Question on exponential decay?

[aftershock]

Hi everyone,

There's something that's kind of been bugging me about applying exponential decay formulas to real world phenomena. For example let's say the discharging of a parallel plate capacitor. Let's consider the negative plate. As it discharges excess electrons leave the plate. The charge falls off exponentially and we model this mathematically by an exponential decay formula.

But wouldn't there be a time while the amount of charge leaving is less than the elementary charge? We know energy is quantized and it seems to me that the exponential decay model completely fails when we get around to the capacitor holding a charge of 1e.

 

[Rap]

Hi everyone,

There's something that's kind of been bugging me about applying exponential decay formulas to real world phenomena. For example let's say the discharging of a parallel plate capacitor. Let's consider the negative plate. As it discharges excess electrons leave the plate. The charge falls off exponentially and we model this mathematically by an exponential decay formula.

But wouldn't there be a time while the amount of charge leaving is less than the elementary charge? We know energy is quantized and it seems to me that the exponential decay model completely fails when we get around to the capacitor holding a charge of 1e.

That's right. The exponential decay formula only holds for a large number of electrons.

 

[aftershock]

That's right. The exponential decay formula only holds for a large number of electrons.

That's interesting. Can anyone further elaborate on this? When does it start to fail, and what do we use instead when it does. Does it become a quantum mechanical problem?

 

[Rap]

Before you can talk about "failure", you have to talk about the definition of failure. The quantization produces an error from the exponential decay. If you are measuring n electrons/second, the error will be about sqrt(n) electrons/second. So if you are measuring 1 amp, that's like 10^16 electrons/sec with an error of 10^8 electrons/sec or about 10^-8 amp or 10^-6 percent. The exponential decay will be good. If you are measuring 100 electrons/sec the error will be 10 electrons/sec or 10 percent. The exponential decay is not so good. Pick a percentage error that you call "failure" and you can figure out at what current that error will occur. For high error rates, it becomes a statistical problem. I think (not sure) that the electrons will have a Poisson distribution and you have to talk about the probability of measuring a certain number of electrons per second. It will depend on your measuring device too - if it cannot count individual electrons, then you have to take that into account. Depending on your particular setup, this might be enough, but maybe not, you may have to start doing QM calculations as well.

 

[jtbell]

Yes, for small numbers, the Poisson distribution is the appropriate one to use.