1549380323-Statistical Mechanics Theory and Molecular Simulation

Adiabatic dynamics 353

SinceZY(q 1 ,...,qn) is the potential of mean force for the reaction coordinates, thefree
energy hypersurfaceA(q 1 ,...,qn) is, by definition,

A(q 1 ,...,qn) =−

1

β

lnZY(q 1 ,...,qn), (8.10.19)

but from eqn. (8.10.18), it follows that

A(q 1 ,...,qn) =−

1

βq

lnPadb(q 1 ,...,qn) + const, (8.10.20)

which is eqn. (8.10.5). The constant in the second term comes from factors dropped
in eqn. (8.10.17) and is irrelevant to the overall free energy hypersurface.
As an illustration of adiabatic dynamics, consider a simple problem with two de-
grees of freedomxandy. The potential is chosen to be a double well inxcoupled
linearly to a harmonic oscillator iny:

U(x,y) =D 0

(

x^2 −a^2

) 2

+

1

2

ky^2 +λxy. (8.10.21)

The free energy profileA(x) for this system can be derived analytically with the result

-2 -1 0 1 2

x

0

5

10

15

20

25

A

(x

)

Bare potential

Analytical free energy

mx= 300 my

mx= 10 my

Fig. 8.7Free energy profile for the potential in eqn. (8.10.21) showntogether with the
bare double-well potential and the results of two simulations using the adiabatic free energy
approach. The simulations correspond toTx= 10Tyandmx= 10myormx= 300my.