1549380323-Statistical Mechanics Theory and Molecular Simulation

Isokinetic ensemble 205

a single kinetic-energy constraint based on the virial theorem, since the number of
degrees of freedom isdN−1 rather thandN.
A simple yet effective integrator for the isokinetic equations can be obtained by
applying the Liouville operator approach. As usual, we begin by writingthe total
Liouville operator

iL=

∑N

i=1

[

pi

mi

·∇ri+

(

Fi−

[∑N

j=1(Fj·pj)/mj

2 K

]

pi

)

·∇pi

]

(4.12.14)

as the sum of two contributionsiL=iL 1 +iL 2 , where

iL 1 =

∑N

i=1

pi

mi

·∇ri

iL 2 =

∑N

i=1

(

Fi−

[∑N

j=1(Fj·pj)/mj

2 K

]

pi

)

·∇pi. (4.12.15)

The approximate evolution of an isokinetic system over a time ∆tis obtained by acting
with a Trotter factorized operator exp(iL∆t) = exp(iL 2 ∆t/2) exp(iL 1 ∆t) exp(iL 2 ∆t/2)
on an initial condition{p(0),r(0)}. The action of each of the operators in this factor-
ization can be evaluated analytically (Zhang, 1997; Minaryet al., 2003). The action
of exp(iL 2 ∆t/2) can be determined by first solving the coupled first-order differential
equations

dpi,α

dt

=Fi,α−

[∑N

j=1(Fj·pj)/mj

2 K

]

pi,α

=Fi,α−h ̇(t)pi,α (4.12.16)

withr 1 ,...,rN(and henceFi,α) held fixed. Here, we explicitly index both the spatial
components (α= 1,...,d) and particle numbersi = 1,...,N. The solution to eqn.
(4.12.16) can be expressed as

pi,α(t) =

pi,α(0) +Fi,αs(t)

s ̇(t)

, (4.12.17)

wheres(t) is a general integrating factor:

s(t) =

∫t

0

dt′exp[h(t′)]. (4.12.18)

By substituting into the time derivative of the constraint condition

∑N

i=1pi·p ̇i/mi= 0,
we find thats(t) satisfies a differential equation of the form