1549380323-Statistical Mechanics Theory and Molecular Simulation

124 Microcanonical ensemble

From the Jacobi identity, it follows that

{{F(x),H 2 (x)},H 1 (x)}=

−{{H 1 (x),F(x)},H 2 (x)}−{{H 2 (x),H 1 (x)},F(x)}. (3.13.8)

Substituting eqn. (3.13.8) into eqn. (3.13.7) yields, after some algebra,

[iL 1 ,iL 2 ]F(x) =−{F(x),{H 1 (x),H 2 (x)}}=−{F(x),H 3 (x)}, (3.13.9)

from which we see thatiL 3 ={...,H 3 (x)}.
A similar analysis can be carried out for each of the termsCkin eqn. (3.13.4).
Thus, for example, the operatorC 1 corresponds to a HamiltonianH ̃ 1 (x) given by

H ̃ 1 (x) =^1

24

{H 2 + 2H 1 ,{H 2 ,H 1 }}. (3.13.10)

Consequently, each operatorCkcorresponds to a HamiltonianH ̃k(x), and it follows
that the operatoriL+

∑∞

k=1∆t

2 kCkis generated by a HamiltonianH ̃(x; ∆t) of the

form

H ̃(x; ∆t) =H(x) +

∑∞

k=1

∆t^2 kH ̃k(x). (3.13.11)

This Hamiltonian, which appears as a power series in ∆t, is exactly conserved by the
factorized operator appearing on the left side of eqn. (3.13.4). Note thatH ̃(x; ∆t)→
H(x) as ∆t→0. The existence ofH ̃(x; ∆t) guarantees the long-time stability of
trajectories generated by the factorized propagator provided∆tis small enough that
H ̃andHare not that different, as the example of the harmonic oscillator above makes
clear. Thus, care must be exercised in the choice of ∆t, as the radius of convergence
of eqn. (3.13.11) is generally unknown.

3.14 Illustrative examples of molecular dynamics calculations

In this section, we present a few illustrative examples of molecular dynamics calcu-
lations (in the microcanonical ensemble) employing symplectic numerical integration
algorithms. We will focus primarily on investigating the properties of the numerical
solvers, including accuracy and long-time stability, rather than on the direct calculation
of observables (we will begin discussing observables in the next chapter). Throughout
the section, energy conservation will be measured via the quantity

∆E(δt,∆t,∆T,...) =

1

Nstep

N∑step

k=1

∣

∣

∣

∣

Ek(δt,∆t,∆T,...)−E(0)

E(0)

∣

∣

∣

∣, (3.14.1)

where the quantity ∆E(δt,∆t,∆T,...) depends on however many time steps are em-
ployed.Nstepis the total number of complete time steps taken (a “complete” timestep
is defined as an application of the full factorized classical propagator),Ek(δt,∆t,∆T,...)
is the energy obtained at thekth step, andE(0) is the initial energy. Eqn. (3.14.1)
measures the average absolute relative deviation of the energy from its initial value
(which determines the energy of the ensemble). Thus, it is a stringent measure of
energy conservation that is sensitive to drifts in the total energyover time.