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the complexity of the system. This technique has recently been applied to lattice
gauge theory. Such equation gives another microcanonical formulation, which
was discussed in the context of continuum field theory by Scientists.
Monte Carlo methods are used as computational tools in many areas of chemi-
cal physics. Although this technique has been largely associated with obtaining
static, or equilibrium properties of model systems, Monte Carlo methods may
also be used to study dynamical phenomena. Often, the dynamics and coopera-
tively leading to certain structural or configurational properties of matter are not
completely amenable to a macroscopic continuum description. On the other hand,
molecular dynamics simulations describing the trajectories of individual atoms or
molecules on potential energy hyper surfaces are not computationally capable of
probing large systems of interacting particles at long times. Thus, in a dynami-
cal capacity, Monte Carlo methods are capable of bridging the ostensibly large
gap existing between these two well-established dynamical approaches, since the
“dynamics” of individual atoms and molecules are modeled in this technique, but
only in a coarse-grained way representing average features which would arise
from a lower-level result. The application of the Monte Carlo method to the study
of dynamical phenomena requires a self-consistent dynamical interpretation of
the technique and a set of criteria under which this interpretation may be practi-
cally extended. In recent publications, certain inconsistencies have been identi-
fied which arise when the dynamical interpretation of the Monte Carlo method
is loosely applied. These studies have emphasized that, unlike static properties,
which must be identical for systems having identical model Hamiltonians, dy-
namical properties are sensitive to the manner in which the time series of events
characterizing the evolution of a system is constructed. In particular, Monte Carlo
studies comparing dynamical properties simulated away from thermal equilib-
rium have revealed differences among various sampling algorithms. These stud-
ies have underscored the importance of using a Monte Carlo sampling procedure
in which transition probabilities are based on a reasonable dynamical model of
a particular physical phenomenon under consideration, in addition to satisfying
the usual criteria for thermal equilibrium. Unless transition probabilities can be
formulated in this way, a relationship between Monte Carlo time and real time
cannot be clearly demonstrated. In many Monte Carlo studies of time-dependent
phenomena, results are reported in terms of integral Monte Carlo steps, which
obfuscate a definitive role of time. Ambiguities surrounding the relationship of
Monte Carlo time to real time preclude rigorous comparison of simulated results
to theory and experiment, needlessly restricting the technique. Within the past few
years, the idea that Monte Carlo methods can be used to simulate the Poisson pro-
cess has been advanced in a few publications and some Monte Carlo algorithms,
which are implicitly based on this assumption, have been used. An attractive pros-
pect, since within the theory of Poisson processes the relationship between Monte
Carlo time and real time can be clearly established.
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