Q. Levy flights- consider a random walk of N steps in one-dimension. suppose that the steps are completely uncorrelated, and that the distribution of step size a is given by phi(a)= k*1/(1+abs(a)^alpha).

Where alpha>0 and k are constants.

we want to know the distribution p(r,n) where r is the sum of a(n) where n is from 1 to N.

Note a(n) is a subscript n.

(a) write down the form of p(r,n) expected from the central limit theorem. State parameters of p in terms of integrals of phi. For what range of alpha might there be a problem?

(b) Compute the distribution p(r,n) directly from phi(a) for alpha=2, and determine its explicit form for all r's. How does a walker typically go after n steps? Describe in words how this is possible given the form of phi.