1549380323-Statistical Mechanics Theory and Molecular Simulation

Molecular dynamics 241

transformed the incompressible equations (5.8.6) into compressibleones; the compress-
ibility of eqns. (5.9.1) now leads to an incorrect volume dependence in the phase space
measure. Applying the rules of Section 4.9 for analyzing non-Hamiltonian systems, we
find that the compressibility of eqns. (5.9.1) is

κ=

∑N

i=1

[∇ri·r ̇i+∇pi·p ̇i] +

∂V ̇

∂V

=dN

pǫ

W

−dN

pǫ

W

+d

pǫ

W

=d

pǫ

W

=

V ̇

V

=

d

dt

ln

(

V

V 0

)

. (5.9.2)

Thus, the functionw(x) = ln(V/V 0 ) and the phase space metric becomes

√

g =
exp(−w) =V 0 /V. The inverse volume dependence in the phase space measure leads to
an incorrect volume distribution. The origin of this problem is the volume dependence
of the transformation leading to eqns. (5.9.1).
We can make the compressibility vanish, however, by a minor modification of eqns.
(5.9.1). All we need is to add a term that yields an extra−dpǫ/Win the compressibility.
One way to proceed is to modify the momentum equation and add a term to thepǫ
equation to ensure conservation of energy. If the momentum equation is modified to
read

p ̇i=F ̃i−

(

1 +

d

Nf

)

pǫ

W

pi, (5.9.3)

whereNf is the number of degrees of freedom (dN−Nc) withNcthe number of
constraints, then the compressibilityκwill be zero, as required for a proper isobaric
ensemble. Here,F ̃iis the total force on atomiincluding any forces of constraint. If
Nc= 0, thenF ̃i=Fi=−∂U/∂ri. In addition, if thepǫequation is modified to read:

p ̇ǫ=dV(P(int)−P) +

d

Nf

∑N

i=1

p^2 i

mi

, (5.9.4)

then eqns. (5.9.1), together with these two modifications, will conserve eqn. (5.8.7).
Since, eqns. (5.9.1), together with eqns. (5.9.3) and (5.9.4), possess the correct phase
space metric and conserved energy, they can now be coupled to a thermostat in or-
der to generate a true isothermal-isobaric ensemble. Choosing theNos ́e–Hoover chain
approach of Section 4.10, we obtain the equations of motion