11. Lévy-stable distributions
LAST TIME

• Rolf Luchsinger showed that Fokker-Planck (Langevin) method helps us understand molecular motors.
• There are a number of other important applications e.g. stochastic resonance.
• Need also to discuss limitations on
  validity of approach :
  • System size expansion $\Rightarrow$ Assume each microscopic processes has little effect. $\Rightarrow$ Rare events don't matter.
  • Processes implicitly taken to be Markovian. i.e. probability distribution for time-evolution of state beyond t only depends on state of system at t.
• Today want to relax first assumption:
  
  General reference Montroll and West [1987].

Much emphasis on Gaussian processes

• Fluctuations about stationary steady state tend to exhibit Gaussian distribution!
• Central limit theorem $\Rightarrow$ Gaussian is "attractor" for random walk when mean and variance of individual steps finite.
• Probability for sum of two independent Gaussian processes.
  
  x=x₁+x₂
  
  
  is Gaussian with $\langle x\rangle=\langle x_1\rangle +\langle x_2\rangle$ and

$$\sigma^2=\langle x^2\rangle-\langle x\rangle^2=\sigma_1^2+\sigma_2^2$$

Augustin Cauchy and Paul Lévy asked:
What distributions share the property that

$$P(x)=\int dx_1P_1(x_1)P_2(x-x_1)$$

is of the same "type" as P₁ and P₂?
Non-Gaussian such distributions Lévy-stable.

Example:
CAUCHY DISTRIBUTION:

$$P(x)=\frac{b}{\pi[(x-a)^2+b^2]}$$

Using contour integration one finds that

$$\frac{b_1+b_2}{\pi[(x-a_1-a_2)^2+(b_1+b_2)^2]}$$

$$=\int_{-\infty}^{\infty}dx_1\frac{b_1b_2} {\pi^2[(x_1-a_1)^2+b_1^2][(x-x_1-a_2)^2+b_2^2]}$$

i.e. the parameters a and b add under
convolution! Cauchy distribution does not have a finite variance!

$$\int dx x^2P(x)=\infty$$

On the other hand derivation of Fokker-Planck equation from jump moments

$$a_n=\frac{1}{\delta t}\int P(q-r,t+\delta t\vert q,t)r^ndr$$

assumed that in limit $\delta t\rightarrow 0, a_1, a_2$ remain finite, while $a_n\rightarrow 0$ in limit.


In probability theory Fourier transform of P(x) called characteristic function

$$p(k)=\int_{-\infty}^{\infty} P(x) e^{ikx}dx=\langle e^{ikx}\rangle$$

$$P(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}p(k)dk$$

The last line follows from the properties of the Dirac $\delta$-function:

$$\int_{-\infty}^{\infty}f(x)\delta(x)=f(0)$$

$$\delta(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{ikx}dk$$

Not all functions p(k) can serve.
Normalization condition imposes constraint

p(0)=1

We must also have

$$p(k)\leq\int_{-\infty}^{\infty}\vert P(x)\vert\vert e^{ikx}\vert dx=1$$

Requirement that $P(x)\geq 0$ imposes further restrictions on characteristic function.

EXAMPLES:
Characteristic function of Gaussian

$$P(x)=\frac{\exp (-\frac{x^2}{2\sigma^2})}{\sqrt{2\pi\sigma^2}}\;;\; p(k)=\exp(-\frac{k^2\sigma^2}{2})$$

Cauchy distribution

$$P(x)=\frac{b}{\pi[(x-a)^2+b^2]}$$

p(k)=e^-b|k|

Fourier transform of convolution integral

$$\int_{-\infty}^{\infty}dx_1\int_{-\infty}^{\infty}dx_2P_1(x_1)P_2(x_2)\exp(ik(x_1+x_2)$$

=p(k)=p₁(k)p₂(k)

Note that if

p₁=p₂=e^-b'|k|

p(k) will be on the same form with b=2b'
b may be complex, as long as p^*(k)=p(-k) but we must have Re b>0
How does the Fourier transform of e^-b|k| look?

Consider translationally invariant random walks

$$P(x_2,t_2\vert x_1,t_1)\equiv P(x_2-x_1,t_2-t_1)$$

What is the most general form of P(x,t) that is positive definite and normalized and satisfies the chain condition

$$P(x,t)=\int dy P(x-y,t-t_1)P(y,t_1)$$

for all intermediate times t₁?

A necessary and sufficient condition on P(x,t),
t>0 is that its characteristic function can be written either on the form

$$p(k,t)=\exp[ikt(v+bc\ln\vert k\vert)-b\vert k\vert t]$$

with $-\pi/2\leq c\leq\pi/2$ or

$$p(k,t)=\exp(-ivkt-bt\vert k\vert^\alpha[1-i\omega k/\vert k\vert])$$

where v is real, $0\leq\alpha\leq2$, $\alpha\neq 1$, b>0, $\omega$ is real and $\vert\omega\vert\leq\vert\tan(\pi\alpha/2)\vert$.
Theorem due to Khintchine and Lévy, proof in Gnedenko and Kolmogorov [1954].

In most cases the inverse Fourier transforms cannot be carried out analytically. An interesting special case is $v=0,\;\alpha=1/2,\;\\ bt=1, \;\omega=1$. One finds

$$P(x)=\left\{\begin{array}{lcl}\frac{\exp-\frac{1}{2x}}{(2\pi)^{1/2}x^{3/2}} &for& x>0\\ 0&&x<0 \end{array}\right.$$

This function can be normalized, but has
neither mean nor variance.











ASYMPTOTIC BEHAVIOR Can show that inverse Fourier transform of

$$p(k)=\exp -b\vert k\vert^\alpha$$

aproaches

$$P(x)\propto \frac{1}{x^{\alpha+1}};\;x\rightarrow \infty$$

if $0<\alpha<2$
while cumulative distribution
(probability that $x\geq z$) satisfies

$$C(z)\propto z^{-\alpha};\; z\rightarrow\infty$$

Power law implies scaling!
Rescale $z\Rightarrow \lambda z$.
Rescale $C\Rightarrow \lambda^{-\alpha} C$
Power law distribution does not change!

SUMMARY

• Not all random walks approach a Gaussian distribution
• Chain condition
  
  $$P(x,t)=\int dy P(x-y,t-t_1)P(y,t_1)$$
  
  
  allow for Lévy distributions with
  characteristic functions
  
  $$p(k)=e^{-b\vert k\vert^\alpha};\;0\leq\alpha\leq 2$$
  
  
  "some restrictions apply".
• Asymptotically $P(x)\propto x^{-\alpha-1}$
• Distributions have no variance if $\alpha <2$ and no mean if $\alpha<1$.
• Lévy walk self-affine.
• Lévy dust characterized by hierarchical arrangements of clusters

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