Radioactive decay versus compound interest

[nomadreid]

The fact that radioactive decay and continuous compound interest end up with the same formula (with the "rate" being negative in the former and positive in the latter) seems to me to be more a result of the ubiquity of the exponential function in solving differential equations than any common physical principle underlying both processes --the mechanism for the interest requires a "memory" and is dependent on the total, whereas the probability for each particle cannot be dependent on any memory or influence from other particles . But I could easily be wrong. If I am, could someone elucidate this principle?

 

[Pengwuino]

I wouldn't say either requires a memory. Suppose you have something at 10% compound interest. $1 is going to turn into $1.10 after that period regardless if it's the only dollar in the account or if it's part of $1000 in the account. The new money created really doesn't care where it came from. That new $.10 in the $1 account is going to earn 10% interest itself and generate $0.01 the next period. All that matters is that it exists and the same process will happen to it as happened to the full $1 earlier.

So in neither situation is the actual $.10 or a single particle dependent on the total. That $.10 will earn $.01 more regardless if there is $1.10 in the account or $1,000.10 in the account.

I think your line of thinking is missing the idea that the particle acts independent of what's going on in the whole, but not realizing that a dollar will act independent of what's going on in the whole as well.

 

[JeffKoch]

dN/dt is proportional to N in both cases, which is why you get exponentials. I don't think there's anything more mystical than this - in both cases the rate of change dN/dt depends on how much you have, N, at that particular time t.

 

[nomadreid]

Thanks for the answers, Pengwuino and JeffKoch.

 

[pmacias]

I can confirm that the similarity in formula between radioactive decay and compound interest is indeed a result of the ubiquitous nature of the exponential function in solving differential equations. This does not necessarily imply a common physical principle underlying both processes.

The mechanism for compound interest does require a "memory" in the sense that the interest is calculated based on the total amount of money, and this total is continuously changing as interest is added. On the other hand, the probability for each particle to decay in radioactive decay is not dependent on any memory or influence from other particles. Each particle has a constant probability of decaying at any given time, regardless of the number of particles present.

It is important to note that while the formulas may be similar, the underlying principles and mechanisms for these processes are fundamentally different. Compound interest is a result of financial transactions and investments, while radioactive decay is a natural phenomenon governed by the laws of physics.

In terms of elucidating a principle, I would say that the exponential function is a powerful tool for expressing change over time in a variety of systems. It is not limited to just radioactive decay and compound interest, but can also be applied to population growth, chemical reactions, and many other processes. The similarity in formula between these seemingly disparate processes highlights the versatility and importance of the exponential function in scientific and mathematical applications.