356 Free energy calculations
is subsequently filled in, and so forth until the entire landscape is “flat.” When this
state is achieved, the accumulated time-dependent potential is used to construct the
free energy profile.
In order to see how such a dynamics can be constructed, consideronce again the
probability distribution function in eqn. (8.6.4). SinceP(s 1 ,...,sn) is an ensemble
average
P(s 1 ,...,sn) =〈n
∏α=1δ(fα(r 1 ,...,rN)−sα)〉
, (8.11.1)
we can replace the phase space average with a time average over a trajectory as
P(s 1 ,...,sn) = lim
T→∞1
T
∫T
0dt∏nα=1δ(fα(r 1 (t),...,rN(t))−sα), (8.11.2)under the assumption of ergodic dynamics. In the metadynamics approach, we express
theδ-function as the limit of a Gaussian function as the width goes to 0 andthe height
is goes to infinity:
δ(x−a) = lim
σ→∞1
√
2 πσ^2e−(x−a)(^2) / 2 σ 2
. (8.11.3)
Using eqn. (8.11.3), eqn. (8.11.2) can be rewritten as
P(s 1 ,...,sn) =lim
T→∞
lim
∆s→ 01
√
2 π∆s^2 T∫T
0dt∏nα=1exp[
−
(sα−fα(r 1 (t),...,rN(t)))^2
2∆s^2]
. (8.11.4)
Thus, for finiteTand ∆s, eqn. (8.11.4) represents an approximation toP(s 1 ,...,sn),
which becomes increasingly accurate asTincreases and the Gaussian width ∆sde-
creases. For numerical evaluation, the integral in eqn. (8.11.4) is written as a discrete
sum so that the approximation becomes
P(s 1 ,...,sn)≈1
√
2 π∆s^2 TN∑− 1
k=0exp[
−
∑nα=1(sα−fα(r 1 (k∆t),...,rN(k∆t)))^2
2∆s^2]
. (8.11.5)
Eqn. (8.11.5) suggests an intriguing bias potential that can be added to the original
potentialU(r 1 ,...,rN) to help the system sample the free energy hypersurface while
allowing for a straightforward reconstruction of this surface directly from the dynamics.
Consider a bias potential of the form
UG(r 1 ,...,rN,t) =W∑
t=τG, 2 τG,...,exp[
−
∑nα=1(fα(r)−fα(rG(t)))^2
2∆s^2]
, (8.11.6)
wherer≡r 1 ,...,rN, as usual, andrG(t) is the time evolution of the complete set of
Cartesian coordinates up to timetunder the action of the potentialU+UG, andτG