Half-life Simulation

Wednesday, November 11, 2020
By: Matthew Doucette

From a conversation with Ryan Young, I was curious why radioactive materials have a half-life. Why do they decay exponentially, which appears to be dependent on mass? Why does the mass matter, when particles decay individually? Why not linear decay? And, it is true that particles decay independent of the other particles.

Is there a connection to the overall mass somehow, even though particles decay individually? It turns out, yes.

It has to do with quantum mechanics. The "quantum" in quantum mecahnics, specificially. Which means discrete values. Particles cannot partially decay. They have to be not decayed or full decayed, and no where in-between. This is an example of the "quanta" and "quantization" in quantum mechanics. Interestingly, if particles could decay partially and each at the same rate, materials would decay linearly. It is the quantum decay that is random, regardless of how long they have existed for, that gives exponential decay. Those of you strong in mathematics and simulations already understand. This was my epiphany. It came to me while walking down Victoria Park, in Truro, Nova Scotia, Canada!

I decided to code it out:

Half-life Simulation:

Half life Simulation v2 - EMERGENT Exponential Decay (1,000,000 particles)

Quote:

Half-life Simulation.

Note that the exponential decay rate is EMERGENT. It is not coded in. Every particle has an equal chance to decay at every moment in time, independent of its time-of-life, as per real-world measurements described by quantum mechanics. The only mathematics being performed is the equal and linearly consistent chance for each particle to decay, independent of time and each other. The overall exponential decay of the mass is emergent, not programmed into the system. It also explains the often unintuitive idea that somehow a mass of radioactive particles somehow "knows" to decay faster when it is larger, as if somehow the particles know how big they are collectively. This is explained by particle physics in that when the mass is larger, there are more particles and therefore more chances for particles to decay (more rolls of the dice, one per each particle) although not more chance for any individual particle to decay. Makes sense?

Purpose:
To explore why half-life decay is exponential and not linear. Why does decay depend on quantity or mass? Why doesn't 100g of substance decay by 50g, and then by another 50g in the same amount of time? Etc.

Inspiration:
A good friend of mine from Acadia University, mentioned half-life as an analogy to aggregated data, and it piqued my interest.

Research:
I read on reddit that a chemistry class teacher could not explain why half-life was exponential decay rather than linear, and I was stuck at the same place. Although the answer given was correct, I did not intuitively understand it. Hence, I explored it more.

- wikipedia.org/wiki/Half-life
- xona.com/simulations (more of our simulation work)

--
Matthew Doucette (Xonatron)
matthewdoucette.com/