Introduction
Fundamentals Of Random Walks
The Simple Isotropic Random Walk
A Brw With Waiting Times
Random Walks With A Barrier
Crws And The Telegraph Equation
Reference
Random Walk
a particle repeatedly
moves in all directions
Brownian motion
continuous irregular motion of
individual pollen particles
Robert Brown(1828 )
Brownian Motion Analysis
Einstein- Smoluchowski Equation (1905,1916)
x2 = 2Dt (D is diffusion coefficient)
Average Particles Actions (Probability)
Langevin Equation(1908)

Single Particle Action (F=ma)
Introduction
Fundamentals Of Random Walks
The Simple Isotropic Random Walk
A Brw With Waiting Times
Random Walks With A Barrier
Crws And The Telegraph Equation
Reference
Uncorrelated
the direction of movement is completely
independent of the previous directions
Unbiased
the 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
Introduction
Fundamentals Of Random Walks
The Simple Isotropic Random Walk
A Brw With Waiting Times
Random Walks With A Barrier
Crws And The Telegraph Equation
Reference
Simple Isotropic Random Walk
Consider a walker moving on an 1-D
infinite uniform lattice




One Dimensional
Fixed 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
Simple Isotropic Random Walk
Consider probability p(x , t)
 x is distance form x=o
 t is number of time step
Simple Isotropic Random Walk
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.
Simple Isotropic Random Walk
Simple Isotropic Random Walk
For large n, this converges to a normal
(or Gaussian) distribution
 after 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
Simple Isotropic Random Walk
mean location E(Xt)=0
the absence of a preferred direction or bias
mean square displacement (MSD)
E(Xt2)=2Dt
a system or process where the signal propagates
as a wave in which MSD increases linearly with t2
Introduction
Fundamentals Of Random Walks
The Simple Isotropic Random Walk
A Brw With Waiting Times
Random Walks With A Barrier
Crws And The Telegraph Equation
Reference
A Brw With Waiting Times
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.
A Brw With Waiting Times
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)l
it 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.
A Brw With Waiting Times
P(x, t+τ )= P(x, t)(1-l-r)+P(x-δ,t)r+
P(x+δ,t)l
 expressed it as a Taylor series about (x, t)
 Fokker–Planck equation
 Special case D is constant
A Brw With Waiting Times
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
 E(Xt2)~ t2 is like wave

 σt 2=2Dt is a standard diffusive process
Introduction
Fundamentals Of Random Walks
The Simple Isotropic Random Walk
A Brw With Waiting Times
Random Walks With A Barrier
Crws And The Telegraph Equation
Reference
Random Walks With A Barrier
a walker reaching the barrier
turn around and move away in the
opposite direction
or absorbed in barrier
Random Walks With A Barrier
1-D random walk process that
satisfies the drift–diffusion
Random Walks With A Barrier
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
Random Walks With A Barrier
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
Introduction
Fundamentals Of Random Walks
The Simple Isotropic Random Walk
A Brw With Waiting Times
Random Walks With A Barrier
Crws And The Telegraph Equation
Reference
Crws And The Telegraph Equation
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)
Crws And The Telegraph Equation
CRW at each time step,turning
events occur as a Poisson process
with rate λ
Crws And The Telegraph Equation
Expanding these as Taylor series and taking
the limit δ, τ->0 such that δ/τ =v gives
telegraph equation:
Crws And The Telegraph Equation
telegraph equation
http://www.math.ubc.ca/~feldman/apps/telegrph.pdf
small t (i.e. t=1/ λ), E(Xt2)~O(v2t2) is a wave
propagation process;
for large t, E(Xt2)~ 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