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Monte Carlo computer simulations are now a standard tool in
scientific fields such as condensed-matter physics, including
surface-physics and applied-physics problems (metallurgy,
diffusion, and segregation, etc. ), chemical physics, including
studies of solutions, chemical reactions, polymer statistics, etc.,
and field theory. With the increasing ability of this method to
deal with quantum-mechanical problems such as quantum spin systems
or many-fermion problems, it will become useful for other questions
in the fields of elementary-particle and nuclear physics as well.
The large number of recent publications dealing either with
applications or further development of some aspects of this method
is a clear indication that the scientific community has realized
the power and versatility of Monte Carlo simula tions, as well as
of related simulation techniques such as "molecular dynamics" and
"Langevin dynamics," which are only briefly mentioned in the
present book. With the increasing availability of recent
very-high-speed general-purpose computers, many problems become
tractable which have so far escaped satisfactory treatment due to
prac tical limitations (too small systems had to be chosen, or too
short averaging times had to be used). While this approach is
admittedly rather expensive, two cheaper alternatives have become
available, too: (i) array or vector processors specifical ly suited
for wide classes of simulation purposes; (ii) special purpose
processors, which are built for a more specific class of problems
or, in the extreme case, for the simulation of one single model
system."
Monte Carlo methods have been a tool of theoretical and
computational scientists for many years. In particular, the
invention and percolation of the algorithm of Metropolis,
Rosenbluth, Rosenbluth, Teller, and Teller sparked a rapid growth
of applications to classical statistical mechanics. Although
proposals for treatment of quantum systems had been made even
earlier, only a few serious calculations had heen carried out. Ruch
calculations are generally more consuming of computer resources
than for classical systems and no universal algorithm had--or
indeed has yet-- emerged. However, with advances in techniques and
in sheer computing power, Monte Carlo methods have been used with
considerable success in treating quantum fluids and crystals,
simple models of nuclear matter, and few-body nuclei. Research at
several institutions suggest that they may offer a new approach to
quantum chemistry, one that is independent of basis ann yet capable
of chemical accuracy. That. Monte Carlo methods can attain the very
great precision needed is itself a remarkable achievement. More
recently, new interest in such methods has arisen in two new a~as.
Particle theorists, in particular K. Wilson, have drawn attention
to the rich analogy between quantum field theoty and statistical
mechanics and to the merits of Monte Carlo calculations for lattice
gauge theories. This has become a rapidly growing sub-field. A
related development is associated with lattice problems in quantum
physics, particularly with models of solid state systems. The~ is
much ferment in the calculation of various one-dimensional problems
such as the'Hubbard model.
Deals with the computer simulation of complex physical sys- tems
encounteredin condensed-matter physics and statistical mechanics as
well as in related fields such as metallurgy, polymer
research,lattice gauge theory and quantummechanics.
In the seven years since this volume first appeared. there has been
an enormous expansion of the range of problems to which Monte Carlo
computer simulation methods have been applied. This fact has
already led to the addition of a companion volume ("Applications of
the Monte Carlo Method in Statistical Physics", Topics in Current
Physics. Vol . 36), edited in 1984, to this book. But the field
continues to develop further; rapid progress is being made with
respect to the implementation of Monte Carlo algorithms, the
construction of special-purpose computers dedicated to exe cute
Monte Carlo programs, and new methods to analyze the "data"
generated by these programs. Brief descriptions of these and other
developments, together with numerous addi tional references, are
included in a new chapter , "Recent Trends in Monte Carlo
Simulations" , which has been written for this second edition.
Typographical correc tions have been made and fuller references
given where appropriate, but otherwise the layout and contents of
the other chapters are left unchanged. Thus this book, together
with its companion volume mentioned above, gives a fairly complete
and up to-date review of the field. It is hoped that the reduced
price of this paperback edition will make it accessible to a wide
range of scientists and students in the fields to which it is
relevant: theoretical phYSics and physical chemistry , con
densed-matter physics and materials science, computational physics
and applied mathematics, etc.
Monte Carlo methods have been a tool of theoretical and
computational scientists for many years. In particular, the
invention and percolation of the algorithm of Metropolis,
Rosenbluth, Rosenbluth, Teller, and Teller sparked a rapid growth
of applications to classical statistical mechanics. Although
proposals for treatment of quantum systems had been made even
earlier, only a few serious calculations had heen carried out. Ruch
calculations are generally more consuming of computer resources
than for classical systems and no universal algorithm had--or
indeed has yet-- emerged. However, with advances in techniques and
in sheer computing power, Monte Carlo methods have been used with
considerable success in treating quantum fluids and crystals,
simple models of nuclear matter, and few-body nuclei. Research at
several institutions suggest that they may offer a new approach to
quantum chemistry, one that is independent of basis ann yet capable
of chemical accuracy. That. Monte Carlo methods can attain the very
great precision needed is itself a remarkable achievement. More
recently, new interest in such methods has arisen in two new a~as.
Particle theorists, in particular K. Wilson, have drawn attention
to the rich analogy between quantum field theoty and statistical
mechanics and to the merits of Monte Carlo calculations for lattice
gauge theories. This has become a rapidly growing sub-field. A
related development is associated with lattice problems in quantum
physics, particularly with models of solid state systems. The~ is
much ferment in the calculation of various one-dimensional problems
such as the'Hubbard model.
This introduction to Monte Carlo methods seeks to identify and
study the unifying elements that underlie their effective
application. Initial chapters provide a short treatment of the
probability and statistics needed as background, enabling those
without experience in Monte Carlo techniques to apply these ideas
to their research.
The book focuses on two basic themes: The first is the importance
of random walks as they occur both in natural stochastic systems
and in their relationship to integral and differential equations.
The second theme is that of variance reduction in general and
importance sampling in particular as a technique for efficient use
of the methods. Random walks are introduced with an elementary
example in which the modeling of radiation transport arises
directly from a schematic probabilistic description of the
interaction of radiation with matter. Building on this example, the
relationship between random walks and integral equations is
outlined. The applicability of these ideas to other problems is
shown by a clear and elementary introduction to the solution of the
Schrodinger equation by random walks.
The text includes sample problems that readers can solve by
themselves to illustrate the content of each chapter.
This is the second, completely revised and extended edition of the
successful monograph, which brings the treatment up to date and
incorporates the many advances in Monte Carlo techniques and their
applications, while retaining the original elementary but general
approach.
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