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Let's now compute the probability distribution of
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a 1-dimensional random walk in the continuum limit.
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So let's start with our master equation from the
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previous slide for the probability distribution.
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I'm going to write it in a slightly different form that
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facilitates taking the continuum limit.
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So let's look at p(x), but instead of looking at it
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time t, let me look at time t+1. So instead of having
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1, we're going to write the time increment as dt which
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will allow us to take the limit as dt goes to 0.
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So the master equation from the previous
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slide will be p(x,t+dt) is one half for a probability
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of hopping from x-1 to x. But instead of writing
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x-1 here, we'll write x-dx comma t, plus one half
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p(x+dx, t). Let's now take a Taylor series expansion
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of this equation. So, I'll write p(x,t) plus dt
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to partial derivative dp by dt to higher-ordered
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terms. And on the right-hand side I have one
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half and the (x-dx) term when we expand it
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in a Taylor series will be p(x,t-dx