Transcript Markov Chains and Osmosis

An Introduction to Markov Chains
Homer and Marge repeatedly play a
gambling game. Each time they play,
the probability that Homer wins is 0.4, and
the probability that Homer loses is 0.6.
They only have 4 cents between them.
A “Drunkard’s Walk”
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P(Homer wins) = .4
P(Homer loses) = .6
Homer and Marge both start with $2
A Markov Chain is a mathematical
model for a process which moves
step by step through various states.
In a Markov chain, the probability
that the process moves from any
given state to any other particular
state is always the same, regardless
of the history of the process.
A Markov chain consists of states
and transition probabilities.
Each transition probability is the probability
of moving from one state to another in one
step. The transition probabilities are
independent of the past, and depend only on
the two states involved. The matrix of
transition probabilities is called the
transition matrix.
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P(Homer wins) = .4
P(Homer loses) = .6
Homer and Marge both start with $2
The states are the amount of money Homer has
If P is the
transition matrix for
a Markov Chain,
then the nth power
of P gives the
probabilities of
going from state to
state in exactly n
steps.
If the vector v represents the initial state,
then the probabilities of winding up in the
various states in exactly n steps are exactly
v times the nth power of P .
When they both start with $2, the
probability that Homer is ruined is 9/13.
If Homer starts with $ x and Marge starts with
$ N-x, and P(Homer wins) = p,
P(Homer loses) = q, then the probability
Homer is ruined is
Suppose you bet on red in roulette.
P(win) = 18/38 = 9/19; P(lose) = 10/19.
Suppose you and the house
each have $10
Now suppose you have $ 10
and the house has $20
Now suppose you and the house each have
$100.
Andrei Markov (1856-1922)
Paul Eherenfest: Diffusion model, early 1900s
Statistical interpretation of the second law of thermodynamics:
The entropy of a closed system can only increase.
Proposed the “Urn Model” to explain diffusion.
Albert Einstein, 1905
Realized Brownian motion would provide a magnifying
glass into the world of the atom. Brownian motion has
been extensively modeled by Markov Chains
Particles are separated by a semipermeable membrane, which they
can pass through in either direction.
Suppose that there are N black
particles inside the membrane, and N
white particles outside the membrane.
Each second, one random molecule
goes from outside the membrane to
inside, and vice versa.
Osmosis
There are N+1 states, given by the
number of white molecules inside.
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5 molecules
5
N
molecules
N molecules
If this process runs for a while, an interesting
question is: How much time, on average, is the
process in each state?
A Markov chain with transition matrix P is said
to be regular if some power of P has all positive
entries for some n. In a regular Markov chain, it
is possible to get from any state to any other
state in n steps.
The Markov chain for our osmosis process
is regular. Even starting with all black
particles inside, if a white particle entered at
every step, then the process would pass
from zero white inside through all possible
states.
For a regular Markov chain, the amount of
time the process spends in each state is
given by the fixed probability vector,
which is the vector a such that Pa = a.
Moreover, for any probability vector w,
No matter what the starting state, if the
process runs for a long time, the probability of
being in a given state is given by a.
In the long run, the fraction of time the process spends in
each state is given by the fixed probability vector.
For N particles, the fixed vector is:
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1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
Fixed vectors
N=4
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.0005
(1/70, 16/70, 36/70, 16/70, 1/70)
N = 10
.000005
.0005
.01
.08
.24
.34
.24
.08
.01
.0005
.000005
Now suppose 1000 molecules
The percent of the time that
there are between 225 and 275
black molecules inside is 0.999.
The percent of the time that there are
either fewer than 100 black or more
than 400 black molecules inside is
If the universe is 15 billion years old, the
average amount of time that a system with
500 molecules will have fewer than 100
black or more than 400 black molecules
inside the membrane is