1549380323-Statistical Mechanics Theory and Molecular Simulation

Umbrella sampling 345

histogram obtained from each molecular dynamics or Monte Carlo simulation. Then,
the biased distribution is estimated by

P ̃(q,s(k))≈^1

nk∆q

H ̃k(q), (8.8.8)

wherenkis the number of configurations sampled in thekth simulation and ∆qis
the bin width used to compute the histogram. The statistical errorin the biased
distribution for thekth umbrella window is then ̃σ^2 k =ǫk(q)H ̃k(q)/(nk∆q), where
ǫk(q) measures the deviation as a function ofq between the numerically sampled
distribution and the true distributionP(q) in thekth umbrella window. The error in
Pk(q) is then given by the square of the unbiasing factor

σ^2 k= e−^2 β(Ak−A^0 )e^2 βW(q,s

(k))

σ ̃^2 k. (8.8.9)

We aim to minimize the total error

σ^2 =

∑n

k=1

Ck^2 (q)σk^2 (8.8.10)

subject to eqn. (8.8.7). The constraint can be imposed by means ofa Lagrange mul-
tiplierλ; the error function to be minimized is

Σ^2 =

∑n

k=1

Ck^2 (q)e−^2 β(Ak−A^0 )e^2 βW(q,s

(k))ǫk(q)H ̃k(q)

(nk∆q)

−λ

(n

∑

k=1

Ck(q)− 1

)

. (8.8.11)

Thus, setting the derivative∂Σ^2 /∂Ck(q) = 0 and solving forCk(q) in terms of the
Lagrange multiplier, we find

Ck(q) =

λnk∆q

2 ǫk(q)H ̃k(q)e−^2 β(Ak−A^0 )e^2 βW(q,s(k))

. (8.8.12)

The Lagrange multiplier is now determined by substituting eqn. (8.8.12) into eqn.
(8.8.7). This yields

λ

∑n

k=1

nk∆q

2 ǫk(q)H ̃k(q)e−^2 β(Ak−A^0 )e^2 βW(q,s(k))

= 1 (8.8.13)

so that

λ=

1

∑n

k=1nk∆q/[2ǫk(q)H ̃k(q)e−^2 β(Ak−A^0 )e^2 βW(q,s

(k))]. (8.8.14)

Substituting the Lagrange multiplier back into eqn. (8.8.12) gives thecoefficients as

Ck(q) =

nk/[ǫk(q)H ̃k(q)e−^2 β(Ak−A^0 )e^2 βW(q,s

(k))

]

∑n

j=1nj/[ǫj(q)

H ̃j(q)e−^2 β(Aj−A^0 )e^2 βW(q,s(j))]. (8.8.15)

At this point, we make two vital assumptions. First, we assume thatthe error
functionǫk(q) is the same in allnumbrella windows, which is tantamount to assuming