Where does the heat go?
Tonight's idea
Imagine five equal pieces of a metal bar. Give each piece a number of heat points. The points are a simple model of how much thermal energy each piece holds.
At each step, every inside cell keeps half its points and sends one-quarter to each neighbour. An end cell has only one neighbour, so it keeps three-quarters and sends one-quarter inward. Work out every new value from the old row, then change them all together.
Hot spikes shrink while nearby cold pieces warm. This spreading and smoothing is called diffusion. Computers model heat in the same way, with far more, much smaller cells.
Insulated ends let nothing escape. Every heat point is kept or passed inside the bar, so the total cannot change.
Watch one: flatten a spike
Start with all 100 points in the middle. Cover the bottom row and predict it before reading on.
For step 1, the middle keeps 50 and gives 25 each way. For step 2, the left end receives 25 ÷ 4 = 6.25. The next cell keeps 12.5 and receives 12.5 from the middle, making 25. The middle keeps 25 and receives 6.25 from each side, making 37.5. The other side mirrors it.
Your turn
- Find step 3 after 6.25, 25, 37.5, 25, 6.25. Remember that an end keeps three-quarters, and use only the old row until all five new values are ready.
- Add the five values in every row, including your step 3. Why must the total stay 100 without needing to add it each time?
- Start a new insulated bar at 100, 0, 0, 0, 100. Find the next two rows. When does the middle cell first warm up?
- Now cold surroundings hold both end cells at 0. Start from 0, 25, 50, 25, 0. The usual next row would put 6.25 at each end, but those points escape instead. Write the new row and its total. How much heat left the bar?
Stretch: where does it settle?
Continue the original insulated bar for a long time. The cells get closer and closer to one shared value. What must that value be? Explain why a settled row cannot have neighbouring cells with different values.
Answers
- Step 3 is 10.9375, 23.4375, 31.25, 23.4375, 10.9375. For example, the left end is ¾ × 6.25 + ¼ × 25 = 10.9375. The next cell is ¼ × 6.25 + ½ × 25 + ¼ × 37.5 = 23.4375.
- Every row totals 100. Each cell divides all its old points into kept points and passed points. With insulated ends, every passed point enters another cell in the bar. Nothing is created, counted twice or lost.
- The rows are 75, 25, 0, 25, 75, then 62.5, 31.25, 12.5, 31.25, 62.5. The middle first warms at the second step, receiving 6.25 from each neighbouring cell.
- The new row is 0, 25, 37.5, 25, 0, with total 87.5. The missing 100 − 87.5 = 12.5 points are the 6.25 lost at each end.
Stretch. The shared value must be 100 ÷ 5 = 20. If neighbours differed, the hotter one would still pass more heat toward the colder one than it received back, so the row would still be changing. Equal neighbours cause no net change. The values approach 20, 20, 20, 20, 20; the model need not land there after any particular finite step.