Page 320 -

Dynamic Monte Carlo

methods 7

In the first six chapters ofthis book, wehaveconcentrated onequilibrium
statistical mechanics and related computational-physics approaches, no- 7.1 Random sequential
deposition 309
tably the equilibrium Monte Carlo method.These and other approaches
7.2 Dynamic spin algorithms 313
allowed usto determine partitionfunctions, energies, superfluid densi-
7.3 Disks on the unit sphere 321
ties, etc. Physical time played a minorrole, as the observables were gen-
erally time-independent.Likewise, Monte Carlo “time”was treated as of Exercises 333
secondary interest, if notanuisance: westrove only to make things hap- References 335
pen as quickly as possible, that is, to havealgorithms converge rapidly.
The moment has come to reach beyond equilibrium statistical mechan-
ics, and to explore time-dependent phenomena such as the crystallization
of hard spheres after a sudden increase in pressure orthe magnetic re-
sponse of Ising spins to an external fieldswitched onatsome initial time.
The localMonte Carlo algorithm often provides an excellent framework
forstudying dynamical phenomena.
The conceptual difference between equilibriumand dynamic Monte
Carlo methods cannotbe overemphasized.In the first case, wehavean
essentially unrestricted choice ofa priori probabilities, since we only want
to generate independent configurations x distributed with a probability
π(x),in whatever waywechoose, butas fast as possible.Indynamic
calculations, the time dependence becomes the main object of ourstudy.
Wefirst lookat this difference between equilibriumand dynamics in the
case of the random-sequential-depositionproblem of Chapter 2, where
apowerful dynamic algorithm perfectly implements the faster-than-the-
clock paradigm.Wethen discuss dynamic Monte Carlo methods forthe
Ising model and encounter the main limitation ofthe faster-than-the-
clock approach, the futility problem.
In the final section of this chapter, weapply a Monte Carlo method
called simulated annealing, an important tool forsolving difficult opti-
mizationproblems, mostly withoutany relationto physics.Inthisap-
proach, a discrete orcontinuous optimizationproblem is mapped onto
an artificial physical system whose ground state (at zero temperature or
infinite pressure) contains the solutionto the original task.This ground
state is slowly approached through simulation.Simulated annealing will
be discussed formonodisperse and polydisperse hard disks onthe surface
of a sphere, under increasing pressure.It works prodigiously in one case,
where the disks end upcrystallizing, butfailsin another case, where
they settleinto aglassy state.