The Fermi distribution, also known as the Fermi-Dirac distribution, is a probability distribution that describes the distribution of particles in a system at thermal equilibrium. It is used to calculate the probability of a particle occupying a particular energy state in a system of fermions, which are particles with half-integer spin such as electrons.
The fixed particle canonical ensemble is a thermodynamic ensemble used to describe a system of particles with a fixed number of particles. This ensemble is particularly useful for studying systems with a fixed number of fermions, such as electrons in a metal.
The Fermi distribution is derived by maximizing the entropy of the system subject to constraints such as the number of particles and total energy. This results in a distribution that follows the Fermi-Dirac statistics, which takes into account the exclusion principle for fermions.
The derivation of the Fermi distribution assumes that the system is in thermal equilibrium, that the particles are non-interacting, and that the system is in a fixed volume. It also assumes that the energy levels are discrete and that the particles follow the Fermi-Dirac statistics.
The Fermi distribution is used in various fields of physics, such as solid state physics, nuclear physics, and astrophysics. It is used to calculate the electronic properties of metals, the energy levels in nuclei, and the behavior of matter in extreme conditions, such as in white dwarfs and neutron stars.