Computational Physics

8.5 Molecular dynamics methods for different ensembles 229

Furthermore we use the relationδ[f(s)]=δ(s−s 0 )/f′(s)withf(s 0 )=0 and set
g= 3 N+1, so that we can rewriteEq. (8.86)as

Z=

1

N!

∫

dps

∫

ds

∫

dP′

∫

dR

s^3 N+^1

( 3 N+ 1 )kBT

×δ

(

s−exp

[

−

H 0 (P′,R)+p^2 s/ 2 Q−E

( 3 N+ 1 )kBT

])

=

1

( 3 N+ 1 )kBT

1

N!

∫

dps

∫

dP′

∫

dRexp

[

−

H 0 (P′,R)+p^2 s/ 2 Q−E

kBT

]

.

(8.88)

The dependence onpsis simply Gaussian and integrating over this coordinate we
obtain

Z=

1

3 N+ 1

(

2 πQ

kBT

) 1 / 2

exp(E/kBT)Zc (8.89)

whereZcis the canonical partition function:

Zc=

1

N!

∫

dP′

∫

dRexp[−H 0 (P′,R)/kBT]. (8.90)

It follows that the expectation value of a quantityAwhich depends onRandPis
given by

〈A(P/s,R)〉=〈A(P′,R)〉c (8.91)

where〈···〉cdenotes an average in the canonical ensemble. The ergodic hypothesis
relates this ensemble average to a virtual-time average.
The Lagrangian equations of motion for therican be obtained by eliminating
the momenta from (8.83a):

d^2 ri

dt^2

=−

1

ms^2

∇iV(R)−

2

s

dri

dt

ds

dt

. (8.92)

In this equation the ordinary force term is recognised with a factor 1/s^2 in front
and with an additional friction term describing the coupling to the heat bath. The
factor 1/s^2 is consistent with the relation between real and virtual-time dt′=dt/s
given above. Together with the definitionsP′=P/sandp′s=ps/s, this leads to