The achievement of Bose-Einstein condensation in ultra-cold vapours of alkali atoms has given enormous impulse to the study of dilute atomic gases in condensed quantum states inside magnetic traps and optical lattices. High purity and easy optical access make them ideal candidates to investigate fundamental issues on interacting quantum systems. This review presents some theoretical issues which have been addressed in this area and the numerical techniques which have been developed and used to describe them, from mean-field models to classical and quantum simulations for equilibrium and dynamical properties. The attention given in this article to methods beyond standard mean-field approaches should make it a useful reference point for future advances in these areas.

These volumes collect the lecture notes of the course a oeAn introduction to computational physicsa held in the academic year 2000/01 for students of the University of Pisa and Scuola Normale Superiore at the level of the last two-year undergraduates in physics and chemistry. The second part deals with various types of particle methods, both deterministic and stochastic, used in modern applications of computer simulations in physics and related disciplines.

These volumes collect the lecture notes of the course a oeAn introduction to computational physicsa held in the academic year 2000/01 for students of the University of Pisa and Scuola Normale Superiore at the level of the last two-year undergraduates in physics and chemistry. Grid methods are the tool of the trade for the solution of ordinary and partial differential equations and consequently they represent a a oemusta for anyone dealing with computational science. With grid methods, a major distinction is made between methods which do not require matrix algebra and those which do.

"Both superb and essential... Succi, with clarity and wit, takes us from quarks and Boltzmann to soft matter - precisely the frontier of physics and life." Stuart Kauffman, MacArthur Fellow, Fellow of the Royal Society of Canada, Gold Medal Accademia Lincea We live in a world of utmost complexity, outside and within us. There are thousand of billions of billions of stars out there in the Universe, a hundred times more molecules in a glass of water, and another hundred times more in our body, all working in sync to keep us alive and well. At face value, such numbers spell certain doom for our ability to make any sense at all of the world around and within us. And yet, they don't. Why, and how - this book endeavours to provide an answer to these questions with specific reference to a selected window of the physics-biology interface. The story unfolds over four main Parts. Part I provides an introduction to the main organizational principles which govern the functioning of complex systems in general, such as nonlinearity, nonlocality and ultra-dimensions. Part II deals with thermodynamics, the science of change, starting with its historical foundations laid down in the 19th century, and then moving on to its modern and still open developments in connection with biology and cosmology. Part III deals with the main character of this book, free energy, and the wondrous scenarios opened up by its merger with the modern tools of statistical physics. It also describes the basic facts about soft matter, the state of matter most relevant to biological organisms. Finally, Part IV discusses the connection between time and complexity, and its profound implications on the human condition, i.e. the one-sided nature of time and the awareness of human mortality. It concludes with a few personal considerations about the special place of emotions and humility in science.

Introduction to Parallel Computational Fluid Dynamics

Flowing matter is all around us, from daily-life vital processes (breathing, blood circulation), to industrial, environmental, biological, and medical sciences. Complex states of flowing matter are equally present in fundamental physical processes, far remote from our direct senses, such as quantum-relativistic matter under ultra-high temperature conditions (quark-gluon plasmas). Capturing the complexities of such states of matter stands as one of the most prominent challenges of modern science, with multiple ramifications to physics, biology, mathematics, and computer science. As a result, mathematical and computational techniques capable of providing a quantitative account of the way that such complex states of flowing matter behave in space and time are becoming increasingly important. This book provides a unique description of a major technique, the Lattice Boltzmann method to accomplish this task. The Lattice Boltzmann method has gained a prominent role as an efficient computational tool for the numerical simulation of a wide variety of complex states of flowing matter across a broad range of scales; from fully-developed turbulence, to multiphase micro-flows, all the way down to nano-biofluidics and lately, even quantum-relativistic sub-nuclear fluids. After providing a self-contained introduction to the kinetic theory of fluids and a thorough account of its transcription to the lattice framework, this text provides a survey of the major developments which have led to the impressive growth of the Lattice Boltzmann across most walks of fluid dynamics and its interfaces with allied disciplines. Included are recent developments of Lattice Boltzmann methods for non-ideal fluids, micro- and nanofluidic flows with suspended bodies of assorted nature and extensions to strong non-equilibrium flows beyond the realm of continuum fluid mechanics. In the final part, it presents the extension of the Lattice Boltzmann method to quantum and relativistic matter, in an attempt to match the major surge of interest spurred by recent developments in the area of strongly interacting holographic fluids, such as electron flows in graphene.

In recent years, stylized forms of the Boltzmann equation, now going by the name of "Lattice Boltzmann equation" (LBE), have emerged, which relinquish most mathematical complexities of the true Boltzmann equation without sacrificing physical fidelity in the description of many situations involving complex fluid motion. This book provides the first detailed survey of LBE theory and its major applications to date. Accessible to a broad audience of scientists dealing with complex system dynamics, the book also portrays future developments in allied areas of science (material science, biology etc.) where fluid motion plays a distinguished role.