7 Classical equilibrium statistical mechanics
7.1 Basic theory
In this chapter we briefly review the theory of classical statistical mechanics with
emphasis on those issues which are relevant to computer simulations. We shall
assume that the reader has some background in thermodynamics and statistical
mechanics; for further reading, numerous textbooks are available [ 1 – 8 ].
Statistical mechanics concerns the study of systems with many (in principle infin-
itely many) degrees of freedom. The degrees of freedom are usually the positions
and momenta of particles, or magnetic moments (‘spins’). We restrict ourselves to
classical systems for which all degrees of freedom commute. The space spanned by
the degrees of freedom is calledphase space– every point in phase space represents
a particular configuration of the system. In the course of time, the system follows
a path in phase space, determined by the equations of motion. We are obviously
not interested in the values of all these degrees of freedom as a function of time:
only the time averages of physical quantities such as pressure are measurable. This
is because our measuring devices (thermometers, barometers) respond relatively
slowly; hence they give a time average of the physical quantity of interest. How-
ever, even if we could perform an instantaneous measurement of some quantity
we would find a result very close to the time average of that quantity as a result
of the law of large numbers, which teaches us that if a quantity is composed ofN
uncorrelated contributions, fluctuations in that quantity are of order 1/
√
N. This
implies that for typical macroscopic physical quantities (such as the temperature of
your cup of tea) for whichNis of the order of 10^24 , the fluctuations are as small
as∼ 10 −^12 if we neglect correlations. If correlations extend over∼100 particles,
the number of uncorrelated contributions is∼ 1024 / 100 = 1022 , so the fluctuations
remain extremely small.
Computer simulations always sample relatively few degrees of freedom, since
only a restricted amount of data can be stored in memory: system sizes in simulations
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