254 Molecular dynamics simulations
velocities have very peculiar values). The collision time for pairi,jis found by
|rij+tvij|=σ.
This is a quadratic equation which yields two solutions for the collision timet. The
first time after the current time must be chosen and recorded as the collision time of
pairij.
The simulation is now constructed as follows. At the beginning, the particles are
released from a lattice with velocities according to a Boltzmann distribution. For all
N(N− 1 )/2 pairs, the collision times are calculated and stored in a sorted list. The
first element of this list contains the first collision to take place. For this collision we
calculate the new velocities and positions. Then each pair containing at least one of
the two collision partners is removed from the list. Their new collision times are
calculated and added again to the list in such a way that the latter remains sorted
with respect to the collision times.
(b) How does the simulation time scale with the number of particles?
(c) Explain why the kinetic energy of the hard sphere system is rigorously constant.
In order to calculate pressures we must adapt the virial theorem to this system.
The virial theorem for smooth forces reads
βP
ρ
= 1 +
1
3 NkBT〈N
∑i= 1ri·Fi〉
.The problem is that the force acts over an infinitely small time during which it
has an infinite value. Show that for this case the virial theorem reads
βP
ρ
= 1 +
1
N〈v^2 〉t∑collisionsvij·rij,where the sum is over the collisions taking place within the sampling timet.
8.4 (a) Show that the Verlet algorithm can be written in the form:
(
p(t+h/ 2 )
x(t+h))
=(
p(t−h/ 2 )+hF[x(t)]
x(t)+hp(t−h/ 2 )+h^2 F[x(t)])
.(b) Find the Jacobian matrix of this map and show that the Verlet algorithm is
symplectic.
8.5 Consider a time-evolution operator acting on vectors in two dimensions, which is
described by the symplectic operator exp(tAD):
z(t)=exp(tAD)z( 0 ),
z=(p,x)=(z 1 ,z 2 ).
(a) Show that symplecticity implies that
∂A 1
∂p
=−
∂A 2
∂x
.