26)thus guaranteeing convergence in the limitM→∞. Since the variance (second) term
in eqn. (7.2.26), decreases as 1/
√
M, efficient convergence relies on making this vari-
ance as small as possible, which is one of the challenges in designing Monte Carlo