Central Limit Theorem

• Paul Levy [1927] generalized the C.L.T.

  • Variance: s 2 = <x2> - <x> 2 = 2Dt
  • Diffusion: D = (<x2> - <x> 2) / 2T
  • Probability Distribution Function, P : <xn>=€dx xnP(x)

• We just need a different PDF to get a fat tail.

  • P(x)~x-m
  • if m < 3, then <x2> = â and s 2 ~ t 1<g<2
  • well, we lost the 2nd moment, but we have a tail.

• What does this do to the physics? What happened to the entropy (or is the energy)?