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MIT OpenCourseWare http://ocw.mit.edu 6.055J / 2.038J The Art of Approximation in Science and Engineering Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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Chapter 9 Discretization 9.1 Diaper usage 9.2 Pendulum period 9.3 Random walks Random walks are everywhere. Do you remember the card game War? How long does it last, on average? A molecule of neurotransmitter is released from a vesicle. Eventually it binds to the synapse, and your leg twitches. How long does it take to get there? On a winter day, you stand outside wearing only a thin layer of clothing. Why do you feel cold? These physical situations are examples of random walks. In a physical random walk, for example a gas molecule moving and colliding, the walker moves a variable distance and can move in any direction. This general situation is complicated. Fortunately, the essential features of the random walk do not depend on these complicated details. Simplify by discarding the generality. The generality arises from the continuous degrees of freedom: the direction is continuous and the distance between collisions is continuous. So, discretize the direction and the distance: Assume that the particle travels a fixed distance between collisions and that it can move only along the coordinate axes. Furthermore, ana- lyze the special case of one-dimensional motion before going to the more general cases of two- and three-dimensional motion. In this discretized, one-dimensional model, a particle starts at the origin and moves along a line. At each tick it moves left or right with probability 1 / 2 in each direction. Let the position after n steps be x n , and the expected position after n steps be h x n i . Because the random walk is unbiased – because moving in each direction is equally likely – the expected position remains constant: h x n i = h x n 1 i . So h x i , the so-called first moment of the position, is an invariant. However, it is not a fascinating invariant because it does not tell us much that we do not already understand intuitively.
92 6.055 / Art of approximation Given that the first moment is not interesting, try the next-most-complicated moment: the second moment h x 2 i . This analysis is easiest in special cases. Suppose that after a while wandering, the particle has arrived at 7 , i.e. x = 7 . At the next tick it will be at either x = 6 or x = 8 . Its expected squared position – not its squared expected position! – has become h x 2 i = 2 1 6 2 + 8 2 = 50 .

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