I'm curious about how to clearly define the sample size when calculating the half-life of particles. My understanding is that the half life is statistical in nature and that for very small samples, the fluctuations in the number of particles that will decay can get quite large.
"there will be fluctuations, and once the number of particles remaining drops to two or one or zero, those fluctuations will be very very large. But what is fluctuating is your measurement of the half life, not the true theoretical half life itself."
But, what defines the sample size? Is the sample just what you have in the jar? If so, why? Why does your sample not include the particles that are in the jar next to the jar you're measuring? Or in the jar in the next room? Or in a jar half way around the world? Where and why does the line get drawn?
Hypothetically, if you were to take your sample into deep space, separating it by a distance (in light seconds or light years or light whatevers) that is greater than the length of the half-life of your sample, so that the decay must be casually disconnected from every other particle of the same type, would the statistical nature of the decay rate still apply? Or would the fluctuations in the number of particles that decay decrease?
For example, let us posit that you have two particles with a half-life of one minute and you've separated them from any other particles of the same time by two light minutes. Traditionally, we would assume that there is a certain probability that one of the particles will decay after one minute has elapsed. There is a lower probability that both or neither of them decayed during that time. But, if ALL of particles within the light-cone (as defined by the half-life) are in your sample, does the statistical nature of the half-life still apply?