1549380323-Statistical Mechanics Theory and Molecular Simulation

Jarzynski’s equality 325

generate a distributionZ(λ,β) as a function ofλin which the physical coordinates
sample essentially all of their configuration space at temperatureTat eachλ, leading
to

Z(λ,β) =

∫

dNre−βU(r^1 ,...,rN,λ). (8.3.8)

In the extreme limit, whereλis fixed at each value, this is certainly the correct
distribution function if the physical degrees of freedom are properly thermostatted.
The adiabatic decoupling approximates the more extreme situation of fixedλ. Eqn.
(8.3.8) leads to an important quantity known as thepotential of mean forceinλ,
obtained from−(1/β) lnZ(λ,β). In the limit of adiabatic decoupling, the potential of
mean force becomes an effective potential on whichλcan be assumed to move quasi-
independently from the physical degrees of freedom. Note that−(1/β) lnZ(λ,β) is
also equal to the free energy profileA(λ) we originally sought to determine. Using the
potential of mean force, we can construct an effective Hamiltonianforλ:

Heff(λ,pλ) =

p^2 λ

2 mλ

−

1

β

lnZ(λ,β). (8.3.9)

Now, ifλis thermostatted to a temperatureTλ, then a canonical distribution in eqn.
(8.3.9) at temperatureTλwill be generated. This distribution takes the form

Padb(λ,pλ,β,βλ)∝e−βλHeff(λ,pλ), (8.3.10)

whereβλ= 1/kTλ, and the “adb” subscript indicates that the distribution is valid
in the limit of adiabatic decoupling ofλ. Integrating overpλyields a distribution
P ̃adb(λ,βλ,β)∝[Z(λ,β)]βλ/β, from which the free energy profile can be computed as

A(λ) =−kTλlnP ̃adb(λ,β,βλ) =−kTlnZ(λ,β) + const. (8.3.11)

Thus, apart from a trivial additive constant, the free energy profile can be computed
from the distributionP ̃adb(λ) generated by the adiabatic dynamics. Note that eqn.
(8.3.11) closely resembles eqn. (8.3.3), the only difference being the prefactor ofkTλ
rather thankT. Despite the fact that lnP ̃adb(λ,β,βλ) is multiplied bykTλ, the free
energy profile is obtained at the correct ensemble temperatureT. As an example,
Abrams and Tuckerman (2006) employed the AFED approach to calculate the hy-
dration free energies of alanine and serine side-chain analogs using the CHARMM22
force field (MacKerellet al., 1998) in a bath of 256 water molecule. These simulations
requiredkTλ= 40kT= 12,000 K,mλone-thousand times the mass of an oxygen atom
and could obtain the desired free energies, 1.98 kcal/mol (for the alanine analog) and
-4.31/kcal/mol (for the serine analog), to within an error of 0.25 kcal/mol in 1-2 ns
using a time step of 1.0 fs. In order to keep theλdegree of freedom in the range [0,1],
reflecting boundaries were placed atλ=−ǫand 1 +ǫwhereǫ= 0.01.

8.4 Jarzynski’s equality and nonequilibrium methods

In this section, we investigate the connection between free energy and nonequilibrium
work. We have already introduced the work–free–energy inequality in eqn. (8.2.9),