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X and the City by John A. Adam

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Appendix 10

RANDOM WALKS AND THE DIFFUSION EQUATION

 

 

 

 

 

Consider the motion of non-interacting “point particles” along the x-axis only, starting at time t = 0 and x = 0 and executing a random walk. The particles can be whatever we wish them to be: molecules, pollen, cars, inebriated people, white-tailed deer, rabbits, viruses, or anything else as long as there are sufficiently many for us to be able to assume a continuous distribution, and yet not so dense that they interact and interfere with one another (though this may seem rather like “having our cake and eating it”!). In its simplest description the particle motion is subject to the following constraints or “rules”:

1. Every τ seconds, each particle moves to the left or the right, ...

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