Monday, January 26, 2015

Statistical Mechanics, Part 2

[This blog post is part of two series. The first series, titled Statistical Mechanics, is a series of lessons that I have developed to illustrate how curriculum and instruction can be designed to encourage vertical learning. We start by drilling down from our understanding of diffusion to develop a dynamic model of diffusion grounded in random particle motion using statistical mechanics. We then build on top of this foundation to construct deeper understandings of fluid flow, heat transfer, and electrical circuits. You can navigate through the lessons in this series using the links below. These lessons are then part of a larger series, Defining Vertical Learning.]

Part 1  |  Part 2  |  Part 3  |  Part 4  |  Part 5  |  Part 6  |  Part 7  |  Summary

Modeling Diffusion

For most of us, diffusion makes sense because we have lots of firsthand experience with it. We see diffusion happening in our everyday lives. It seems natural and intuitive. But few of us understand the local mechanisms that drive diffusion. We know that particles flow from regions of high concentration to regions of low concentration, but we don’t know how or why. Some of us even have an inaccurate and misleading view of diffusion on a molecular scale, even though we know that the particles of red dye are moving randomly and not being guided in any specific direction.

An inaccurate and misleading visualization
of diffusion on a molecular scale

In this lesson, we are going to drill down and model diffusion using random particle motion. Pennies will represent red dye particles, and to keep things simple for now, pennies will only move along one dimension, to the left or to the right. To determine the direction a penny moves, we will flip it. If the penny comes up tails, it moves to the left. If it comes up heads, it moves to the right. It doesn’t get any more random than that!

Place your pennies in a cup. Shake them up and then dump
them out. The pennies that come up tails are passed to the
left. The pennies that come up heads are passed to the right.

Next, we are going to set up a number of discrete regions in a row. Pennies will move either left or right, from region to region.

At the start of the round, there are six pennies in region 3,
four pennies in region 4, and three pennies in region 5.

Shake up the pennies in each
region in a cup and dump them out.

Pass the tails to the left and the heads to the right.

At the start of the next round, there are five pennies
in region 4, three from region 3 and two from region 5.

Any pennies in the leftmost region that come up tails will stay where they are, and any pennies in the rightmost region that come up heads will stay where they are. These are our boundary conditions. At the start of the simulation, all of the pennies will be in the leftmost region. This is our initial condition.

In this simulation, I am using 200 pennies. The more pennies you use, the smoother the results. But you also want to make sure that the number of pennies you use doesn’t take too much effort to flip, sort, and pass. When I did this in a classroom of 32 students, I set up the simulation so that there were eight regions in a row, with one student per region. This gave us four rows. Then, instead of treating each row as a separate simulation, we combined the data so that all of the pennies were in a single simulation.

Pennies moving randomly left and right, region to region, over time

The red bars in the graph above represent the number of pennies in each region. As you can see, the pennies are initially concentrated in the leftmost region, but as they move randomly left and right, they are slowly becoming evenly distributed.

Here is another way to visualize the same data that was inspired by a molecular
diffusion animation created by a commenter, Steve. The model simulates diffusion
in one dimension, which you can think of as a sealed pipe with red dye in one end.

If you run this simulation with pennies, the participants are actually fairly shocked at the end. During the simulation, you are simply receiving pennies, shaking them up and dumping them out on a table, and then passing them on. After each round, you count the number of pennies in your region and report the results. You may, from time to time, glance at the regions next to you, but you generally have no idea what is happening across the entire simulation. All you see are your pennies, which are moving randomly with no direction or sense of their surroundings. You may know that the pennies are suppose to end up evenly distributed, but you can’t see how your actions contributed to that happening. It’s a bit like a thousand people each painting the roof of their house a random color, and then flying overhead and seeing that, when taking all the roofs together, you’ve collectively painted the Mona Lisa without meaning to. Okay, that would be way cooler, but you get the idea!

On an intellectual level, we know that simple local mechanisms on the molecular scale generate the complex macroscopic behaviors that we see everyday. They have to, since particles don’t have brains and there’s no one coordinating them. But most of us don’t know that on a visceral level. For most of us, this simulation serves as an introduction to a brave new world. Have we figured out how the random motion of pennies caused those pennies to diffuse? Not yet. But we saw it happen right in front of us, and we will.

Part 1  |  Part 2  |  Part 3  |  Part 4  |  Part 5  |  Part 6  |  Part 7  |  Summary

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