Page 322 -

7.1 Random sequential deposition 309

7.1 Random sequential deposition

A number ofdynamic models in statistical physics do notcall onequi-
libriumconcepts such as the Boltzmann distributionand equiprobabil-
ity. From Chapter 2, weare already familiar with one ofthese models,
that ofrandomsequential deposition.This model describes hard disks
which are deposited, one after another, at randompositions in an ini-
tially empty square.Disks stick to the regionin which they have been
deposited if they do not overlap with anyother disk placed earlier;oth-
erwise, they are removed, leaving the state ofthe system unchanged.It
is instructiveto implement randomsequential deposition (see Alg. 7.1
(naive-deposition)) and to compare this dynamic Monte Carlo process
with Alg. 2.7 (direct-disks),the equilibrium direct-sampling method
forhard disks.

procedure naive-deposition
k ← 1
for t =1, 2,... do
⎧
⎪ x k ← ran(x min ,x max )
⎪
⎨ y k ← ran (y min ,y max)
⎪
⎪
if ( min l<k [dist(x k , x l )] > 2r)then

⎪
⎪ output {x k ,t}
⎪
⎪
⎩
k ← k +1
——
Algorithm 7.1 naive-deposition. Depositing hard disks of radius r on
a deposition region delimited by {x min,y min} and {x max,y max}.
As seen in earlier chapters, equilibrium Monte Carlo problems possess
a stationary probability distribution π(x). Incontrast, dynamic pro-
cesses such as randomsequential deposition are defined through a rule.
In fact, the ruleis all there is to the model.
Randomsequential depositionraisesseveralquestions.Forexample,
we would like to compute the stopping time t s for each sample, the
time after which it becomes impossibleto place an additional disk in
the system.Wenotice that Alg. 7.1 (naive-deposition) is unableto
decide whether the current simulationtime issmaller or larger than the
stopping time.
We would like to investigate the structure of the final state, after t s .
More generally, we would like to compute the ensemble-averaged density
ofthe system as a function oftime, upto the stopping time, fordifferent
sizes of the depositionregion.To compare the time behavior ofdifferent-
sized systems, wemust rescalethe time as τ = t/(depositionarea). This
lets uscompare systems which haveseen the samenumber ofdeposition
attempts per unit area.Inthe limit ofa large area L × L,the rescaled
stopping time τ s diverges, and it becomes difficultto study the late
stages of the depositionprocess.