Computational Physics

230 Molecular dynamics simulations

the equations of motion in real variables:

dri′

dt′

=

p′i

m

(8.93a)

dp′i

dt′

=−∇iV(R)−sp′sp′i/Q (8.93b)

ds

dt′

=s′^2 p′s/Q (8.93c)

dp′s

dt′

=

(∑

i

p

′ 2

i/m−gkBT

)

/s−s^2 p

′ 2

s/Q. (8.93d)

Although these equations are equivalent to the equations for the virtual variables,
there is a slight complication in the evaluation of averages. The point is that we
have used the ergodic theorem for the canonical Hamiltonian expressed in virtual
variables(P,R,t,s,ps)in order to relatevirtual-timeaverages to ensemble averages.
The real time steps however are not equidistant and time averaging in real time is
therefore not equivalent to averaging in virtual-time. Fortunately the two can be
related. Expressing the real timet′as an integral over virtual-timeτaccording to
t′=

∫t

0 dτ/swe obtain

lim

t′→∞

1

t′

∫t′

0

A(P/s,R)dτ′= lim

t′→∞

t

t′

1

t

∫t

0

A(P/s,R)dτ/s

=

[

lim

t′→∞

1

t

∫t

0

A(P/s,R)dτ/s

]/(

lim

t′→∞

1

t

∫t

0

dτ/s

)

=〈A(P/s,R)/s〉/〈 1 /s〉. (8.94)

It can be verified (see Problem 8.9) that the latter expression coincides with the
canonical ensemble average if we putgequal to 3Ninstead of 3N+1. This means
that if we carry out the simulation usingEqs. (8.93)withg= 3 N, real-time averages
are equivalent to canonical averages.
Hoover[37]showed that by definingζ=sp′s/Q, Eqs. (8.93) can be reduced to
the simpler form

dri′

dt′

=

p′i

m

;

dp′i

dt′

=Fi−ζp′i; (8.95)

dζ

dt′

=

(∑

ip

′^2

i

m

−gkBT

)

/Q, (8.96)

and takinggequal to the number of degrees of freedom, i.e. 3N, he was able to show
that the distributionf(P,R,ζ)is phase space conserving, i.e. it satisfies Liouville’s
equation.