Computational Physics

228 Molecular dynamics simulations

First we derive the equations of motion in the usual way:

dri

dt

=

∂H

∂pi

=

pi

ms^2

(8.83a)

ds

dt

=

∂H

∂ps

=

ps

Q

(8.83b)

dpi

dt

=−

∂H

∂ri

=−∇iU(R)=−

∑

i<j

∇iU(ri−rj) (8.83c)

dps

dt

=−

∂H

∂s

=

(∑

ip

2

i

ms^2

−gkBT

)/

s. (8.83d)

We have used the notation∂H/∂pi=∇piH, etc. The partition function of the total
system (i.e. including heat bath degree of freedoms) is given by the expression:

Z=

1

N!

∫

dps

∫

ds

∫

dP

∫

dR

×δ





∑

i

p^2 i

2 ms^2

+

1

2

∑

ij,i
=j

U(rij)+

p^2 s

2 Q

+gkBTln(s)−E



. (8.84)

Integrations

∫

dRand

∫

dPare over all position and momentum degrees of freedom.
We now rescale the momentapi:

pi

s

=p′i, (8.85)

so that we can rewrite the partition function as

Z=

1

N!

∫

dps

∫

ds

∫

dP′

∫

dR

×s^3 Nδ





∑

i

p′^2 i

2 m

+

1

2

∑

ij,i
=j

U(rij)+

p^2 s

2 Q

+gkBTln(s)−E



. (8.86)

We define the HamiltonianH 0 in terms ofRandP′as

H 0 =

∑

i

p′^2 i

2 m

+

1

2

∑

ij,i
=j

U(rij). (8.87)