14. Return time distribution. Continuous time random walk(continued).
LAST TIME

• Started to discuss situations where individual processes were not Poisson distributed and/or were not independent.
• Introduced "continuous time random walk".
• Interested in the following distributions:
  • $P_n({\bf x})=$ probability that walker has arrived at x after n steps, starting at the origin.
  • $P({\bf x},t)=$ probability that a walker is at x at time t.
  • $\psi(\tau)=$ waiting time distribution
    $\tau=$ waiting time at each site.

For discussions that follows need the following closely related distributions:

• First passage time distribution $\Rightarrow$ Distribution of n when system first reaches x starting at the origin.
• Return distribution $\Rightarrow$ Distribution of n when system first returns to its starting point.
• Escape probability $\Rightarrow$ Probability that a system will never return to starting point.

Consider walk on lattice of dimension d, n is the number of jumps:

$$P_{n+1}({\bf x})=\sum_ep({\bf e})P_n({\bf x-e})$$

e vector to near neighbor.
$p({\bf e})=$ probability of jump to neighbor ${\bf e}$.
Write:

$$P_n({\bf x})=\sum_{n'=1}^{n}\Pi_{n'}({\bf x})P({\bf x},n\vert{\bf x},n')$$

$\Pi_n$ is the first passage time distribution.
Assume translational invariance

$$P_n({\bf x})=\sum_{n'=1}^{n}\Pi_{n'}({\bf x})P_{n-n'}(0)$$

Introduce generating functions

$$G({\bf x},\lambda)=\delta_{{\bf x},0}+\sum_{n=1}^{\infty}P_n({\bf x})z^n$$

$$\Gamma({\bf x},z)=\sum_{n=1}^\infty\Pi_n({\bf x})z^n$$

$$G({\bf x},z)=\delta_{{\bf x},0}+\sum_{n=1}^{\infty}\sum_{n'=1}^n z^{n'}\Pi_{n'}({\bf x})z^{n-n'}P_{n-n'}(0)$$

$$=\delta_{{\bf x},0}+G(0,z)\Gamma({\bf x},z)$$

We find:

$$\Gamma(0,z)=1-\frac{1}{G(0,z)}$$

$$\Gamma({\bf x},z)=\frac{G({\bf x},z)}{G(0,z)};\;\;{\bf x}\neq 0$$

Probability that a particle will return to homesite at any time is

$$\Gamma(0,1)=\Pi_1(0)+\Pi_2(0)+\Pi_3(0)+\cdots$$

If $G(0,1)<\infty$ there is a finite escape
probability.
If $G(0,1)=\infty$ the system will "always" return to the home site


Last time we calculated

$$G({\bf x},z)=\frac{1}{(2\pi)^d}\int_{BZ}d^dk\frac{\exp(- i{\bf x\cdot k})} {1-z \lambda({\bf k})}$$

where BZ stands for Brillouin zone and

$$\lambda({\bf k})=\sum_{\bf e}p({\bf e})\exp(i{\bf k\cdot e})$$

If the walk is unbiased we must have for small values of k

$$1-\lambda(k)\propto k^2$$

(Technical note: We have implicitly assumed that the walk takes place on a Bravais lattice for which all lattice sites are equivalent. There must then be a neighbor -e for every neighbor e and we can rewrite the defining equation for $\lambda({\bf k})$

$$\lambda({\bf k})=\sum_{\bf e}p({\bf e})\cos({\bf k\cdot e})$$

Clearly $\lambda(0)=1$ and there will be no odd powered terms in an expansion of $\lambda({\bf k})$ in powers of k. If the lattice is not Bravais i.e. has a basis we must take into account that there inequivalent sites in the unit cell of the lattice. The notational complications this introduces represents an unnecessary difficulty for our present discussion, in the continuum limit where the discretenes of the lattice is neglected $1-\lambda(k)\propto k^2$ must still hold)
For a d-dimensional lattice

$$d^dk\propto k^{d-1}dk$$

and we conclude that the integral over k in the expression for $G({\bf x},z)$ will be divergent for small k (infrared divergence) for $d\leq 2$.
We conclude:

• For d>2 a particle starting at origin will always escape.
• For $d\leq 2$ diffusing particles will keep returning to origin.

