1549380323-Statistical Mechanics Theory and Molecular Simulation

204 Canonical ensemble

α=

∑N

∑i=1Fi·pi/mi

N

i=1p

2

i/mi

. (4.12.8)

When eqn. (4.12.8) is substituted into eqn. (4.12.5), the equations of motion for the
isokinetic ensemble become

r ̇i=

pi

mi

p ̇i=Fi−

[∑N

j=1Fj·pj/mj

∑N

j=1p

2

j/mj

]

pi. (4.12.9)

Because eqns. (4.12.9) were constructed to preserve eqn. (4.12.3), they manifestly
conservethe kinetic energy; however, that eqn. (4.12.3) is a conservation law of the
isokinetic equations of motion can also be verified by direct substitution. Eqns. (4.12.9)
are non-Hamiltonian and can, therefore, be analyzed via the techniques Section 4.9.
In order to carry out the analysis, we first need to calculate the phase space com-
pressibility:

κ=

∑N

i=1

[∇ri·r ̇i+∇pi·p ̇i]

=

∑N

i=1

∇pi·

{

Fi−

[∑N

j=1Fj·pj/mj

∑N

j=1p

2

j/mj

]

pi

}

=−

(dN−1)

∑N

i=1Fi·pi/mi

2 K

=

(dN−1)

2 K

dU(r 1 ,...,rN)

dt

. (4.12.10)

Thus, the functionw(x) is just (dN−1)U(r 1 ,...,rN)/ 2 K, and the phase space metric
becomes √
g= e−(dN−1)U(r^1 ,...,rN)/^2 K. (4.12.11)

Since the equations of motion explicitly conserve the total kinetic energy

∑N

i=1p

2
i/mi,
we can immediately write down the partition function generated by the equations of
motion:

Ω =

∫

dNpdNre−(dN−1)U(r^1 ,...,rN)/Kδ

(N

∑

i=1

p^2 i

mi

−(dN−1)kT

)

. (4.12.12)

The analysis shows that if the constant parameterKis chosen to be (dN−1)kT, then
the partition function becomes

Ω =

∫

dNpdNre−βU(r^1 ,...,rN)δ

(N

∑

i=1

p^2 i

mi

−(dN−1)kT

)

, (4.12.13)

which is the partition function of the isokinetic ensemble. Indeed, the constraint con-
dition

∑N

i=1p

2

i/mi= (dN−1)kTis exactly what we would expect for a system with