108 Microcanonical ensemble
then cycling through the constraints again to compute a new increment to the multi-
plier until convergence is reached as was proposed for the positionupdate step. The
latter iterative procedure is known as the RATTLE algorithm (Andersen, 1983). Once
converged multipliers are obtained, the final velocity update is performed by substi-
tuting into eqn. (3.9.17). The SHAKE algorithm can be used in conjunction with the
Verlet and velocity Verlet algorithms while RATTLE is particular to velocity Verlet.
For other numerical solvers, constraint algorithms need to be adapted or tailored for
consistency with the particulars of the solver.
3.10 The classical time evolution operator and numerical integrators
Thus far, we have discussed numerical integration in a somewhat simplistic way, relying
on Taylor series expansions to generate update procedures. However, because there
are certain formal properties of Hamiltonian systems that should be preserved by
numerical integration methods, it is important to develop a formal structure that
allows numerical solvers to be generated more rigorously. The framework we seek is
based on the classical time evolution operator approach, and we willreturn to this
framework repeatedly throughout the book.
We begin by considering the time evolution of any functiona(x) of the phase space
vector. Ifa(x) is evaluated along a trajectory xt, then in generalized coordinates, the
time derivative ofa(xt) is given by the chain rule
da
dt=
∑^3 N
α=1[
∂a
∂qα
q ̇α+∂a
∂pα
p ̇α]
. (3.10.1)
Hamilton’s equations
q ̇α=∂H
∂pα, p ̇α=−∂H
∂qα(3.10.2)
are now used for the time derivatives appearing in eqn. (3.10.1), which yields
da
dt=
∑^3 N
α=1[
∂a
∂qα∂H
∂pα−
∂a
∂pα∂H
∂qα]
={a,H}. (3.10.3)The bracket{a,H}appearing in eqn. (3.10.3) is the Poisson bracket from eqns. (1.6.19)
and (1.6.20). Eqn. (3.10.3) indicates that the Poisson bracket betweena(x) andH(x)
is a generator of the time evolution ofa(xt).
The Poisson bracket allows us to introduce an operator on the phase space that
acts on any phase space function. Define an operator,iL, wherei=
√
−1, byiLa={a,H}, (3.10.4)whereLis known as theLiouville operator. Note thatiLcan be expressed abstractly
asiL={...,H}, which means “take whatever functioniLacts on and substitute it