Recurring themes in (condensed matter/nano) physics: Exponential decay laws

It's been a little while (ok, 1.6 years) since I made a few posts about recurring motifs that crop up in physics, particularly in condensed matter and at the nanoscale.  Often the reason certain mathematical relationships crop up repeatedly in physics is that they are, deep down, based on underlying assumptions that are very simple.  One example common in all of physics is the idea of exponential decay, that some physical property or parameter often ends up having a time dependence proportional to \(\exp(-t/\tau)\), where \(\tau\) is some characteristic timescale.

Buffalo Bayou cistern.  (photo by Katya Horner).

Why is this time dependence so common?  Let's take a particular example.  Suppose we are in the remarkable cistern, shown here, that used to store water for the city of Houston.   If you go on a tour there (I highly recommend it - it's very impressive.), you will observe that it has remarkable acoustic properties.  If you yell or clap, the echo gradually dies out by (approximately) exponential decay, fading to undetectable levels after about 18 seconds (!).  The cistern is about 100 m across, and the speed of sound is around 340 m/s, meaning that in 18 seconds the sound you made has bounced off the walls around 61 times.  Each time the sound bounces off a wall, it loses some percentage of its intensity (stored acoustic energy).

That idea, that the decrease in some quantity is a fixed fraction of the current size of that quantity, is the key to the exponential decay, in the limit that you consider the change in the quantity from instant to instant (rather than taking place via discrete events).    Note that this is also basically the same math that is behind compound interest, though that involves exponential growth.