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84 Hard disks and spheres

A collision occurs when the norm ofthe spatial distance vector

x k (t) − x l (t) = ∆ x +∆ v ·(t − t 0 ) (2.1)

∆x(t) x k (t 0 )−x l (t 0 ) v k −v l
equalstwice the radius σ of the disks (see Fig. 2.2). This can happen at
two times t 1 and t 2 , obtained by squaring eqn (2.1), setting |∆x| = 2σ,
and solving the quadratic equation

2
2
2
2
−(∆ x · · · ∆ v ) ± (∆ x · · · ∆ v ) − (∆ v ) ((∆ x ) − 4σ )
t 1,2 = t 0 + . (2.2)
(∆ v ) 2
The two disks will collide in the future only if the argument ofthe square
rootis positiveand if they are approaching each other ((∆ x · · · ∆ v ) < 0,
see Alg. 2.2 (pair-time)). The smallest ofall the pair collisiontimes
obviously gives the nextpaircollisionin the wholesystem (see Alg. 2.1
(event-disks)). Analogously, the parameters for the next wall collision
are computed froma simpletime-of-flight analysis (see Fig. 2.3,and
Alg. 2.1 (event-disks) again).
procedure pair-time
input ∆ x (≡ x k (t 0 ) − x l (t 0 ))
input ∆ v (≡ v k − v l =0)
2
2
2
2
Υ ← (∆ x · · · ∆ v ) −|∆ v | (|∆ x | − 4σ )
if (Υ > 0 and (∆ x · · · ∆ v ) < 0)then
√
( 2 3
t pair ← t 0 − (∆ x · · · ∆ v )+ Υ /∆ 2
v
else

t pair ←∞
output t pair
——
Algorithm 2.2 pair-time. Pair collision time for two particles starting
at time t 0 from positions x k and x l, and with velocities v k and v l.

t 0 t 1 t 2

Fig. 2.3 Computing the wall collision time t wall =min(t 1,t 2).

Continuing ourimplementation of Alg. 2.1 (event-disks), wenow
compute the velocities after the collision: collisions with a wall ofthe
box lead to a change ofsign ofthe velocity component normal to the