Random Processes in Physics

This succinct book is unique in that it covers topics starting from the very basics of probability theory leading up to applications in (equilibrium) statistical physics, quantum theory and complex systems. The overarching aim of the book is to present probabilistic concepts, in a way accessible to physics undergraduates and with applications in physics in mind. The book is split into three parts: Part I: Fundamentals - The starting point of the book will present basic concepts in probability theory including discussions about general motivation, the difficulties in dealing with probabilities and a number of basic examples. This part will include some of the general mathematical concepts underpinning probability theory. This will be followed by a discussion of joint probabilities and topics relating to Bayes theorem. Lastly I will discuss how computers generate (pseudo-) random numbers, a topic which is key in any modern view on random processes for physicists. Part II: Specific probability distributions - The second part of the book will focus on the probability distributions most important for applications in physics including discussions about the Central Limit Theorem, the Gaussian distribution, Poissonian distributions and the connection with exponential random variables. Chapters on Levy flights, selfsimilarity, power law distributions (including applications) and on extreme value statistics (e.g. Gumbel and Weibull distributions) will complete Part II of the book. Part III: Applications in and beyond physics - The final part of the book will explore specific applications of random process modelling in physics and in selected adjacent disciplines, specifically applications in quantum theory, viewed specifically from the perspective of random processes and probability theory. Chapters will contain a derivation of the standard ensembles of classical statistical physics, from the point of view of information theory. This is then extended to quantum statistical physics and density matrices. Finally we will discuss simple Markov chains and Monte Carlo simulations, as well as applications of probability theory to a selected set of topics in complex systems. These will be described at a level suitable for undergraduate students in the first and second year.