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This volume is the third edition of the first-ever elementary book
on the Langevin equation method for the solution of problems
involving the translational and rotational Brownian motion of
particles and spins in a potential highlighting modern applications
in physics, chemistry, electrical engineering, and so on. In order
to improve the presentation, to accommodate all the new
developments, and to appeal to the specialized interests of the
various communities involved, the book has been extensively
rewritten and a very large amount of new material has been added.
This has been done in order to present a comprehensive overview of
the subject emphasizing via a synergetic approach that seemingly
unrelated physical problems involving random noise may be described
using virtually identical mathematical methods in the spirit of the
founders of the subject, viz., Einstein, Langevin, Smoluchowski,
Kramers, etc. The book has been written in such a way that all the
material should be accessible both to an advanced researcher and a
beginning graduate student. It draws together, in a coherent
fashion, a variety of results which have hitherto been available
only in the form of scattered research papers and review articles.
Presenting in a coherent and accessible fashion current results in
nanomagnetism, this book constitutes a comprehensive, rigorous and
readable account, from first principles of the classical and
quantum theories underlying the dynamics of magnetic nanoparticles
subject to thermal fluctuations.Starting with the Larmor-like
equation for a giant spin, both the stochastic (Langevin) equation
of motion of the magnetization and the associated evolution
(Fokker-Planck) equation for the distribution function of the
magnetization orientations of ferromagnetic nanoparticles
(classical spins) in a heat bath are developed along with their
solution (using angular momentum theory) for arbitrary
magnetocrystalline-Zeeman energy. Thus, observables such as the
magnetization reversal time, relaxation functions, dynamic
susceptibilities, etc. are calculated and compared with the
predictions of classical escape rate theory including in the most
general case spin-torque-transfer. Regarding quantum effects, which
are based on the reduced spin density matrix evolution equation in
Hilbert space as is described at length, they are comprehensively
treated via the Wigner-Stratonovich formulation of the quantum
mechanics of spins via their orientational quasi-probability
distributions on a classically meaningful representation space.
Here, as suggested by the relevant Weyl symbols, the latter is the
configuration space of the polar angles. Hence, one is led, by
mapping the reduced density matrix equation onto that space, to a
master equation for the quasi-probability evolution akin to the
Fokker-Planck equation which may be solved in a similar way. Thus,
one may study in a classical-like manner the evolution of
observables with spin number ranging from an elementary spin to
molecular clusters to the classical limit, viz. a nanoparticle. The
entire discussion hinges on the one-to-one correspondence between
polarization operators in Hilbert space and the spherical harmonics
allied to concepts of spin coherent states long familiar in quantum
optics.Catering for the reader with only a passing knowledge of
statistical and quantum mechanics, the book serves as an
introductory text on a complicated subject where the literature is
remarkably sparse.
Our original objective in writing this book was to demonstrate how
the concept of the equation of motion of a Brownian particle - the
Langevin equation or Newtonian-like evolution equation of the
random phase space variables describing the motion - first
formulated by Langevin in 1908 - so making him inter alia the
founder of the subject of stochastic differential equations, may be
extended to solve the nonlinear problems arising from the Brownian
motion in a potential. Such problems appear under various guises in
many diverse applications in physics, chemistry, biology,
electrical engineering, etc. However, they have been invariably
treated (following the original approach of Einstein and
Smoluchowski) via the Fokker-Planck equation for the evolution of
the probability density function in phase space. Thus the more
simple direct dynamical approach of Langevin which we use and
extend here, has been virtually ignored as far as the Brownian
motion in a potential is concerned. In addition two other
considerations have driven us to write this new edition of The
Langevin Equation. First, more than five years have elapsed since
the publication of the third edition and following many suggestions
and comments of our colleagues and other interested readers, it
became increasingly evident to us that the book should be revised
in order to give a better presentation of the contents. In
particular, several chapters appearing in the third edition have
been rewritten so as to provide a more direct appeal to the
particular community involved and at the same time to emphasize via
a synergetic approach how seemingly unrelated physical problems all
involving random noise may be described using virtually identical
mathematical methods. Secondly, in that period many new and
exciting developments have occurred in the application of the
Langevin equation to Brownian motion. Consequently, in order to
accommodate all these, a very large amount of new material has been
added so as to present a comprehensive overview of the subject.
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