126 Microcanonical ensemble
-3 -2.5 -2 -1.5 -1
log 10 (Dt)-8
-7
-6
-5
-4
-3
-2
log10(D
E
)
Velocity Verlet
RESPA
RESPA(l=0.9)
(l=0.99)Fig. 3.6Logarithm of the energy conservation measure in eqn. (3.14.1) vs. logarithm of the
time step for a harmonic oscillator withm= 1,ω= 2 using the velocity Verlet algorithm
(solid line), RESPA withλ= 0.9 and a fixed small time step ofδt= 10−^4 and variable
large time step (dashed line), and RESPA withλ= 0.99 and the same fixed small time step
(dotted-dashed line).
significantly better energy conservation than the single time step case. A similar plot
forλ= 0.99 is also shown in Fig. 3.6. These examples illustrate the fact that the
RESPA method becomes more effective as the separation in time scales increases.
3.14.2 The Lennard–Jones fluid
We next consider a system ofNidentical particles interacting via a pair-wise additive
potential of the form
U(r 1 ,...,rN) =∑N
i=1∑N
j=i+14 ǫ[(
σ
|ri−rj|) 12
−
(
σ
|ri−rj|) 6 ]
. (3.14.3)
This potential, known as the Lennard–Jones potential (Lennard-Jones, 1924) is often
used to describe the Van der Waals forces between simple rare-gasatom systems as
well as in more complex systems. The numerical integration of Hamilton’s equations
r ̇i=pi/m,p ̇i=−∇iUrequires both the specification of initial conditions, which was
discussed in Section 3.8.3, as well as boundary conditions on the simulation cell. In this
case, periodic boundary conditions are employed as a means of reducing the influence
of the walls of the box. When periodic boundary conditions are employed, a particle
that leaves the box through a particular face reenters the system at the same point of