G(0,1) has been evalauted analytically for the body centered, face centered and simple cubic lattices
The return probability is found to be:

$$0.256318237\cdots\;(fcc)$$

$$0.282229983\cdots\;(bcc)$$

$$0.340537330\cdots\;(sc)$$

To find the asymptotic behavior of the first return distribution for $d\leq 2$ look at how $G(0,\lambda)$ diverges as $\epsilon=\lambda-1\Rightarrow 0$

$$G(0,1+\epsilon)\propto\epsilon^{-1/2}\;\;for\; d=1$$

$$G(0,1+\epsilon)\propto\ln\frac{1}{\epsilon}\;\;for\;d=2$$

One can show that this means that the probability that a walk has not yet returned home after n steps is

$$\propto n^{-1/2}\;\; for \;d=1$$

$$\propto \frac{1}{\ln n}\;\; for\;d=2$$

Unbiased, continuous, Gaussian, random walk in one dimension

In this case we can calculate the first passage time distribution explicitly:
Let x₁<x<x₂. A random walk in one dimension from x₁ to x₂ must pass through x at least once.
P(x₂-x₁,t) probability of motion from x₁ to x₂ in time t
$\Pi(x-x_1,\tau)$ probability of arriving for the first time at x at time $\tau$.

$$P(x_2-x_1,t)=\int_0^tP(x_2-x,t-\tau)\Pi(x-x_1,\tau)d\tau$$

L(x,u)= Laplace transform of P(x,t)
$\Lambda(x,u)=$ Laplace transform of $\Pi(x,\tau)$
Find

$$L(x_2-x_1,t)=L(x_2-x,u)\Lambda(x-x_1,u)$$

For an unbiased, Gaussian random walk

$$P(x,t)=\frac{\exp(-\frac{x^2}{4Dt})}{\sqrt{4\pi Dt}}$$

with Laplace transform

$$L(x,u)=\frac{\exp(-\sqrt{x^2u/D})}{\sqrt{4uD}}$$

We find

$$\Lambda(x-x_1,u)=\exp(-\sqrt{(x-x_1)^2u/D})$$

Performing an inverse Laplace transform we find

$$\Pi(x,t)=\frac{1}{t}\sqrt{\frac{x^2}{4\pi Dt}}\exp[-\frac{x^2}{4Dt}]$$

The first passage time distribution is on the Levy- Smirnov form for a stable Lévy distribution with $\alpha=1/2$. We have encountered this distribution before (Lecture 11)

Note that in the continuos case we cannot take the limit $x\rightarrow 0$!

Returning to the continuous time random walk:distributions P_n(x), P( x,t), $\psi(\tau)$
Let us assume that P_n is well behaved

$$\langle x^2\rangle\propto n$$

If the waiting time distribution has a mean

$$\langle \tau\rangle =\int_0^{\infty}\tau\psi(\tau)$$

successive jumps occur on the average at the rate $1/\tau_0$ and

$$\langle x^2\rangle\propto t$$

If the waiting time distribution doesn't have a mean:

$$\psi(\tau)\approx \tau_0\alpha/\tau^{-1-\alpha}\;large\;\tau$$

$$0<\mu\leq 1$$

the time for n jumps typically $\propto\tau_0n^{1/\mu}$

$$\langle x^2\rangle\propto t^\mu\; if \;0<\mu<1$$

and

$$\langle x^2\rangle\propto t/\ln t\; if \;\mu=1$$

Subdiffusive behavior!

Two examples of subdiffusive behavior were discussed in class:

• Photoconductivity in amorphous materials. The example was taken from Montroll and West [1987], see also Montroll and Scher and Montroll [1975].
• Diffusion on comb-like structures. The example was taken from Bouchaud and Georges [1990]

SUMMARY
We have discussed

• The escape probability for diffusion on a lattice. Found that in one or two dimensions a diffusing particle will "always" return to home site, but in three or more dimensions there is a nonzero probability of escape.
• The return distribution in one and two dimensions.
• The first passage time distribution for continuous random walk in 1-d.
• Anomalous diffusion when waiting distribution did not have a mean.

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15. Random walk in fractal time. Correlated walks