1549380323-Statistical Mechanics Theory and Molecular Simulation

130 Microcanonical ensemble

can be seen that the temperature exhibits regular fluctuations, leading to a well-defined
thermodynamic temperature of 300 K. The figure also shows that the energy is well
conserved over this run. The CPU time needed for one step of molecular dynamics
using the RESPA integrator is nearly the same as the time that would be required for
a single time step method such as velocity Verlet with a 1.0 fs time step because of the
low computational overhead of the bonding and bending forces compared to that of a
full force calculation. Hence, the gain in efficiency using the RESPA solver is very close
to a factor of 6. In order to examine the conformational changestaking place in the
dipeptide over the run, we plot the dihedral anglesφandψin Fig. 3.10 as functions
of time. Different values of these angles correspond to different stable conformations
of the alanine dipeptide. We see from the figure that the motion of these angles is
characterized by local fluctuations about these stable conformations with occasional
abrupt transitions to a different stable conformation. The fact that such transitions
are rare indicates that the full conformational space is not adequately sampled on the
1.5 ns time scale of the run. The problem of enhancing conformational sampling in
molecular dynamics will be treated in greater detail in Chapter 8.

0 500 1000 1500

t (ps)

-200

-100

0

100

200

f

0 500 1000 1500

t (ps)

-400

-300

-200

-100

0

y

Fig. 3.10Trajectories of the anglesφ(left) andψ(right) over a 1.5 ns run of the alanine
dipeptide in water.

3.15 Problems

3.1. Consider the standard Hamiltonian for a system ofNidentical particles

H=

∑

i=1

p^2 i

2 m

+U(r 1 ,...,rN).

a. Show that the microcanonical partition function can be expressed in the

form