PHY452H1S Basic Statistical Mechanics. Problem Set 2: Generating functions and diffusion
Posted by peeterjoot on January 26, 2013
This is an ungraded set of answers to the problems posed.
The usual diffusion equation for the probability density in one dimension is given by
where is the diffusion constant. Define the Fourier components of the probability distribution via
This is useful since the diffusion equation is linear in the probability and each Fourier component will evolve independently. Using this, solve the diffusion equation to obtain in Fourier space given the initial .
Assuming an initial Gaussian profile
obtain the probability density at a later time . (NB: Fourier transform, get the solution, transform back.) Schematically plot the profile at the initial time and a later time.
A small modulation on top of a uniform value
Let the probability density be proportional to
at an initial time . Assume this is in a box of large size , but ignore boundary effects except to note that it will help to normalize the constant piece, assuming the oscillating piece integrates to zero. Also note that we have
to assume to ensure that the probability density is positive. Obtain at a later time . Roughly how long does the modulation take to decay away? Schematically plot the profile at the initial time and a later time.
Inserting the transform definitions we have
We conclude that
If the Fourier transform of the distribution is constant until time , so that , we can write
The time evolution of the distributions transform just requires multiplication by the decreasing exponential factor .
Propagator for the diffusion equation
We can also use this to express the explicit time evolution of the distribution
Our distribution time evolution is given by convolve with a propagator function
For we can complete the square, finding that this propagator is
A schematic plot of this function as a function of for fixed is plotted in (Fig1).
For the Gaussian of 1.3 we compute the initial time Fourier transform
The time evolution of the generating function is
and we can find our time evolved probability density by inverse transforming
For this is
As a check, we see that this reproduces the value as expected. A further check using Mathematica applying the propagator 1.0.12, also finds the same result as this manual calculation.
This is plotted for in (Fig2) for a couple different times .
Boxed constant with small oscillation
The normalization of the distribution depends on the interval boundaries. With the box range given by we have
With an even range for box this is unity.
To find the distribution at a later point in time we can utilize the propagator
Let’s write this as
Applying a change of variables for the first term, we can reduce it to a difference of error functions
Following Mathematica, lets introduce a two argument error function for the difference between two points
Using that our rectangular function’s time evolution can be written
For , and , this is plotted in (Fig3). Somewhat surprisingly, this difference of error functions does appear to result in a rectangular function for small .
The time evolution of this non-oscillation part of the probability distribution is plotted as a function of both and in (Fig4).
For the sine piece we can also find a solution in terms of (complex) error functions
This is plotted for , , and in (Fig5).
The diffusion of this, again for , , and is plotted in (Fig6). Again we see that we have the expected sine for small .
Putting both the rectangular and the windowed sine portions of the probability distribution together, we have the diffusion result for the entire distribution
It is certainly ugly looking! We see that the oscillation die off is dependent on the term. In time
that oscillation dies away to of its initial amplitude. This dispersion is plotted at times and for , and in (Fig7).
Similar to the individual plots of and above, we plot the time evolution of the total probability dispersion in (Fig8). We see in the plots above that the rectangular portion of this distribution will also continue to flatten over time after most of the oscillation has also died off.
An easier solution for the sinusoidal part
After working this problem, talking with classmates about how they solved it (because I was sure I’d done this windowed oscillating distribution the hard way), I now understand what was meant by “ignore boundary effects”. That is, ignore the boundary effects in the sinusoid portion of the distribution. I didn’t see how we could ignore the boundary effects because doing so would make the sine Fourier transform non-convergent. Ignoring pesky ideas like convergence we can “approximate” the Fourier transform of the windowed sine as
Now we can inverse Fourier transform the diffusion result with ease since we’ve got delta functions. That is
Question: Generating function
The Fourier transform of the probability distribution defined above is called the “generating function” of the distribution. Show that the -th derivative of this generating function at the origin is related to the -th moment of the distribution function defined via . We will later see that the “partition function” in statistical mechanics is closely related to this concept of a generating function, and derivatives of this partition function can be related to thermodynamic averages of various observables.
This entry was posted on January 26, 2013 at 12:42 am and is filed under Math and Physics Learning.. Tagged: diffusion equation, dispersion, fourier transform, Gaussian, Generating function, moment, PHY452H1S, probability distribution, propagator, Statistics mechanics. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.