Computational Physics

170 Classical equilibrium statistical mechanics

are always much smaller than those of experimental systems.^1 Furthermore, a time
average of a physical quantityAis given by

A ̄= lim

T→∞

1

T

∫T

0

A(t)dt, (7.1)

and we want to obtain results in a finite amount of time! In a molecular dynam-
ics simulation (see Chapter 8), the typical simulation time is of the order of
10 −^9 –10−^6 seconds, far below the time in which most measuring devices sample
physical quantities. The results of such simulations can only be representative if
the spatial correlations extend over ranges smaller than the system size and if the
correlation time of the system is smaller than the simulation time. Sometimes it is
possible to extract useful information from simulations of systems with a size much
smaller than the correlation length by extrapolation – this is done in the finite-size
scaling method which will be discussed in Section 7.3.2. In this chapter, we shall
almost exclusively be concerned with systems in equilibrium.

7.1.1 Ensembles

If a system is thermally and mechanically insulated, the internal energy will remain
unchanged in the course of time. If the system is not insulated, it will eventually
take on the temperature of its surroundings (we assume that the surroundings have
a constant temperature). Such physical quantities, which are either kept fixed or
whose average value is controlled externally, are calledsystem parameters. Differ-
ent experimental circumstances correspond to different parameters being kept fixed.
In the theory of statistical physics, these cases correspond to differentensembles.
We shall see that adapting the simulation techniques for classical many-particle
systems (Monte Carlo and molecular dynamics) to these experimental situations is
a nontrivial problem – that is why we consider the ensemble theory in some detail
in this section.
The fundamental postulate, or assumption, of statistical mechanics pertains to
systems with fixed energyE, volumeVand particle numberN(in magnetic sys-
tems, instead of the volumeV, the external magnetic fieldHis kept constant).
The fundamental postulate says that all states accessible to the system and hav-
ing a prescribed energy, volume and number of particles are equally likely to be
visited in the course of time (the ergodic hypothesis). This leads to an identifica-
tion of the time averageA ̄(7.1) of the physical quantityAwith a uniform average
over all accessible states – the latter is denoted as〈A〉. Denoting the states byX,

(^1) A notable exception is formed by the so-called mesoscopic systems which contain typically 10 (^2) to 10 5
particles.