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.
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.
This volume is the second edition of the first-ever elementary book on the Langevin equation method for the solution of problems involving the Brownian motion in a potential, with emphasis on modern applications in the natural sciences, electrical engineering and so on. It has been substantially enlarged to cover in a succinct manner a number of new topics, such as anomalous diffusion, continuous time random walks, stochastic resonance etc, which are of major current interest in view of the large number of disparate physical systems exhibiting these phenomena. The book has been written in such a way that all the material should be accessible to an advanced undergraduate or beginning graduate student. It draws together, in a coherent fashion, a variety of results which have hitherto been available only in the form of research papers or scattered 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.
The book is suitable for a lecture course on the theory of Brownian motion, being based on final year undergraduate lectures given at Trinity College, Dublin. Topics that are discussed include: white noise; the Chapman-Kolmogorov equation ? Kramers-Moyal expansion; the Langevin equation; the Fokker-Planck equation; Brownian motion of a free particle; spectral density and the Wiener-Khintchin theorem ? Brownian motion in a potential application to the Josephson effect, ring laser gyro; Brownian motion in two dimensions; harmonic oscillators; itinerant oscillators; linear response theory; rotational Brownian motion; application to loss processes in dielectric and ferrofluids; superparamagnetism and nonlinear relaxation processes.As the first elementary book on the Langevin equation approach to Brownian motion, this volume attempts to fill in all the missing details which students find particularly hard to comprehend from the fundamental papers contained in the Dover reprint ? Selected Papers on Noise and Stochastic Processes, ed. N Wax (1954) ? together with modern applications particularly to relaxation in ferrofluids and polar dielectrics.
This book offers to the reader a self-contained treatment and systematic exposition of the real-valued theory of a nonabsolute integral on measure spaces. It is an introductory textbook to HenstockKurzweil type integrals defined on abstract spaces. It co
The Civil War obliterated Americas past, along with many of the founders visions of what America should be. Replacing those visions was the America that we have today. Any true understanding of America, both past and present, must include a specific understanding of this conflict. This work, with a thought-provoking introduction exploring the true causes of the war, traces the entire story of the conflict in a concise monthly summary. In addition to all the major events that shaped the war, key facts that have disappeared from most mainstream texts are also included, such as: Both Presidents Abraham Lincoln and Jefferson Davis lost young sons during the war The legendary Robert E. Lee faced intense southern criticism for military failures in the wars first year U.S. forces battled the Sioux Indians during the war, leading to the largest mass execution in American history A former Ohio congressman was banished to the South by Lincoln for opposing the war Facts are explored and myths are exposed as the conflict is put in its proper chronological perspective. For anyone seeking a general resource guide to the seminal event in American history, this is required reading.
From the ashes of the most terrible war in American history began the agonizing process of restoring the Union that became known as Reconstruction. Like the War Between the States itself, Reconstruction lasted longer and produced more tragedy than ever anticipated. This work explores the era's important events in a year-by-year digest. These events reflect the unintended and tragic consequences of excessive government intervention in the liberties of the people. They also illustrate how such intervention has helped transform America from a constitutional republic to the centralized empire that it is today. Key events that shaped both Reconstruction and subsequent American history include: The subjugation of former Confederates through the military and corrupt state governments, followed by the subjugation of former slaves through Jim Crow laws The new alliance between business and government, which introduced the crony capitalist economic system that flourishes today The rise of organized labor, women's suffrage, and other special interest groups seeking recognition The political intrigues and unprecedented scandals that undermined the people's trust in government The westward expansion that encroached on the land of Native Americans and virtually annihilated their way of life The complex Reconstruction era laid the groundwork that would establish America as a world power by the beginning of the 20th century. The fundamental and permanent changes that both the Civil War and Reconstruction brought to America are explored, as well as how such changes have posed a threat to individual freedom ever since. As a resource guide to a vital yet often misunderstood era in American history, this is essential reading.
In this exciting new work, David Coffey explores Sheridan's relationships with his subordinates and their substantial role in shaping the final year of the Civil War.
The first of two volumes, this book examines constitutionalism in Ireland in the 1930s. Donal K. Coffey places the document and its drafters in the context of a turbulent decade for the United Kingdom, the Commonwealth, and Europe. He considers a series of key issues leading up to its drafting, including the failure of the 1922 Constitution, the rise of nationalism in the 1920s and 1930s, and the abdication of Edward VIII. He sketches the drafting process, examines the roles of individual drafters and their intellectual influences, and considers the Constitution’s public reception, both domestically and internationally. This book illuminates a critical moment in Irish history and the confluence of national, Commonwealth, and international influences that gave rise to it, for scholars of Irish history as well as of legal, constitutional, and Commonwealth history more broadly.
Bear Mountain, once home to Native Americans and early settlers, had evolved by the 20th century into one of the nations most outstanding public parks. Threatened by quarrying operations and the relocation of Sing Sing Prison, this integral section of the Hudson Highlands was saved by the combined efforts of local citizens and the Palisades Interstate Park Commission. Graced by a deep, picturesque lake at the foot of the mountain and accessible by riverboat, train, and automobile, Bear Mountain State Park became a playground for New York City residents. Visionary park management enabled the park to expand, encompassing nature exhibits, a zoo, camps, and restaurants. It continues to be a year-round recreation center featuring hiking, boating, swimming, and cross-country skiing.
By the time of her death in 1912 this enigmatic woman's life had become the stuff of myth, which she had only encouraged. Stories featured her fighting beside her husband in battle while treating the wounded. She claimed to have received a captain's commission for her services and to have been a close friend of President Lincoln, which apparently she was not. One story even placed her in command of a company of troops during Sherman's March to the Sea."--BOOK JACKET.
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