BASIC CONCEPTS AND AN OVERVIEW - Foundations of Nanotechnology: Pore Size in Carbon-Based Nano-Adsorbents - page 115

characteristics of a Monte Carlo simulation: its ergodicity (measured by teq(wi)) i ))
and its pertinence (measured by Ns(wi; i ; I), the average number of independent
samples needed to obtain the information I that we seek). We should choose wi i so
as to minimize the total number of configurations that need to be generated, which
is proportional to teq(w i )Ns(w i ; I). It is easy to specify an ensemble, which would
yield the sought information if independent samples could be drawn from it, but
an ensemble with too much weight at low energies may become fragmented into
“pools” at the bottoms of “valleys” of the energy function, and so have a large
equilibration time. For example, it is well known that at low temperatures the
Metropolis algorithm can get stuck in ordered or glassy phases. Ergodicity may
be improved by sampling instead in a non-physical ensemble with a broad energy
distribution, which allows the valleys to be connected by paths passing through
higher energies. A weight assignment leading to such a distribution cannot in gen-
eral be written as an explicit function of energy alone; rather it is an algorithm's
purpose to find this assignment, which then tells us about the density of states
(E). This reversal (starting with the distribution and finding the weights) of the
usual Monte Carlo process can be achieved using a series of normal simulations,
adjusting the weight wi i after each run so that the resulting energy distribution
w i (E) converges to the desired one. The last application reported here is a simula-
tion of a regular system with frustration, the triangular anti ferromagnetic, on a
48 × 48 parallelogram with periodic boundary conditions. Using 5 runs of 7.4 ×
105 sweeps, we obtained a ground state entropy of0.32320, with a variance of
0.00015, which is consistent with the exact bulk value

0.32307.As computers
have improved in capability, the simulation of large statistical systems governed
by known Hamiltonians has become an important tool of the theoretical physicist.
Applications range from studies of phase transitions in condensed matter to cal-
culation of hadronic properties via lattice gauge theory. Most of these simulations
rely on adaptations of the algorithm of Metropolis. This generates a sequence
of configurations via a Markovian process such that ultimately the probability
of encountering any given configuration in the sequence is proportional to the
Boltzmann weight function, S(C), where is the energy for a statistical mechanics
problem or the action for a quantum field theory simulation. Thus one obtains
a sample of configurations, which dominate the partition function sum or path
integral. An alternative simulation technique is the molecular dynamics or micro-
canonical method. This begins with a set of equations for a dynamical evolution,
which conserves the total energy. Upon numerical integration the system will flow
through phase space in a hopefully ergodic manner (Indeed, non-ergodic behavior
would represent a fascinating exception to the generic case.). Such a program
does not explicitly depend on an inverse temperature, which is determined dy-
namically by measuring, say, the average kinetic energy, which by equipartition
should be IkT per degree of freedom. Note that the conventional microcanonical
simulations make no use of random numbers, which are effectively generated by

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