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12
Symplectic and Reversible Integrators
time reversible. Again the non unitarity is due to the fact that the two factorized exponential operators
are non commuting. We can overcome this problem by halving the time step and using the approximant:
e(A+B)t
≃
eAt/2 eBt/2 eBt/2 eAt/2 = eAt/2 eBt eAt/2 .
(2.21)
The resulting propagator is clearly unitary, therefore time reversible, and is also correct to the second
order [74]. Thus, requiring that the product of the exponential operator be unitary, automatically leads to
more accurate approximations of the true discrete time propagator [75, 74]. Applying the same argument
to the propagator (2.18) we have
eiL∆t
=
e(q̇ ∂q +ṗ ∂p )∆t ≃ eṗ ∂p ∆t/2 eq̇ ∂q ∆t eṗ ∂p ∆t/2 + O(∆t3 ).
∂
∂
∂
∂
∂
(2.22)
The action of an exponential operator e(a∂/∂x) on a generic function f (x) trivially corresponds to the Taylor
expansion of f (x) around the point x at the point x + a, that is
ea∂/∂x f (x)
= f (x + a).
(2.23)
Using Eq. (2.23), the time reversible and symplectic integration algorithm can now be derived by acting
with our Hermitian operator Eq. (2.22) onto the state vector at t = 0 to produce updated coordinate and
momenta at a later time ∆t. The resulting algorithm is completely equivalent to the well known velocity
Verlet:
p(∆t/2) =
q(∆t)
=
p(∆t)
=
p(0) + F (0)∆t/2
p(∆t/2)
q(0) +
∆t
m
p(∆t/2) + F (∆t)∆t/2.
(2.24)
We first notice that each of the three transformations obeys the symplectic condition Eq. (2.8) and has a
Jacobian determinant equal to one. The product of the three transformation is also symplectic and, thus,
phase volume preserving. Finally, since the discrete time propagator (2.22) is unitary, the algorithm is
time reversible.
One may wonder what it is obtained if the operators q̇∂/∂q and −∂V /∂q(∂/∂p) are exchanged in the
definition of the discrete time propagator (2.22). If we do so, the new integrator is
p(0)
∆t/2
m
= p(0) + F [q(∆t/2)]∆t
p(∆t)
= q(∆t/2) +
∆t/2.
m
q(∆t/2) = q(0) +
p(∆t)
q(∆t)
(2.25)
This algorithm has been proved to be equivalent to the so-called Leap-frog algorithm [76]. Tuckerman
et al. [19] called this algorithm position Verlet which is certainly a more appropriate name in the light of
the exchanged role of positions and velocities with respect to the velocity Verlet. Also, Eq. (2.21) clearly
shows that the position Verlet is essentially identical to the Velocity Verlet. A shift of a time origin by
∆t/2 of either Eq. (2.25) or Eq. (2.24) would actually make both integrator perfectly equivalent. However,
as pointed out in Ref. [20], half time steps are not formally defined, being the right hand side of Eq. (2.21)
an approximation of the discrete time propagator for the full step ∆t. Velocity Verlet and Position Verlet,
therefore, do not generate numerically identical trajectories although of course the trajectories are similar.
We conclude this section by saying that is indeed noticeable that using the same Liouville formalism
different long-time known schemes can be derived. The Liouville approach represent therefore a unifying
treatment for understanding the properties and relationships between stepwise integrators.
2.3
Potential Subdivision and Multiple Time Steps Integrators
for NVE Simulations
The ideas developed in the preceding sections can be used to build multiple time step integrators. Multiple
time step integration is based on the concept of reference system. Let us now assume that the system