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Hard disks and spheres 2

In the first chapter ofthis book, weconsidered simpleproblems in statis-
tics: pebbles on the beach, needles falling butnever rolling, and people 2.1 Newtonian deterministic
mechanics 83
strolling onheliports by night.Wenow move onto study model systems
in physics—particles with positions, velocities, and interactions—that 2.2 Boltzmann’s statistical
mechanics 92
obey classical equations ofmotion.To understand how physical systems
2.3 Pressure and the
can be treated with the tools of statistics and simulated with Monte Boltzmann distribution 108
Carlo methods, weshall consider the hard-sphere model, which lies at 2.4 Large hard-sphere
the heart of statistical mechanics.Hard spheres, which are idealizations systems 119
ofbilliard balls, in free space orinabox,behaveasfree particles when- 2.5 Cluster algorithms 122
ever they are notin contact with other particles or with walls, and obey Exercises 128
simplereflectionrules oncontact. References 130
The hard-sphere model played a crucial role in the genesis ofsta-
tistical mechanics.Since the early days of machine computing, in the
1950s, and upto the present day, the hard-sphere model has spurred
the development ofcomputer algorithms, and both the explicit numeri-
cal integration of Newton’sequations and the Markov-chain Monte Carlo
algorithm were first tried out onthis model. Weshall use such algorithms
to illustrate mechanics and statistical mechanics, and to introduce the
fundamental concepts of statistical mechanics: the equiprobability prin-
ciple, the Boltzmann distribution, the thermodynamic temperature, and
the pressure.Weshall also be concerned with the practical aspects of
computations and witness the problems of Markov-chain algorithms at
high densities.Weshall conclude the chapter with a first discussion of
sophisticated cluster algorithms which are commonto many fields of
computational physics.
In the hard-sphere model,all configurations havethe same potential
energy and there is no energetic reasonto prefer any configuration over
anyother. Only entropic effects come into play. Inspite ofthis restric-
tion, hard spheres and disks show a rich phenomenology and exhibit
phase transitions fromthe liquid to the solid state.These “entropic
transitions”were once quite unsuspected, and then hotly debated, be-
fore they ended uppoorly understood, especially in two dimensions.
The physics ofentropywill appear in several places in this chapter, to
be taken up again in earnest in Chapter 6.