Distribution of Electric oscillator's in Boltzmann's theory

Question

While I was reading the book "In Search of Schrodinger's Cat" I found an interesting excerpt on how Max Planck used Boltzmann's statistical equations to solve the Blackbody radiation problem. The book mentions that there will be very few electric oscillators at very high energy end and at the lower end, electric oscillators would not have that much energy to add up to any significance. So most will be on the middle range. But my question is why and how this distribution of electric oscillators comes into picture. I mean why can't be the this distribution skewed, so that we may have electric oscillators concentrated at high ends. More specifically how this probability distribution was derived by Boltzmann? ( I am a layman interested in Physics so please be simple in your explanation)

Answer 1

The reasoning behind the Boltzmann distribution, in broad qualitative (and therefore not precise) terms goes like this.

Say we have $N=10$ particles and they share a total energy $E=100$, which, to make things easier, we may take to be discrete so that the energy of a particle may be 1, 2 etc. Then we ask, what is the most likely distribution of energy among these particles knowing nothing else but the information I gave already?

Consider a few possible scenarios:

1. All particles have energy 10
2. One particle has energy 91 and the remaining 9 particles have energy 1 each
3. One particle has energy 20, one particle has energy 16 and the remaining 8 particles have energy 8 each
4. $\cdots$

Not all of these are equally likely. Case 1, for example is too restrictive, there is only one way to divide the energy equally to all particles. Case 2 is certainly more likely because we have ten choices for the lucky particle that gets to have 91 units of energy. Case 3 is even more flexible because there are more ways to choose which particle gets what energy.

It turns out that having a lot of particles with small energy generally gives us more ways to make a distribution happen. If $N$ and $E$ are small enough you can work out all the possibilities, but beware, the possibilities increase dramatically fast with increasing $N$ and $E$.

When we do the numbers (see here, for example), it turns out that the most likely distribution is exponential, i.e., lots of particles with relatively small energy and few particles with large energy. This is the Boltzmann distribution.