1549380323-Statistical Mechanics Theory and Molecular Simulation

114 Microcanonical ensemble

iL=

∑N

i=1

pi

mi

·

∂

∂ri

+

∑N

i=1

Fi·pi. (3.10.34)

If we writeiL=iL 1 +iL 2 withiL 1 andiL 2 defined in a manner analogous to eqn.
(3.10.13), then it can be easily shown that the Trotter factorization of eqn. (3.10.22)
yields the velocity Verlet algorithm of eqns. (3.8.7) and (3.8.9) because all of the terms
iniL 1 commute with each other, as do all of the terms iniL 2. It should be noted that
if a system is subject to a set of holonomic constraints imposed via the SHAKE and
RATTLE algorithms, the symplectic and time-reversibility propertiesof the Trotter-
factorized integrators are lost unless the iterative solutions for the Lagrange multipliers
are iterated tofullconvergence.
Before concluding this section, one final point should be made. It is not necessary
to grind out explicit finite difference equations by applying the operators in a Trotter
factorization analytically. Note that the velocity Verlet algorithm can be expressed as
a three-step procedure:

p(∆t/2) =p(0) +

∆t

2

F(x(0))

x(∆t) =x(0) +

∆t

m

p(∆t/2)

p(∆t) =p(∆t/2) +

∆t

2

F(x(∆t)). (3.10.35)

This three-step procedure can also be rewritten to resemble actual lines of computer
code

p=p+ 0. 5 ∗∆t∗F

x=x+ ∆t∗p/m

Recalculate the force

p=p+ 0. 5 ∗∆t∗F. (3.10.36)

Eqns. (3.10.35) and (3.10.36) could also be cast in terms ofv= p/mas in eqn.
(3.10.32). The third line in (3.10.36) involves a call to some function or subroutine
that updates the force from the new positions generated in the second line. When
written this way, the specific instructions are: i) perform a momentum translation; ii)
follow this by a position translation; iii) recalculate the force using thenew position; iv)
use the new force to perform a momentum translation. Note, however, that these are
just the steps required by the operator factorization scheme ofeqn. (3.10.23): The first
operator that acts on the phase space vector is exp[(∆t/2)F(x)∂/∂p], which produces
the momentum translation; the next operator exp[∆t(p/m)∂/∂p] takes the output of
the preceding step and performs the position translation; since this step changes the
positions, the force must be recalculated; the last operator exp[(∆t/2)F(x)∂/∂p] pro-
duces the final momentum translation using the new force. The fact that instructions
in computer code can be written directly from the operator factorization scheme, by-
passing the lengthy algebra needed to derive explicit finite-difference equations, is an