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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 t

Langevin equations. --- Brownian motion processes. --- Langevin's equations --- Stochastic differential equations --- Wiener processes --- Brownian movements --- Fluctuations (Physics) --- Markov processes

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The Langevin and Generalised Langevin Approach To The Dynamics Of Atomic, Polymeric And Colloidal Systems is concerned with the description of aspects of the theory and use of so-called random processes to describe the properties of atomic, polymeric and colloidal systems in terms of the dynamics of the particles in the system. It provides derivations of the basic equations, the development of numerical schemes to solve them on computers and gives illustrations of application to typical systems.Extensive appendices are given to enable the reader to carry out computations to illustra

Brownian movements. --- Langevin equations. --- Physics. --- Random dynamical systems. --- Langevin equations --- Brownian movements --- Random dynamical systems --- Physics --- Mathematics --- Physical Sciences & Mathematics --- Atomic Physics --- Mathematical Statistics --- Natural philosophy --- Philosophy, Natural --- Dynamical systems, Random --- Langevin's equations --- Physical sciences --- Dynamics --- Differentiable dynamical systems --- Capillarity --- Liquids --- Matter --- Stochastic differential equations --- Properties

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Scientists are increasingly finding themselves engaged in research problems that cross the traditional disciplinary lines of physics, chemistry, biology, materials science, and engineering. Because of its broad scope, statistical mechanics is an essential tool for students and more experienced researchers planning to become active in such an interdisciplinary research environment. Powerful computational methods that are based in statistical mechanics allow complex systems to be studied at an unprecedented level of detail. This book synthesizes the underlying theory of statistical mechanics with the computational techniques and algorithms used to solve real-world problems and provides readers with a solid foundation in topics that reflect the modern landscape of statistical mechanics. Topics covered include detailed reviews of classical and quantum mechanics, in-depth discussions of the equilibrium ensembles and the use of molecular dynamics and Monte Carlo to sample classical and quantum ensemble distributions, Feynman path integrals, classical and quantum linear-response theory, nonequilibrium molecular dynamics, the Langevin and generalized Langevin equations, critical phenomena, techniques for free energy calculations, machine learning models, and the use of these models in statistical mechanics applications. The book is structured such that the theoretical underpinnings of each topic are covered side by side with computational methods used for practical implementation of the theoretical concepts.

Statistical mechanics. --- Statistical physics. --- Quantum theory. --- Molecular dynamics --- Monte Carlo method. --- Langevin equations. --- Feynman integrals. --- Critical phenomena (Physics) --- Mécanique statistique. --- Physique statistique. --- Dynamique moléculaire. --- Bioinformatique. --- Thermodynamique. --- Théorie quantique. --- Monte-Carlo, Méthode de. --- Langevin, Équation de. --- Feynman, Intégrales de. --- Phénomènes critiques (physique) --- Manuels d'enseignement supérieur. --- Simulation methods --- Mécanique statistique. --- Dynamique moléculaire. --- Théorie quantique. --- Monte-Carlo, Méthode de. --- Langevin, Équation de. --- Feynman, Intégrales de. --- Phénomènes critiques (physique) --- Manuels d'enseignement supérieur.

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The study of fractional integrals and fractional derivatives has a long history, and they have many real-world applications because of their properties of interpolation between integer-order operators. This field includes classical fractional operators such as Riemann–Liouville, Weyl, Caputo, and Grunwald–Letnikov; nevertheless, especially in the last two decades, many new operators have also appeared that often define using integrals with special functions in the kernel, such as Atangana–Baleanu, Prabhakar, Marichev–Saigo–Maeda, and the tempered fractional equation, as well as their extended or multivariable forms. These have been intensively studied because they can also be useful in modelling and analysing real-world processes, due to their different properties and behaviours from those of the classical cases.Special functions, such as Mittag–Leffler functions, hypergeometric functions, Fox's H-functions, Wright functions, and Bessel and hyper-Bessel functions, also have important connections with fractional calculus. Some of them, such as the Mittag–Leffler function and its generalisations, appear naturally as solutions of fractional differential equations. Furthermore, many interesting relationships between different special functions are found by using the operators of fractional calculus. Certain special functions have also been applied to analyse the qualitative properties of fractional differential equations, e.g., the concept of Mittag–Leffler stability.The aim of this reprint is to explore and highlight the diverse connections between fractional calculus and special functions, and their associated applications.

Caputo-Hadamard fractional derivative --- coupled system --- Hadamard fractional integral --- boundary conditions --- existence --- fixed point theorem --- fractional Langevin equations --- existence and uniqueness solution --- fractional derivatives and integrals --- stochastic processes --- calculus of variations --- Mittag-Leffler functions --- Prabhakar fractional calculus --- Atangana–Baleanu fractional calculus --- complex integrals --- analytic continuation --- k-gamma function --- k-beta function --- Pochhammer symbol --- hypergeometric function --- Appell functions --- integral representation --- reduction and transformation formula --- fractional derivative --- generating function --- physical problems --- fractional derivatives --- fractional modeling --- real-world problems --- electrical circuits --- fractional differential equations --- fixed point theory --- Atangana–Baleanu derivative --- mobile phone worms --- fractional integrals --- Abel equations --- Laplace transforms --- mixed partial derivatives --- second Chebyshev wavelet --- system of Volterra–Fredholm integro-differential equations --- fractional-order Caputo derivative operator --- fractional-order Riemann–Liouville integral operator --- error bound --- n/a --- Atangana-Baleanu fractional calculus --- Atangana-Baleanu derivative --- system of Volterra-Fredholm integro-differential equations --- fractional-order Riemann-Liouville integral operator