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This volume is the second edition of the 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 other areas.

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.

For teachers and high-school and university students, this work addresses the problem of teaching the electronic structure and chemical bonding of atoms and molecules. It presents the outcomes of thorough investigations of some teaching methods as well as an unconventional didactic approach.

Provides an understanding of the dynamics of electromagnetism; and marshals the evidence that in certain precisely defined topological conditions, electromagnetic theory (Maxwell's theory) must be extended or generalized in order to provide an explanation and understanding of unusual electromagnetic phenomena.

This text presents conceptual and experimental evidence showing that Heisenberg's uncertainty relations are not valid in all cases. It also derives a more general set of uncertainty relations, from the replacement of the Fourier nonlocal and nontemporal paradigm by wavelet local analysis.

The first section of the book is devoted to discussion of basic concepts of chemistry and physics. The second section applies these concepts and extends them to three classes of materials that make the laptop possible: liquid crystals, polymers, and semiconductors.

Divided into three parts in the book, the first three parts of this book formulate Newton dynamics from the Liouville equation. In the third part, this forms the basis for implementing the uncertainty postulate to formulate quantum mechanics. This book is useful for courses in classical and quantum mechanics, and atomic and molecular physics.