introduction fundamentals of random walks the simple isotropic random walk a brw with waiting times...
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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
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
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
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)
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