The Gibbs paradox arises when considering the entropy change of a system during a reversible process:
where f(E) is the probability that a state with energy E is occupied, EF is the Fermi energy, k is the Boltzmann constant, and T is the temperature. The Gibbs paradox arises when considering the entropy
The Fermi-Dirac distribution describes the statistical behavior of fermions, such as electrons, in a system: EF is the Fermi energy
At very low temperatures, certain systems can exhibit a Bose-Einstein condensate, where a macroscopic fraction of particles occupies a single quantum state. k is the Boltzmann constant
where μ is the chemical potential. By analyzing the behavior of this distribution, we can show that a Bose-Einstein condensate forms when the temperature is below a critical value.