BASIC CONCEPTS AND AN OVERVIEW - Foundations of Nanotechnology: Pore Size in Carbon-Based Nano-Adsorbents - page 114

1.3.1 MONTE CARLO SIMULATION, APPLICATION, IMPORTANCE
AND ITS AIMS
In a Monte Carlo simulation we attempt to follow the 'time dependence' of a mod-
el for which change, or growth, does not proceed in some rigorously pre-denned
fashion (according to Newton's equations of motion) but rather in a stochastic
manner which depends on a sequence of random numbers which is generated dur-
ing the simulation method, in a simulation cell that design for special purposes.
With a second, different sequence of random numbers the simulation will not give
identical results but will yield values, which agree with those, obtained from the
first sequence to within some 'statistical error.' A very large number of different
problems fall into this category: in percolation an empty lattice is gradually filled
with particles by placing a particle on the lattice randomly with each 'tick of the
clock.' Considering problems of statistical mechanics, we may be attempting to
sample a region of phase space in order to estimate certain properties of the mod-
el, although we may not be moving in phase space along the same path, which
an exact solution to the time dependence of the model would yield. Remember
that the task of equilibrium statistical mechanics is to calculate thermal averages
of (interacting) many-particle systems. Monte Carlo simulations can do that, tak-
ing proper account of statistical errors and their effects in such systems. Many of
these models will be discussed in more detail in later chapters so we shall not pro-
vide further details here. Since the accuracy of a Monte Carlo estimate depends
upon the thoroughness with which phase space is probed, improvement may be
obtained by simply running the calculation a little longer to increase the number
of samples. Unlike in the application of many analytic techniques (perturbation
theory for which the extension to higher order may be prohibitively difficult), the
improvement of the accuracy of Monte Carlo results is possible not just in prin-
ciple but also in practice.
The method of Monte Carlo simulation has proved very useful for studying
the thermodynamic properties of model systems with moderately many degrees
of freedom. The idea is to sample the system's phase space stochastically, using a
computer to generate a series of random configurations. We take the phase space
to consist of N discrete states (with label i), though the method applies equally
to continuous systems. Often only a tiny fraction of the phase space (the part at
low energy) is relevant to the properties being studied, due to the strong variation
of the Boltzmann equation) in the canonical ensemble (CE). It is then helpful to
sample in an ensemble (with relative weights wi i and absolute probabilities pi i =
w i /P j w j ), which is concentrated on this region of phase space. The Metropolis
algorithm samples directly in the CE, and is good at determines many physical
properties [12, 17, 29].
The price to be paid for this is that successive configurations are not indepen-
dent (typically they have a single microscopic difference), but instead form a Mar-
kov chain with some equilibration time teq(w i ).We may distinguish two important