Simulating Hamiltonian Dynamics

Geometric integrators are time-stepping methods, designed such that they exactly satisfy conservation laws, symmetries or symplectic properties of a system of differential equations. In this book the authors outline the principles of geometric integration and demonstrate how they can be applied to provide efficient numerical methods for simulating conservative models. Beginning from basic principles and continuing with discussions regarding the advantageous properties of such schemes, the book introduces methods for the N-body problem, systems with holonomic constraints, and rigid bodies. More advanced topics treated include high-order and variable stepsize methods, schemes for treating problems involving multiple time-scales, and applications to molecular dynamics and partial differential equations. The emphasis is on providing a unified theoretical framework as well as a practical guide for users. The inclusion of examples, background material and exercises enhance the usefulness of the book for self-instruction or as a text for a graduate course on the subject.

The simulation of physical systems requires a simplified, hierarchical approach which models each level from the atomistic to the macroscopic scale. From quantum mechanics to fluid dynamics, this book systematically treats the broad scope of computer modeling and simulations, describing the fundamental theory behind each level of approximation. Berendsen evaluates each stage in relation to its applications giving the reader insight into the possibilities and limitations of the models. Practical guidance for applications and sample programs in Python are provided. With a strong emphasis on molecular models in chemistry and biochemistry, this 2007 book will be suitable for advanced undergraduate and graduate courses on molecular modeling and simulation within physics, biophysics, physical chemistry and materials science. It will also be a useful reference to all those working in the field. Additional resources for this title including solutions for instructors and programs are available online at www.cambridge.org/9780521835275.

Concise and pedagogical textbook that covers all the topics necessary for a graduate-level course in dynamics based on Hamiltonian methods.

This is both a textbook and a monograph. It is partially based on a two-semester course, held by the author for third-year students in physics and mathematics at the University of Salerno, on analytical mechanics, differential geometry, symplectic manifolds and integrable systems. As a textbook, it provides a systematic and self-consistent formulation of Hamiltonian dynamics both in a rigorous coordinate language and in the modern language of differential geometry. It also presents powerful mathematical methods of theoretical physics, especially in gauge theories and general relativity. As a monograph, the book deals with the advanced research topic of completely integrable dynamics, with both finitely and infinitely many degrees of freedom, including geometrical structures of solitonic wave equations. Contents: Analytical Mechanics: The Lagrangian Coordinates; Hamiltonian Systems; Transformation Theory; The Integration Methods; Basic Ideas of Differential Geometry: Manifolds and Tangent Spaces; Differential Forms; Integration Theory; Lie Groups and Lie Algebras; Geometry and Physics: Symplectic Manifolds and Hamiltonian Systems; The Orbits Method; Classical Electrodynamics; Integrable Field Theories: KdV Equation; General Structures; Meaning and Existence of Recursion Operators; Miscellanea; Integrability of Fermionic Dynamics. Readership: Physicists and mathematicians.

This book details the necessary numerical methods, the theoretical background and foundations and the techniques involved in creating computer particle models, including linked-cell method, SPME-method, tree codes, amd multipol technique. It illustrates modeling, discretization, algorithms and their parallel implementation with MPI on computer systems with distributed memory. The text offers step-by-step explanations of numerical simulation, providing illustrative code examples. With the description of the algorithms and the presentation of the results of various simulations from fields such as material science, nanotechnology, biochemistry and astrophysics, the reader of this book will learn how to write programs capable of running successful experiments for molecular dynamics.

This book offers an introduction into quantum machine learning research, covering approaches that range from "near-term" to fault-tolerant quantum machine learning algorithms, and from theoretical to practical techniques that help us understand how quantum computers can learn from data. Among the topics discussed are parameterized quantum circuits, hybrid optimization, data encoding, quantum feature maps and kernel methods, quantum learning theory, as well as quantum neural networks. The book aims at an audience of computer scientists and physicists at the graduate level onwards. The second edition extends the material beyond supervised learning and puts a special focus on the developments in near-term quantum machine learning seen over the past few years.

This book focuses on current applications of molecular quantum dynamics. Examples from all main subjects in the field, presented by the internationally renowned experts, illustrate the importance of the domain. Recent success in helping to understand experimental observations in fields like heterogeneous catalysis, photochemistry, reactive scattering, optical spectroscopy, or femto- and attosecond chemistry and spectroscopy underline that nuclear quantum mechanical effects affect many areas of chemical and physical research. In contrast to standard quantum chemistry calculations, where the nuclei are treated classically, molecular quantum dynamics can cover quantum mechanical effects in their motion. Many examples, ranging from fundamental to applied problems, are known today that are impacted by nuclear quantum mechanical effects, including phenomena like tunneling, zero point energy effects, or non-adiabatic transitions. Being important to correctly understand many observations in chemical, organic and biological systems, or for the understanding of molecular spectroscopy, the range of applications covered in this book comprises broad areas of science: from astrophysics and the physics and chemistry of the atmosphere, over elementary processes in chemistry, to biological processes (such as the first steps of photosynthesis or vision). Nevertheless, many researchers refrain from entering this domain. The book "Molecular Quantum Dynamics" offers them an accessible introduction. Although the calculation of large systems still presents a challenge - despite the considerable power of modern computers - new strategies have been developed to extend the studies to systems of increasing size. Such strategies are presented after a brief overview of the historical background. Strong emphasis is put on an educational presentation of the fundamental concepts, so that the reader can inform himself about the most important concepts, like eigenstates, wave packets, quantum mechanical resonances, entanglement, etc. The chosen examples highlight that high-level experiments and theory need to work closely together. This book thus is a must-read both for researchers working experimentally or theoretically in the concerned fields, and generally for anyone interested in the exciting world of molecular quantum dynamics.

This book introduces and explores modern developments in the well established field of Hamiltonian dynamical systems. It focuses on high degree-of-freedom systems and the transitional regimes between regular and chaotic motion. The role of nonlinear normal modes is highlighted and the importance of low-dimensional tori in the resolution of the famous FPU paradox is emphasized. Novel powerful numerical methods are used to study localization phenomena and distinguish order from strongly and weakly chaotic regimes. The emerging hierarchy of complex structures in such regimes gives rise to particularly long-lived patterns and phenomena called quasi-stationary states, which are explored in particular in the concrete setting of one-dimensional Hamiltonian lattices and physical applications in condensed matter systems. The self-contained and pedagogical approach is blended with a unique balance between mathematical rigor, physics insights and concrete applications. End of chapter exercises and (more demanding) research oriented problems provide many opportunities to deepen the reader’s insights into specific aspects of the subject matter. Addressing a broad audience of graduate students, theoretical physicists and applied mathematicians, this text combines the benefits of a reference work with those of a self-study guide for newcomers to the field.