128 Microcanonical ensemble
The implication of this exercise is that a single dynamical trajectory conveys very
little information because a slight change in initial conditions changes the trajectory
completely. In the spirit of the ensemble concept, dynamical observables do not rely
on single trajectories. Rather, as we will explore further in Chapter 13, observables
require averaging over an ensemble of trajectories each with different initial conditions.
Thus, no single initial condition can be given special significance. Despite the sensitive
dependence on initial conditions of the LJ fluid, Fig. 3.7 shows that the energy is well
conserved over a single trajectory. The average value of the energy conservation based
on eqn. (3.14.1), which is around 10−^4 , is typical in molecular dynamics simulations.
0 1 2 3
t (ps)-20
-10
0
10
20
y^1(Å)
0 1 2 3 4 5 6
t (ps)0.0
5.0 ́ 10
-51.0 ́ 10 -4
1.5 ́ 10
-42.0 ́ 10
-4D
E
Fig. 3.7(Left) Theycoordinate for particle 1 as a function of time for two identical
Lennard–Jones systems whose initial conditions differ by only 10−^10 % in the position of
a single particle. (Right) The energy conservation as measured by eqn. (3.14.1) for one of the
two systems. The light-grey background shows the instantaneous fluctuations of the summand
in eqn. (3.14.1).
3.14.3 A realistic example: The alanine dipeptide in water
As a realistic example of a molecular dynamics calculation, we consider the alanine
dipeptide in water. An isolated alanine dipeptide is depicted in Fig. 3.8. The solvated
alanine dipeptide is one of the most studied simple peptide systems, both theoretically
and experimentally, as it provides important clues about the conformational variability
and thermodynamics of more complex polypeptides and biological macromolecules. At
the same time, the system is simple enough that its conformational equilibria can be
mapped out in great detail, which is important for benchmarking new models for the
interactions. Fig. 1.11 shows a schematic of the alanine dipeptide, which has been
capped at both ends by methyl groups. In the present simulation,a force field of
the type given in eqn. (3.11.1) is employed with the parameters corresponding to
the CHARMM22 model (MacKerellet al., 1998). In addition, water is treated as
a completely rigid molecule, which requires three internal distance constraints (see