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Project 1
Investigation of Anomalous Diffusion
Introduction
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In the early 19th century Robert Brown discovered that
particles suspended in a fluid exhibited a random motion,
which we have come to call “Brownian Motion”.
This phenomenon occurs because of the random motions
of the particles that make up the fluid. Theoretical work in
the early 20th century allowed us to connect this observable
behavior to the physics of gases and liquids.
Today, the work is being extended to look at individual
molecules within cells, with some interesting and surprising
results.....
See E. Barkai et al Physics Today Vol. 65 No. 8 p.29 (2012)
Example of Perrin (1908)
Measurements of Brownian Motion
Individual Particle Tracks
Brownian Motion
Probability Distribution from Particle Tracks
fit to a gaussian distribution
Brownian Motion is well modeled as
a Random Walk
Useful Statistics
Ensemble Average Mean Square Distance
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Let d be the distance achieved after a random
walk of a particular number of steps (or time).
For a large number (N) of particles, let di
represent the distance traveled by the ith
particle in its random walk.
The Ensemble Average of the Mean Square
Distance achieved is:
Ensemble Averages
4096 Random Walkers
Linear Scale
Logarithmic Scale
MSD proportional to t
MSD proportional to t1
Useful Statistics
Time Average MSD for a Single Particle
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Let array x contain the positions (xi) of a single
particle at time steps i. Let there be N positions
in the array.
Time average MSD for lag L is mean square
distance between particle positions at time i and
i+L.
Time Averaged Behavior
for Individual Particle Tracks
Ergodic Behavior
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A simple random walk shows the same
behavior for MSD calculated in these two
different ways.
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This behavior is said to be “ergodic.”
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Time Average MSD = Ensemble Average MSD.
This is not so unfamiliar. Consider an experiment in
which we flip N coins once versus one in which we
flip one coin N times. We expect to get the same
answer if N is sufficiently large.
Ergodic behavior is one of the cornerstones of
statistical mechanics.
Can there be Non-Ergodic
Behavior?
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Observations of many
complex physical systems
lead to non-ergodic
behavior.
Tracking individual molecules in cells
Complex behaviors
observed in tracking of
individual molecules in
cells.
Behavior of the “random
walk” that is observed
provides clues to
processes at work.
1 micron
From Weigel et al PNAS 108 6439 (2011)
Tracking Molecules in Cells
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Particles in fluids expected to follow a random
walk and exhibit Normal Diffusion.
Molecules in cells move randomly, but do not
follow the same behavior and are said to
display Anomalous Diffusion.
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Mean Square Distance not simply proportional to
time (number of steps)
Time and Ensemble Averages of behavior lead to
different results! (System is Not Ergodic)
In this project, we seek to calculate simulations
of this phenomenon.
Extending the Random Walk Model
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In our simple random walk,
P(x)
particles took a random step of
fixed length in position for a fixed
step in time. Pretty Simple!!
You get the same behavior for
steps in position and the length Q(t)
of time between steps that are
drawn from a probability
distribution as long as you can
determine an average value over
the entire distribution.
<x>
x
<t>
t
Continuous Time Random Walk
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What happens if the time interval between
steps follows a probability distribution that does
NOT lead to a well defined value?
Consider time steps drawn from a power law
probability distribution:
In this case we get anomalous diffusion.....
Drawing random numbers from an
arbitrary probability distribution
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We'll describe two methods:
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Transformation Method
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Rejection Method
For this project I recommend the transformation
method....
Creating New Distributions
Transformation Method
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Equal areas have equal
probabilities:
| P(x) dx | = | P(y) dy |
Consider:
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P(x) = e-x
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f(x) = 1 - e-x
Select f from uniform
distribution, then:
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x = -ln(1-f)
Creating New Distributions
Rejection Method
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Draw box around
function
REJE
CT
P(x)
Generate UD random
number in x: R1
Generate second UD
random number in y:
R2
If P(R1) > R2: keep
the point.
O
K
R
1
R
2
X
Project Assignment
Step 1
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First lets show that if we take steps spaced by
random times selected from a distribution with a
well determined average value then we just get
a normal random walk behavior.
Let particles sit for a period of time and then
take a step at random time intervals drawn from
the probability distribution:
-t
Q(t) = e
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Show that you get the same random walk
behavior expected from uniform unit time steps.
Step 1 (Continued)
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NOTE: In order to demonstrate the normal diffusion
behavior, we need to keep track of the position as a
function of time. But, the time steps in this simulation are
not uniform! Therefore, we must use our random walk
data to derive the position of the particles at regular
intervals in time in order to compute our statistics.
Position
Particle Track
Sample Times
Time
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Demonstrate normal diffusion by showing the MSD is
proportional to time and that Ensemble and Time averaged
MSD's lead to the same result.
Step 2
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Now lets extend the model to consider what happens
when we draw the step times from a power law
distribution
This one is a little tricky for the “transformation”
method. We must set upper and lower limits on values
of “t” to achieve the proper normalization. Lets set the
lower limit of t to .01 and set the upper limit to a
parameter we'll call the “experiment time”,
corresponding to the length, in time, of the random
walk. Our nominal “experiment time” will be 5000 units
Q(t)
tlo tp
thi
Step 2 (continued)
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Consider a nominal value for alpha of 0.7 and
calculate random walks to compute:
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The ensemble average of Mean Square Distance
versus time in the simulation.
The time average Mean Square Distance for
individual particles as a function of the time lag.
Explore and describe the behavior of this new
random walk compared to the normal case.
Note that what seems to be a modest change in
assumptions for the random walk leads to some
big differences in behavior!
Questions to Consider
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Is the Ensemble Average MSD simply
proportional to time as in the standard random
walk?
What is the functional form of the Ensemble
Average versus time? (Hint: plot results using a
log-log plot)
Is the behavior of the Time Average MSD for an
individual particle the same as the behavior of
an ensemble? (Is the behavior ergodic?)
Do all particles achieve the same MSD in the
same amount of time?
Extra Credit Question
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This new system is said to exhibit “aging”,
meaning that the behavior of the random
walkers depends on the length of the
experiment!
Show that there is a relationship between the
length of the experiment and the mean square
distance that is traveled by a typical particle in a
given time.
Things to include in your report
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Introduction
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Description of the Problem
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Description of the Numerical Method
Tell how you verified that the program works!
Present results of running the program – be sure
to present results from any milestones called for
in assignment. Provide listings of programs.
Present analysis of results; be sure to answer any
specific questions posed.
Conclusion