1549380323-Statistical Mechanics Theory and Molecular Simulation

Time step dependent Hamiltonians 121

eiL∆t≈eiL^5 ∆t/^2

×

[

eiL^4 δt/^2 eiL^3 δt/^2 eiL^2 δteiL^3 δt/^2 eiL^4 δt/^2

]n

×eiL^5 ∆t/^2. (3.12.8)

Note that sinceh 1 (q,p) = 0, the operatoriL 1 does not appear in the integrator.
What is particularly convenient about this decomposition is that the operators
exp(iLkδt/2) fork= 2, 3 ,4, can be applied analytically according to

eiLktq=qcos(ζkt) + sin(ζkt)ck

eiLktp=pcos(ζkt) + sin(ζkt)dk, (3.12.9)

where

ζk=

1

4 Ik

p·ck, (3.12.10)

anddk is defined analogously tock but with the components ofpreplacing the
components ofq. The action of exp(iL 5 ∆t) is just a translationp←p+ (∆t/2)F,
whereF=−∂U/∂q. Milleret al.present several numerical examples that exhibit the
performance of eqn. (3.12.8) on realistic systems, and the interested reader is referred
to the aforementioned paper (Milleret al., 2002) for more details.

3.13 Exactly conserved time step dependent Hamiltonians

As already noted, the velocity Verlet algorithm is an example of a symplectic algo-
rithm orsymplectic map, the latter indicating that the algorithm maps the initial
phase space point x 0 into x∆twithout destroying the symplectic property of classi-
cal mechanics. Although numerical solvers do not exactly conserve the Hamiltonian
H(x), a symplectic solver has the important property that there exists a Hamiltonian
H ̃(x,∆t) such that, along a trajectory,H ̃(x,∆t) remainscloseto the true Hamilto-
nian and isexactlyconserved by the map. By close, we mean thatH ̃(x,∆t) approaches
the true HamiltonianH(x) as ∆t→0. Because the auxiliary HamiltonianH ̃(x,∆t)
is a close approximation to the true Hamiltonian, it is referred to as a “shadow”
Hamiltonian (Yoshida, 1990; Toxvaerd, 1994; Gans and Shalloway, 2000; Skeel and
Hardy, 2001). The existence ofH ̃(x,∆t) ensures the error in a symplectic map will be
bounded. After presenting an illustrative example of a shadow Hamiltonian, we will
indicate how to prove the existence ofH ̃(x,∆t).
The existence of a shadow Hamiltonian does not mean that this Hamiltonian can
be constructed exactly for a general system. In fact, the general form of the shadow
Hamiltonian is not known. Skeel and coworkers have described how approximate
shadow Hamiltonians can be constructed practically (Skeel and Hardy, 2001) and have
provided formulas for shadow Hamiltonians up to 24th order in the time step (Engle