Question
Compute the mean departure of a one-dimensional random walker from its starting point. In particular, use the fact that the mean excursion can be written as $\langle R\rangle=\left(\left\langle n_{\mathrm{r}}\right\rangle-\left\langle n_{\ell}\right\rangle\right) a$ and that the probability distribution for $n_{\mathrm{r}}$ right steps out of a total of $N$ steps is given by the binomial distribution.
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Here, $\langle R\rangle$ is the mean displacement of the random walker, $n_{\mathrm{r}}$ and $n_{\mathrm{l}}$ are the number of steps to the right and left respectively, and $a$ is the step size. Show more…
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A random walker takes a stroll in his 1-dimensional world with probability of making a 'right step' p = 2/3 and a 'left step' q= 1/3. (a) Find the probability of having n1 steps to the right after N number of steps have been made. (b) Find the probability of having n2 steps to the left after N number of steps have been made. (c) Find the expression for the means <n1> and <n2>.
Biased Random Walker in One Dimension: Consider a particle that moves in one dimension, taking steps always in the same direction, but with step lengths s equally likely to be anywhere in the range l - b < s < l + b (b < l). After N steps, what is (a) the mean value of the displacement, x = sum_{i=1}^{N} s_i, from the origin? (b) the dispersion (variance) overline{(x - ar{x})^2} in the displacement? (c) Now suppose that the step length has a Gaussian distribution, such that the probability of a step length between s and s + ds is given by p(s)ds = frac{1}{sqrt{2pi}sigma} exp left[ frac{-(s - l)^2}{2sigma^2} ight] ds . After N steps, what are the mean displacement ar{x} from the origin and the dispersion overline{(x - ar{x})^2} ?
Equation 8.39 characterizes the probability distribution for random walkers. Derive this equation by using the fact that the probability that the walker will be at position $x$ at step $N$ implies that the walker was either at $x-a$ or $x+a$ at step $N-1 .$ In particular, write an equation for $p(x, N)$ in terms of $p(x \pm a, N-1)$ and by Taylor expanding $p(x \pm a, N-1) \approx$ $p(x, N)+$ derivative terms.
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