Workout: The Boltzmann distribution


Read the webpage Boltzmann distribution. Although you are likely to be well familiar with the graph of the exponential function (see powers and exponentials), in the Boltzmann distribution, $P(E) = P_0e^{-E/k_BT}$, the parameter we are familiar with and control, the temperature $T$, is in the denominator of the exponent. This makes it somewhat tricky to see when looking at a graph.


To get some sense of how this works,launch one of the web's graphing calculators, such as Desmos.

Set up

In the first entry space type (or use the entry pad at the bottom), y = Ae^(-ax), and press enter. Then select the button "All" to create sliders for changing a and A.

Click the "+" icon on the right of the screen and move the graph around so you are only looking at the curve for positive values of x.

Your screen should look something like the figure below.

The horizontal axis on your graph represents energy and the vertical axis represents probability, P, as controlled by the Boltzmann factor.

Answer these questions

1. Before you change anything, describe what the curve looks like for positive values of $x$ What is the value of the probability for $x = 0$?

2. Now slide $A$ through various positive values and describe what happens.

3. Put $A$ back equal to 1 and observe what happens as you change $a$. What happens when $a$ is positive and getting larger? Smaller? Pick a value of $x$ to focus on and generalize to a description of what happens to the entire curve.

4. Put $a$ back equal to 1. Now go to your equation and edit it so that instead of having $ax$ in the exponential, you now have $x/T$. (Add the $T$ slider and x-out the $a$ slider.) Now describe what happens as $T$ increases and decreases (always remaining positive). How does this compare to what happened with $a$?

5. What would you have to do to reproduce the graph of the Boltzmann factor as a function of $T$ given in the reading?

Joe Redish 2/11/18


Article 614
Last Modified: April 9, 2019