Chapter 7 reviews the statistical physics of fermions and bosons. Fermions are shown to obey Fermi–Dirac statistics and must comply with the Pauli exclusion principle. Bosons follow Bose–Einstein statistics and do not obey the Pauli principle. The distinction between distinguishable and indistinguishable particles is discussed. Finally, the method of identifying of different nuclei and atoms as either fermions or bosons is presented.
Chapter 7 reviews the statistical physics of fermions and bosons. Fermions are shown to obey Fermi–Dirac statistics and must comply with the Pauli exclusion principle. Bosons follow Bose–Einstein statistics and do not obey the Pauli principle. The distinction between distinguishable and indistinguishable particles is discussed. Finally, the method of identifying of different nuclei and atoms as either fermions or bosons is presented.
Satyendra Nath Bose's attempt to describe the quantum statistical aspects of light consistently in terms of particles, and Einstein's generalisation, lead to the concept of Bosons as a class of quanta obeying `Bose-Einstein statistics'. Their identity as a class came in sharp contrast when the Pauli exclusion principle and the Dirac equation revealed the other class called Fermions, obeying `Fermi-Dirac statistics'. Spin, and spin alone, is the determining factor of the multiparticle behaviour of fundamental quanta. This is the basis of the Spin-Statistics Connection. While it is known that the overall theoretical picture is consistent, the physical reason for the connection is unknown. Further, the class difference is sensitive only to the total spin in a quantum aggregate, as spectacularly seen in superconductivity and superfluidity, and in the Bose-Einstein condensation of neutral atomic gas. Can we grasp the true reason behind the difference in the collective behaviour of Bosons and Fermions? An explorer's journey demanding logical and physical consistency of what we already know takes us to the hidden factors in the relation between spin and the statistics of quanta. The surprising answer is in the domain of gravity, that too, on a cosmic scale.
Quantum mechanics broadly classifies the particles into two categories: (1) fermions and (2) bosons. Fermions are half-integer spin particles, obeying Pauli's exclusion principle and Fermi-Dirac statistics. Whereas bosons are integer spin particles, not obeying Pauli's exclusion principle, and obeying Bose-Einstein statistics. However, there are two exceptions to this standard case: first, anyons, which exist in 2-dimensional systems, and secondly, paraparticles, which can exist in any dimension. Paraparticles follow their non-trivial parastatistics, obeying their generalised exclusion principle. In this paper, we provide a detailed review of the foundations of paraparticle statistics established in \cite{wang2025particle}. We extend this work further and then derive an important expression for the heat capacity of paraparticles for a specific case, which would provide a handle for the experimental detection of paraparticles in appropriate systems.
When identical particles are not localized they can no longer be distinguished . This situation occurs in the case of a gas since the particles of a gas are unconstrained in their motion except by the walls of the container. It also occurs when the particles are so close together that their wave functions overlap, that is when they are free to exchange positions. It should be stressed that this is a typical quantum-mechanical phenomenon, formulated as QM 3. Classical particles are always distinguishable because one can in principle follow each particle along its trajectory. According to quantum mechanics this is impossible even in principle. According to principle QM 3, there are in fact two different kinds of statistics depending on the type of particle: Bose-Einstein statistics for bosons and Fermi-Dirac statistics for fermions . This is related to the fact that fermions satisfy the Pauli exclusion principle ; there can be no more than one particle in each eigenstate.
In this article, we discuss the identity and indistinguishability of quantum systems and the consequent need to introduce an extra postulate in Quantum Mechanics to correctly describe situations involving indistinguishable particles. This is, for electrons, the Pauli Exclusion Principle, or in general, the Symmetrization Postulate. Then, we introduce fermions and bosons and the distributions respectively describing their statistical behaviour in indistinguishable situations. Following that, we discuss the spin-statistics connection, as well as alternative statistics and experimental evidence for all these results, including the use of bunching and antibunching of particles emerging from a beam splitter as a signature for some bosonic or fermionic states.
As a preliminary to discussing the properties of Bose–Einstein and Fermi–Dirac gases, this chapter presents the basic quantum equations that underlie both. The process of building N-body states from single-particle states is described, along with notation necessary to simplify the problem. The average number of particles in a given energy state and chemical potential is derived for fermions and bosons. Although all atoms are either bosons or fermions, and therefore indistinguishable, there are nevertheless real systems that are composed of distinguishable particles. Although all atoms are either bosons or fermions, and therefore indistinguishable, there are nevertheless real systems that are composed of distinguishable particles.
In this article, we discuss the identity and indistinguishability of quantum systems and the consequent need to introduce an extra postulate in Quantum Mechanics to correctly describe situations involving indistinguishable particles. This is, for electrons, the Pauli Exclusion Principle, or in general, the Symmetrization Postulate. Then, we introduce fermions and bosons and the distributions respectively describing their statistical behaviour in indistinguishable situations. Following that, we discuss the spin-statistics connection, as well as alternative statistics and experimental evidence for all these results, including the use of bunching and antibunching of particles emerging from a beam splitter as a signature for some bosonic or fermionic states.