1549380323-Statistical Mechanics Theory and Molecular Simulation

Extended phase space 181

gappearing in eqn. (4.8.1) will be determined by the condition that amicrocanonical
distribution of 2dN+ 2-dimensional phase space ofHNyields acanonicaldistribution
in the 2dN-dimensional physical phase space. The presence ofsin the kinetic energy
is essentially what we would expect for an agent that must scale the kinetic energy
in order to control its fluctuations. The choicegkTlnsas the potential ins, while
seemingly mysterious, is carefully chosen to ensure that a canonical distribution in the
physical phase space is obtained.
In order to see how the canonical distribution emerges fromHN, consider the
microcanonical partition function of the full 2dN+ 2-dimensional phase space:

Ω =

∫

dNrdNpdsdps

×δ

(N

∑

i=1

p^2 i

2 mis^2

+U(r 1 ,...,rN) +

p^2 s

2 Q

+gkTlns−E

)

, (4.8.2)

whereEis the energy of the ensemble. (For clarity, prefactors precedingthe inte-
gral have been left out.) The distribution of the physical phase space is obtained by
integrating oversandps. We first introduce a change of momentum variables:

̃pi=

pi

s

, (4.8.3)

which gives

Ω =

∫

dNrdNp ̃dsdpssdNδ

(N

∑

i=1

̃p^2 i

2 mi

+U(r 1 ,...,rN) +

p^2 s

2 Q

+gkTlns−E

)

=

∫

dNrdNpdsdpssdNδ

(

H(r,p) +

p^2 s

2 Q

+gkTlns−E

)

, (4.8.4)

whereH(r,p) is the physical Hamiltonian

H(r,p) =

∑N

i=1

p^2 i

2 mi

+U(r 1 ,...,rN). (4.8.5)

In the last line of eqn. (4.8.4), we have renamedp ̃iaspi. We can now integrate over
susing theδ-function by making use of the following identity: Given a functionf(s)
that has a single zero ats 0 ,δ(f(s)) can be replaced by

δ(f(s)) =

δ(s−s 0 )

|f′(s 0 )|

. (4.8.6)

Takingf(s) =H(r,p) +p^2 s/ 2 Q+gkTlns−E, the solution off(s 0 ) = 0 is