Theory and Simulation of Random Phenomena

The purpose of this book is twofold: first, it sets out to equip the reader with a sound understanding of the foundations of probability theory and stochastic processes, offering step-by-step guidance from basic probability theory to advanced topics, such as stochastic differential equations, which typically are presented in textbooks that require a very strong mathematical background. Second, while leading the reader on this journey, it aims to impart the knowledge needed in order to develop algorithms that simulate realistic physical systems. Connections with several fields of pure and applied physics, from quantum mechanics to econophysics, are provided. Furthermore, the inclusion of fully solved exercises will enable the reader to learn quickly and to explore topics not covered in the main text. The book will appeal especially to graduate students wishing to learn how to simulate physical systems and to deepen their knowledge of the mathematical framework, which has very deep connections with modern quantum field theory.

Auteur

Ettore Vitali is a Post-Doctoral Research Associate in the Computational Materials Physics Group, Physics Department, College of William and Mary, Williamsburg, VA, USA. In 2005 he obtained his Master's degree in Physics (summa cum laude) at the Università degli Studi di Milano and in 2008 gained a PhD at the same university for a thesis entitled "Superfluid and supersolid phases of 4He". He has since undertaken postdoctoral fellowships at the Centro di Ricerca e Sviluppo DEMOCRITOS, CNR-INFM, Trieste, Italy and the Università degli Studi di Milano. Dr. Vitali's research interests range across condensed matter physics; probability theory, stochastic processes, stochastic differential equations, and numerical simulation techniques; and statistical methods for ill-posed inverse problems. He is the author of 15 peer-reviewed journal articles. Mario Motta is a Post-Doctoral Research Associate in the Computational Materials Physics Group, Physics Department, College of William and Mary, Williamsburg, VA, USA. He gained Bachelor's and Master's degrees in Physics at the University of Milan in 2010 and 2012 and then completed a PhD in Physics, Astrophysics and Applied Physics from the Università degli Studi di Milano for a thesis entitled "Dynamical properties of many-body systems from Quantum Monte Carlo Simulations", which won the Dr. Davide Colosimo Award for worthy PhD students. His research focuses include topics in condensed matter physics; quantum chemistry; probability theory, stochastic processes, and numerical simulations; and the foundations of quantum mechanics.

Davide Emilio Galli is Associate Professor at the Parallel Computing and Condensed Matter Simulations Laboratory, Physics Department, Università degli Studi di Milano, Milan, Italy. He graduated with a degree in Physics from the Università degli Studi di Milano in 1993 and in 1997 completed his PhD at the university. Dr. Galli's interests include Monte Carlo and quantum Monte Carlo simulations of many-particle systems and stochastic processes, the theory of superfluids and quantum liquids and solids, theoretical methods for quantum many-body systems, and stochastic optimization methods. He is the author of almost 60 journal articles and acts as a referee for a number of international scientific journals. He is also an International Advisory Committee member for a series of international conferences on Recent Progress in Many Body Theories (RPMBT/MBT).

Contenu

1 Review of Probability Theory.- 2 Applications to Mathematical Statistics.- 3 Conditional Probability and Conditional Expectation.- 4 Markov Chains.- 5 Sampling of Random Variables and Simulation.- 6 Brownian Motion.- 7 Introduction to Stochastic Calculus and Ito Integral.- 8 Introduction to Stochastic Differential Equations and Applications.- Bibliography.- Solutions.