# Another exponential decay example

In the last couple of weeks, my wife has been having a go at making sourdough bread. One of the defining characteristics of this bread is that it doesn’t use yeast – at least, not directly. The idea is, to start it off rising, you leave it outside for a while, and allow lots of useful microbes to land on it. These then start multiplying, producing carbon dioxide, and allow the bread to rise. Presumably any nasties that land on it get walloped by the baking process (Anyway, I haven’t suffered any ill effects from eating it yet).

But – here’s the clever bit, once you’ve got one lot of dough you can reuse it. With your first batch of dough you make about 3/4 of it into bread, and keep the other quarter till tomorrow. Then, tomorrow, you add in more flour, liquid etc, bake 3/4 of it, but keep a quarter for the next day. And so on. Once you’ve got a nice population of microorganisms in the dough they’ll keep multiplying overnight, and so you can carry out this process indefinitely. I’m told that there are sourdoughs that are 200 years old – that is this quartering process has been going on for a very very long time. (I haven’t checked this out myself, so I might be getting some details wrong here, but hopefully you get the idea).

Now, here’s a question – how much of the original dough is left after 200 years of daily quartering? The answer follows along the much discussed homeopathic lines – nothing. The first day you take 1/4 of what you had, and keep it. The second day, you take 1/4 of that. After N days, you have 1/(4 to the power N) of the original left. With N just 5, (say about a week), you are left with 1/1000 of the original mixture. With N=10, you are left with a thosandth of a thousandth, or a millionth of the original. It doesn’t take long to end up with virtually nothing of the original left. If your starter dough contained every atom in the universe, it would only take about 140 days before there wasn’t a single atom left. So after 200 years, the chances of there being even a single molecule from your first day’s dough left in the bread is for all practical purposes zero. In which case, can you really say that the dough is 200 years old?

It’s just another example of exponential decay, which permeates through so much of physics.

## One Response to “Another exponential decay example”

Calling the dough 200 years old is like the Delphic boat riddle. (If during the voyage you replaced all parts of the boat, do you return on the same vessel you left with?)