The Bose-Einstein Distribution
The Bose-Einstein distribution describes the statistical behavior of integer spin particles (bosons). At low temperatures, bosons can behave very differently than fermions because an unlimited number of them can collect into the same energy state, a phenomenon called “condensation”.
The Fermi-Dirac Distribution
The Fermi-Dirac distribution applies to fermions, particles with half-integer spin which must obey the pauli exclusion principle. Each type of distribution fumction has a normalization term multiplying the exponential in the denominator which may be temperature dependent. For the Fermi-Dirac case, that term is usually written:
The significance of the Fermi energy is most clearly seen by setting T=0. At absolute zero, the probability is =1 for energies less than the Fermi energy and zero for energies greater than the Fermi energy. We picture all the levels up to the Fermi energy as filled, but no particle has a greater energy. This is entirely consistent with the Pauli exclusion principle where each quantum state can have one but only one particle.
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