derive Fermi-Dirac distribution law...?
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The distribution or probability density functions describe the probability that particles occupy the available energy levels in a given system. Of particular interest is the probability density function of electrons, called the Fermi function. The derivation of such probability density functions can be found in one of the many statistical thermodynamics references . However, given the importance of the Fermi distribution function, we will carefully examine an example as well as the characteristics of this function. It is also derived in section 2.5.5. Other distribution functions such as the impurity distribution functions, the Bose-Einstein distribution function and the Maxwell Boltzmann distribution are also provided.
2.5.1 Fermi-Dirac distribution function
 
The Fermi-Dirac distribution function, also called Fermi function, provides the probability of occupancy of energy levels by Fermions. Fermions are half-integer spin particles, which obey the Pauli exclusion principle. The Pauli exclusion principle postulates that only one Fermion can occupy a single quantum state. Therefore, as Fermions are added to an energy band, they will fill the available states in an energy band just like water fills a bucket. The states with the lowest energy are filled first, followed by the next higher ones. At absolute zero temperature (T = 0 K), the energy levels are all filled up to a maximum energy, which we call the Fermi level. No states above the Fermi level are filled. At higher temperature, one finds that the transition between completely filled states and completely empty states is gradual rather than abrupt.
Electrons are Fermions. Therefore, the Fermi function provides the probability that an energy level at energy, E, in thermal equilibrium with a large system, is occupied by an electron. The system is characterized by its temperature, T, and its Fermi energy, EF. The Fermi function is given by:
(2.5.1)
This function is plotted in Figure 2.5.1 at 150, 300 and 600 K.
Figure 2.5.1 :The Fermi function at three different temperatures. 
The Fermi function has a value of one for energies, which are more than a few times kT below the Fermi energy. It equals 1/2 if the energy equals the Fermi energy and decreases exponentially for energies which are a few times kT larger than the Fermi energy. While at T = 0 K the Fermi function equals a step function, the transition is more gradual at finite temperatures and more so at higher temperatures
In quantum physics statics, a branch of physics Fermi- Dirac describes distribution of particles over energy states in system consisting of many identical particles that they obey "Pauli exclusion principle".