The book takes a distinctly “applied” philosophy, explicitly stating that it will use the language and approaches common in physics, and will often eschew rigorous derivations in favor of plausible arguments. There are also interesting historical discussions, and several appendices which cover much of the needed mathematical background concisely. Throughout the text are copious discussions of related ideas and applications, often, though not always, with references. Finally Chapters 7 through 9 cover topics that are more specialized than in general books covering similar topics: anomalous diffusion, a brief but surprisingly deep foray into random matrix theory, and finally a discussion of percolation theory and related tools from statistical physics. long-tailed distributions), with frequent but brief discussions of applications from condensed matter physics and engineering, to cell biology and financial mathematics. Chapters 3 through 6 then survey and connect a variety of standard topics in statistical physics and stochastic analysis from Langevin equations to extreme value statistics and rare events (i.e. Chapters 1 and 2 motivate the study of randomness and introduce important aspects of elementary probability theory such as Markov processes (and chains), random walks, and their connection to PDE. Loosely, I would organize the book into three parts. Often these topics are introduced in the context of a problem from physics, engineering, or biology, which can be both refreshing but a bit disorienting if the reader is completely unfamiliar with the context. While things like random walks, the Fourier Transform, and the Central Limit Theorem are “introduced” in the text, the development is really too rapid for someone completely unfamiliar with these ideas to follow along, at least without augmenting one’s reading with a more standard and systematic text. The (ambitious) opening sentence sums up the philosophy of the book well: “the purposes of this book are to familiarize you with a broad range of examples where randomness plays a key role, develop an intuition for it, and get to the level where you may read a recent research paper on the subject and be able to understand the terminology, the context, and the tools used.” While it is written roughly at an introductory level for many of the topics, it assumes a reasonably sophisticated mathematical background from the intended audience – standard PDE solution methods, linear algebra, multivariable analysis, and reasonable familiarity with undergraduate-level probability. Thinking Probabilistically is a conceptual and problem-focused introduction to a wide range of topics in probability theory, and its connections with a huge range of theoretical and applied fields.
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