Understanding 1D Walks and Their Properties

[superwolf]

My textbook simply states that

"For a simple 1D walk with step size L:
<x^2> = NL^2

So after N steps 99.7% of the particles will be closer then 3L sqrt(N) from the centre"

How does it get from the first to the latter?

 

[lanedance]

Hi Superwolf

$$3\sigma$$ (3 standard deviations) is 99.7% confidence limit for a normal distribution

So the 99.7 and <x^2> seem to point to using variance of a normal distribution

Which shouldn't be too had to get too asuming we know the total length is normally distributed...

Taken from wiki:

In probability theory, the central limit theorem (CLT) states conditions under which the sum of a sufficiently large number of independent random variables

http://en.wikipedia.org/wiki/Central_limit_theorem

So each step can be modeled as a binomial distribution with outcomes (L,-L) and 0.5 chance of success, and the sum giving the average length is then approximated by a normal distributions at large N

 

[superwolf]

So L sqrt(N) is the standard deviation?

 

[lanedance]

i would strat with the definition of variance and work from there
$$\sigma^2 = <(x - \bar{x})^2>$$
where the <> is expectation

it should be a simple matter to get the standard deviation from there, and will probably end up as L sqrt(N), if you convince youself the mean is zero