Kolmogorov/Smoluchowski equation approach to Brownian motion
Tuesday, February 12, 2013
1:53 PM
Readings: Gardiner, Secs. 1.2, 3.8.1, 3.8.2
Einstein
Homework 1 due February 22.
Conditional probability for random variables.
Let's transfer our results for conditional probability for events into conditional probabilities
for random variables.
The standard bridge between events and random variables is to talk about events that are
described by the random variable taking a value for a certain (nice) Borel set.
If we had a discrete random variable Z, then we can form a partition of sample space by simply
decomposing it into the various possible values of Z.
If on the other hand Z is continuously distributed, then the notion of conditional probability,
where one conditions on the value of Z=z becomes technically problematic, and requires a
rather extensive discussion in a measure-theoretic presentation. The reason is that
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We will see that in the end, the formulas are not any more difficult, but the technical considerations
are real -- see Borel paradox, i.e., the wikipedia entry.
The outcome is that the law of total probability has a completely analogous form (in terms of PDFs):
Where the conditional PDF (for absolutely continuously
distributed random variables) is defined:
Now we return to our simplest model for Brownian motion, but now we will take the point of view
of the evolution of probability distributions for where the particle is, rather than describing the
particle position in terms of trajectories.
(don't need to assume the Z are Gaussian for this approach.)
Our first goal is to encode this stochastic update rule in terms of the probability distributions for
the position of the particle.
The positions are taken to be absolutely continuous random variables, so we will describe them
in terms of PDFs.
Apply law of total probability:
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Apply law of total probability:
Let's compute the transition PDF carefully.
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This implies:
The above calculation illustrated two common techniques for working with conditional
probability:
• Exploit the information given by the condition by substituting it as appropriate into the
event whose probability is being computed
• Try to express all remaining random variables in terms of random variables
independent of the condition, so the condition can be removed.
Substituting our last result into the law of total probability, we obtain:
The integral on the right is a convolution integral; these arise generically when one
adds independent random variables. Just as one can work with these in integral
equation theory using Fourier or Laplace transforms, one often works with sums of
independent random variables in probability theory by using analogous concepts,
known as characteristic functions and moment generating functions. And one could
solve the above equation exactly using this procedure, but this is not instructive for
our overall purposes (more complex applications) so we'll skip it. (See Gardiner Sec.
3.8.2). Instead, we'll show how one can derive an evolution equation for the PDF of
the particle position in the limit of continuous time.
The equation we've just written can be thought of as a Kolmogorov forward equation in discrete
time. This is simply an equation that describes how probability distributions evolve when the
underlying stochastic process is a Markov process.
We should supplement this evolution equation with an initial condition. This simply means
specifying
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underlying stochastic process is a Markov process.
We should supplement this evolution equation with an initial condition. This simply means
specifying
Which is the probability density for where the particle started at time t=0. If the initial
position is deterministic, then we can write this using Dirac delta function:
This can't be defined as an ordinary function, but rather can be rigorously defined as
a generalized function (functional):
Now let's see what results if we take a continuous-in-time limit of our discrete-time
Kolmogorov equation.
We will interpret
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Now we take the limit as
Now we will consider both the LHS and RHS in the limit of
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As for the RHS, we must bear in mind that pZ will change
as
There are fundamentally two distinct classes of stochastic processes that result depending
on how one assumes these kicks behave over a small interval of time.
Diffusion process (continuous):
Amplitude of the kicks become small as the
Time interval becomes small.
Jump process:
Amplitude of kicks don't become small; but they
become less likely.
We will proceed with the diffusion process framework, meaning we assume jumps become small as
the time interval between observations becomes small. So with the intuition (which can be
rigorized) that at small
We formally Taylor expand
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Is the matrix of second moments of the kick. It's like the
covariance matrix, except the mean isn't subtracted off.
Why did we stop the Taylor expansion at second order? This will become apparent later -it’s a generic feature of stochastic processes that truncation at second order is much more
natural than at any other order. (see Pawula theorem.)
So putting the above together, we see that for small
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