How can I derive the Fermi-Dirac distribution function using simple mathematics? I am now tired of looking for the derivation on the net.So please help me to understand how actually electrons are distributed between various energy levels?
Okay, so do you understand derivation of thermodynamics of an ideal gas and what partition function is? This derivation can be done through use of principle of maximization of entropy and Lagrange multipliers. Do you know how to derive partition function of an ideal gas with fixed number of particles (canonical ensemble)? Make sure you understand this one and similar derivations for different gases. Try relativistic ideal gas as an exercise. Make sure you understand what partition function is and in general the basic principles of statistical physics and get a feeling of what it is all about. Then the next step would be derivation of grand canonical ensemble, and understanding what is meant by occupation numbers, chemical potential and so on and so forth (I found this part to be least intuitive). If you understand that, now you just need some understanding of QM and exchange symmetries. Now apply your understanding of QM and it's oddities to the derivation of partition function for a gas of fermions (What does Pauli exclusion principle imply about the occupation numbers and partition function derivation?) This is how the course was taught to me at my university and I feel that is the best way to understand it. To be honest, I am a dumdum when it comes to statistics and combinatorics, but learning it in that order I managed to obtain a very decent understanding of the topic myself. I don't know what is your level of understanding. If your level is basic don't expect to be able to jump in and understand it straight away, that is not an easy topic. If you level of knowledge is good but you are still struggling,maybe take look at the grand canonical ensemble in more detail, I personally don't find the derivation of it very intuitive and it keeps giving me headaches. If you are struggling with any of the steps be sure to leave a comment, I will be glad to explain it in more detail.