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# Disclaimer

This is an ungraded set of answers to the problems posed.

## Question: Diffusion

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.

## Answer

Inserting the transform definitions we have

We conclude that

or

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

or

A schematic plot of this function as a function of for fixed is plotted in (Fig1).

Fig1: Form of the propagator function for the diffusion equation

### Gaussian

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 .

Fig2: Gaussian probability density time evolution with diffusion

### 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 .

Fig3: Rectangular part of the probability distribution for very small t

The time evolution of this non-oscillation part of the probability distribution is plotted as a function of both and in (Fig4).

Fig4: Time evolution of the rectangular part of the probability distribution

For the sine piece we can also find a solution in terms of (complex) error functions

This is plotted for , , and in (Fig5).

Fig5: Verification at t -> 0 that the diffusion result is sine like

The diffusion of this, again for , , and is plotted in (Fig6). Again we see that we have the expected sine for small .

Fig6: Diffusion of the oscillatory term

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).

Fig7: Initial time distribution and dispersion of the oscillatory portion to 1/e of initial amplitude

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.

Fig8: Diffusion of uniform but oscillating probability distribution

**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.

## Answer