``Condensation'' in Fermi-Dirac statistics [D. S. Kothari and B. Nath, Nature 151, 420 (1943)], which appears as a natural consequence of Born's reciprocity principle [M. Born, Proc. R. Soc. London A 165, 291 (1938); M. Born, Nature 141, 328 (1938)], is examined from a theoretical perspective. Since fermions obey the Pauli exclusion principle, it is conceptually different from Bose-Einstein condensation, which permits macroscopic occupation of bosons at the single-particle level below a critical temperature. Yet, in accordance with the Cahill and Glauber [Phys. Rev. A 59, 1538 (1999)] formulation for fermionic fields, and in close kinship to bosonic fields, we have shown that in analogy to Bose-Einstein condensation, it is possible to associate an intrinsic notion of symmetry breaking and the thermodynamic ``order parameter'' to characterize the foregoing hitherto unexplored phenomenon in an ideal Fermi-Dirac gas as condensation-like coherence within fermions.
Condensation'' in Fermi-Dirac statistics [D. S. Kothari and B. Nath, Nature 151, 420 (1943)], which appears as a natural consequence of Born's reciprocity principle [M. Born, Proc. R. Soc. London A 165, 291 (1938); M. Born, Nature 141, 328 (1938)], is examined from a theoretical perspective. Since fermions obey the Pauli exclusion principle, it is conceptually different from Bose-Einstein condensation, which permits macroscopic occupation of bosons at the single-particle level below a critical temperature. Yet, in accordance with the Cahill and Glauber [Phys. Rev. A 59, 1538 (1999)] formulation for fermionic fields, and in close kinship to bosonic fields, we have shown that in analogy to Bose-Einstein condensation, it is possible to associate an intrinsic notion of symmetry breaking and the thermodynamic order parameter'' to characterize the foregoing hitherto unexplored phenomenon in an ideal Fermi-Dirac gas as condensation-like coherence within fermions.
In the spirit of Bose–Einstein condensation, we present a detailed account of the statistical description of the condensation phenomena for a Fermi–Dirac gas following the works of Born and Kothari. For bosons, while the condensed phase below a certain critical temperature, permits macroscopic occupation at the lowest energy single particle state, for fermions, due to Pauli exclusion principle, the condensed phase occurs only in the form of a single occupancy dense modes at the highest energy state. In spite of these rudimentary differences, our recent findings [Ghosh and Ray, 2017] identify the foregoing phenomenon as condensation-like coherence among fermions in an analogous way to Bose–Einstein condensate which is collectively described by a coherent matter wave. To reach the above conclusion, we employ the close relationship between the statistical methods of bosonic and fermionic fields pioneered by Cahill and Glauber. In addition to our previous results, we described in this mini-review that the highest momentum (energy) for individual fermions, prerequisite for the condensation process, can be specified in terms of the natural length and energy scales of the problem. The existence of such condensed phases, which are of obvious significance in the context of elementary particles, have also been scrutinized.
In the pair-correlation, or quasi-chemical equilibrium, approximation to statistical mechanics, there appears the possibility of a condensation phenomenon closely analogous to a Bose-Einstein condensation of correlated pairs of particles. However, the expansion used in earlier work to establish this phenomenon fails to yield an adequate approximation at and below the condensation point. In this paper, we give an alternative development, first formally for an arbitrary number of quantum states of the pair, then carry out in detail for the case of only one, or only two, quantum states of the pair. It is shown that, in spite of the Fermi-Dirac statistics of the particles making up the pairs, it is nevertheless possible to accomodate an arbitrarily large number of particles within a single pair state. Furthermore, if two quantum states of the pair are possible, and conditions are such that either the pair state would be occupied by a macroscopic number of particles if it alone were present, then the Pauli exclusion principle works in such a way that the lower one of the two pair states is occupied by a macroscopic number of particles, whereas the occupation number of the upper pair state remains independent of the volume. This result is not restricted to pairs whose mean separation exceeds their internal size, but applies also to pairs which completely overlap each other. The result is therefore of significance in the theory of superconductivity where the electron pairs have an internal size of the order of the Pippard coherence length, which is much larger than the mean separation of their centres of gravity.
Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed
Quantum statistical mechanics is the arithmetic of indistinguishable particles: bosons that pile into a single state and fermions that refuse to share---statistics whose consequences range from the Bose-Einstein condensate, a macroscopic quantum state, to the Fermi gas's pressure that holds white dwarfs against collapse. This article presents a narrative review of the primary literature of that statistics, from Bose's Planck's law and the light quantum hypothesis and Einstein's quantum theory of the monatomic ideal gas with its 1925 second paper predicting the condensation, through Fermi's quantization of the monatomic perfect gas and Dirac's theory of quantum mechanics, which independently founded the Fermi-Dirac statistics, London's identification of the lambda-phenomenon of liquid helium with the Bose-Einstein degeneracy, the experimental vindication of Anderson, Ensher, Matthews, Wieman, and Cornell's observation of Bose-Einstein condensation in a dilute atomic vapor and Davis and colleagues' sodium condensate, DeMarco and Jin's onset of Fermi degeneracy in a trapped atomic gas, and the syntheses of Leggett's fundamental concepts, Pitaevskii and Stringari's Bose-Einstein condensation, and Bloch, Dalibard, and Zwerger's many-body physics with ultracold gases. The synthesis is organized around three themes: the statistics' invention, in which indistinguishability's consequences were computed before any condensate existed; the condensate's realization, in which the dilute atomic vapors delivered Einstein's sixty-year-old prediction; and the many-body era, in which the ultracold gases became quantum simulators of matter's collective behavior. It is concluded that quantum statistics is the theory whose consequences were verified in reverse---predicted abstractly, found in liquid helium, created in atomic vapors---and that its bosons and fermions remain the universe's two grammars.
Abstract Indistinguishability of like particles, and the fermion and boson exchange symmetries discussed.Pauli exclusion principle and features of multi-electron atoms, including selection rules are discussed. Degeneracy pressure and the formation of compact stellar objects is analysed. Quantum exchange force between electrons and its contribution to ferromagnetism is outlined. Fermi-Dirac and Bose-Einstein statistics, includng the chemical potential are derived. The conditions for Bose-Einstein condensation are deduced; condensates and their stability are considered.
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.