IntroductionFundamentals Of Random WalksThe Simple Isotropic Random WalkA Brw With Waiting TimesRandom Walks With A BarrierCrws And The Telegraph EquationReference

Random Walka particle repeatedly

moves in all directions

Brownian motioncontinuous irregular motion of individual pollen particles

Robert Brown(1828 )

Brownian Motion AnalysisEinstein- Smoluchowski Equation (1905,1916)

x2 = 2Dt (D is diffusion coefficient)

Average Particles Actions (Probability)

Langevin Equation(1908) Single Particle Action (F=ma)

IntroductionFundamentals Of Random WalksThe Simple Isotropic Random WalkA Brw With Waiting TimesRandom Walks With A BarrierCrws And The Telegraph EquationReference

Uncorrelatedthe direction of movement is completely independent of the previous directions

Unbiasedthe direction moved at each step is completely random

The Brown motion is uncorrelated & unbiased

Fixed Step length moves a distance δ in a short time τ

Variable Step length Finite variance (Brownian motion)Infinite variance (Lévy flight)

Brownian motion Lévy flight

IntroductionFundamentals Of Random WalksThe Simple Isotropic Random WalkA Brw With Waiting TimesRandom Walks With A BarrierCrws And The Telegraph EquationReference

Consider a walker moving on an 1-D infinite uniform latticeOne DimensionalFixed Step length Uncorrelated Unbiased

The walker starts at the origin (x=0) and then moves a distance δ either left or right in a short time τ

Consider probability p(x , t)x is distance form x=ot is number of time step

The probability a walker will be at a distance mδ to the right of the origin after nτ time steps

This form is Binomial distribution, with mean displacement 0 and variance nδ2.

For large n, this converges to a normal

(or Gaussian) distributionafter a sufficiently large amount of time t=nτ,

the location x=mδ of the walker is normally distributed with mean 0 and variance δ2t/τ. (δ2/2τ = D)

PDF for location of the walker after time t

mean location E(Xt)=0 the absence of a preferred direction or bias

mean square displacement (MSD) E(Xt

2)=2Dt

a system or process where the signal propagates as a wave in which MSD increases linearly with t2

IntroductionFundamentals Of Random WalksThe Simple Isotropic Random WalkA Brw With Waiting TimesRandom Walks With A BarrierCrws And The Telegraph EquationReference

1-D Biased random walks preferred direction (or bias) and a possible waiting time between movement steps.at each time step τ, the walker moves a distance δ to the left or right with probabilities l and r, or stays in the same location (‘waits’) probabilities 1-l-r.

the walker is at location x at time t+τ, then there are three possibilities for its location at time t.

P(x, t+τ)= P(x, t)(1-l-r)+P(x-δ,t)r+ P(x+δ,t)lit was at x and did not move at all.it was at x - δ and then moved to the right.it was at x + δ and then moved to the left.

P(x, t+τ )= P(x, t)(1-l-r)+P(x-δ,t)r+ P(x+δ,t)lexpressed it as a Taylor series about (x, t)

Fokker–Planck equation Special case D is constant

E(Xt2)~ t2 is like wave

σt

2=2Dt is a standard diffusive process

IntroductionFundamentals Of Random WalksThe Simple Isotropic Random WalkA Brw With Waiting TimesRandom Walks With A BarrierCrws And The Telegraph EquationReference

a walker reaching the barrierturn around and move away in the opposite directionor absorbed in barrier

1-D random walk process that satisfies the drift–diffusion

At time t, either the walker has been absorbed or its location has PDF given by p(x, t)

PDF of the absorbing time Ta

the probability of absorption taking place in a finite time (Ta < ∞)

the walker is certain to be absorbed within a finite time

drift towards the barrier (u ≤0)(u >0) probability decreases exponentially as the rate of drift u, or the initial distance x0 from the barrier, increases. if the rate of diffusion D increases, the probability of absorption will increase

IntroductionFundamentals Of Random WalksThe Simple Isotropic Random WalkA Brw With Waiting TimesRandom Walks With A BarrierCrws And The Telegraph EquationReference

Correlated random walks (CRWs) involve a correlation between successive step orientations

CRW is a velocity jump process

population of individuals moving either left or right along an infinite line at a constant speed v

total population density is p(x, t)=a(x, t)+b(x, t). (left + right-moving)

CRW at each time step,turning events occur as a Poisson process with rate λ

Expanding these as Taylor series and taking the limit δ, τ->0 such that δ/τ =v gives

telegraph equation:

telegraph equationhttp://www.math.ubc.ca/~feldman/apps/telegrph.pdf

small t (i.e. t=1/ λ), E(Xt

2)~O(v2t2) is a wave propagation process; for large t, E(Xt

2)~ O(v2t/ λ), which is a diffusion process

Random walk models in biology http://privatewww.essex.ac.uk/~ecodling/Codling_et_al_2008.pdf

Random walk in biology http://rieke-server.physiol.washington.edu/People/Fred/Classes/532/berg_randomwalk_ch1.pdf

Diffusion http://www.che.ilstu.edu/standard/che38056/lecturenotes/380.56chapter13-S06.pdf

Brownian motion http://sciweb.nybg.org/science2/pdfs/dws/Brownian.pdf