In physical chemistry book (Thomas Engel, Philip Reid and Robert G Mortimer), the Boltzmann distribution is derived based on the argument that for large system, the dominant energy configuration is similar to the average energy configuration (Example of 4 Harmonic Oscillators of a macrostate E=4h$\nu$). I'm convinced by this but i have a gap in understanding how this is related to energy distribution of an ensemble of particles. Like I think the macrostate with the most number of microstates will have the highest chance to be occupied so i dont quite understand how a dominant configuration of 1 single macrostate plays a role in energy configuration of many macrostates. I understand the mathematical derivation but I want to understand it intuitively too (with some physical reasons). I think it has sth to do with the number of particles too. But not sure about the exact relationship. Thanks for the help!
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$\begingroup$ Your question is a bit unclear. It could do with some reformatting and some rewording. I'm guessing dominant energy means the energy of the most likely macrostate, and and what you mean by average energy is the expectation value of the energy. I think to resolve this you should just read up on some probability, and it will develop the intuition for this. However, if this is not what you mean (or even if it is and you want and answer from PSE) please reformat your question. $\endgroup$– FizzKicksCommented May 31, 2020 at 22:12
2 Answers
The key idea in deriving the Boltzmann distribution is that the dominant macrostate is the one with highest degeneracy, not lowest energy. This is true, because for extremely large systems the distribution of these possible macrostates narrows down near the mode. Intuitively, you can think about flipping a fair coin 10 times -- oberving 60%-40% heads-tails is hardly unlikely at this sample size, but when you flip it $10^{23}$ times, 60%-40% becomes extremely unlikely, and you expect to obtain something very close 50%-50% -- the most degenerate macrostate.
Also note that 4 harmonic oscillators might be nice to think about, but they absolutely don't constitute a thermodynamic macrostate. You should always relate this approximation to a system with a very large number of particles, because it can and does break down at smaller scales.
I don't know if I understood correctly as a doubt about microstate with max. energy also being the average energy.
For a group of energy ranges $E_k$, the average energy of the system $E_{av}$ not necessarily matches the range with bigger probability. Below there is a Boltzmann distribution where the average is $150$, but the range with the max. probability is $0$ to $75$.
