Random Walks
Lead instructor: Sid Redner
 About the Tutorial:
The goal of this tutorial is to outline some elementary, but beautiful aspects of random walks. Random walks are ubiquitous in nature. They naturally arise in describing the motion of microscopic particles, such as bacteria or pollen grains, whose motion is governed by being buffeted by collisions with the molecules in a surrounding fluid. Random walks also control many type of fluctuation phenomena that arise in finance.
The tutorial begins by presenting examples of random walks in nature and summarizing important classes of random walks. We'll then give a quantitative discussion of basic properties of random walks. We'll show that the root meansquare displacement of a random walk grows as the squareroot of the elapsed time. Next, we will determine the underlying probability distribution of a random walk. In the longtime limit, this distribution is independent of almost all microscopic details of the randomwalk motion. This universality is embodied by the centrallimit theorem. In addition to presenting this theorem, we'll also discuss the anomalous features that arise when the very mild conditions that underlie the centrallimit theorem are not satisfied. Finally, we will show how to recover the diffusion equation as the continuum limit of the evolution equation for the probability distribution of a random walk.
We will then present some basic firstpassage properties of random walks, which address the following simple question: does a random walk reach a specified point for the first time? We will determine the firstpassage properties in a finite interval; specifically, how long does it take for a random walk to leave an interval of length L, and what is the probability to leave either end of the interval as a function of the starting location. Finally, we'll discus the application of firstpassage ideas to reactionrate theory, which defines how quickly diffusioncontrolled chemical reactions can occur.
Note that Complexity Explorer tutorials are meant to introduce students to various important techniques and to provide illustrations of their application in complex systems. A given tutorial is not meant to offer complete coverage of its topic or substitute for an entire course on that topic.
This tutorial is designed for more advanced math students. Math prerequisites for this course are an understanding of calculus, basic probability, and Fourier transforms.
 About the Instructor(s):
Sid Redner is a Resident Faculty Member at the Santa Fe Institute.
Sid Redner received an A.B. in physics from the University of California, Berkeley in 1972 and a Ph.D. in Physics from MIT in 1977. After a postdoctoral year at the University of Toronto, Sid joined the physics faculty at Boston University in 1978. During his 36 years at BU, he served as Acting Chair during two separate terms and also served as Departmental Chair. Sid has been a Visiting Scientist at SchlumbergerDoll Research in the mid 80's, the Ulam Scholar at LANL in 2004, and a sabbatical visitor at Université Paul Sabatier in Toulouse France and at Université PierreetMarieCurie in Paris.
Sid has published more than 250 articles in major peerreviewed journals, as well as two books: the monograph A Guide to FirstPassage Processes (Cambridge Univ. Press, 2001) and the graduate text, jointly with P. L. Krapivsky and E. BenNaim, A Kinetic View of Statistical Physics (Cambridge Univ. Press, 2010). He also a member of the Editorial Board for the Journal of Informetrics, an Associate Editor for the Journal of Statistical Physics, and a Divisional Associate Editor for Physical Review Letters. For more on Sid, visit his website. Read a Q&A with Sid on his Random Walks tutorial here.
 How to use Complexity Explorer:
 How to use Complexity Explorer
 Enrolled students:

2,595
 Prerequisites:

Calculus; Basic Probability; Fourier Transforms
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Syllabus
 Introduction
 Root Mean Square Displacement
 Role of the Spatial Dimension
 Probability Distribution and Diffusion Equation
 Central Limit Theorem
 First Passage Phenomena
 Elementary Applications of First Passage Phenomena
 Final Remarks
 Homework