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 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.
``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 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.
I will review recent work by us on the properties of Bose-Fermi mixtures with a tunable pairing interaction between bosons and fermions. A many-body diagrammatic approach, able to describe the condensed phase of a Bose-Fermi mixture from weak to strong boson-fermion couplings, will be presented [1]. This approach will be validated by comparing it with previous [2] and new dedicated fixed-node diffusion Monte Carlo calculations. By using both methods, a universal behavior of the condensate fraction and bosonic momentum distribution with respect to the boson concentration is found in an extended range of boson-fermion couplings and concentrations. For vanishing boson density, the bosonic condensate fraction reduces to the quasiparticle weight Z of the Fermi polaron studied in the context of polarized Fermi gases, unifying in this way two apparently unrelated quantities. Finally, I will discuss an interesting effect occurring in the molecular limit of the boson- fermion coupling, where the condensation is completely suppressed [3]. This phenomenon is an indirect effect on bosons of the Pauli exclusion principle acting on fermions, and is the counterpart in BoseFermi mixtures of the so called “Sarma phase” discussed for polarized Fermi gases. [1] A. Guidini, G. Bertaina, D. Galli, and P. Pieri, arXiv:1412.2542. [2] G. Bertaina, E. Fratini, S. Giorgini, and P. Pieri, Phys. Rev. Lett. 110, 115303 (2013). [3] A. Guidini, G. Bertaina, E. Fratini, and P. Pieri, Phys. Rev. A 89, 023634 (2014).
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.
Quantum systems consisting of many particles are both interesting from the physics and the mathematics perspective. One aspect that makes them fascinating from the physics point of view is that they can exhibit quantum effects on macroscopic scales such as Bose-Einstein condensation. This phenomenon, first predicted 100 years ago by Bose [19] and Einstein [44], occurs when a dilute gas of bosons is cooled down to very low temperatures. Below a certain critical temperature a condensate forms, meaning that a macroscopic fraction of the particles is in the same one-particle state, and thus, quantum phenomena become relevant on length scales much larger than the individual atom. The occurrence of Bose-Einstein condensation is also related to superfluidity as suggested in [88]. Bose-Einstein condensation has been achieved experimentally independently by the groups of Cornell and Wieman [3] and Ketterle [40] earning them a Nobel prize in 2001. While Bose and Einstein analyzed theoretically the non-interacting gas, an effective theory describing weakly interacting Bose gases was proposed by Bogoliubov in 1947 with the goal of explaining the occurrence of superfluidity in liquid helium [18]. Using assumptions such as that condensation indeed occurs, his theory yields predictions for the low-energy states of the system. Seen from the mathematics perspective, describing the Bose gas is an interesting challenge due to the strong correlations between particles introduced by the interaction making this case much harder to study than the non-interacting one. In particular, a proof that Bose-Einstein condensation occurs in the thermodynamic limit for continuous systems is still out of reach. However, there has been a lot of work regarding the rigorous analysis of bosonic systems in various relevant regimes. This thesis contains three new contributions establishing rigorous results on bosonic systems in different, physically relevant settings. It is based on the articles [37, 27, 31]. The first result concerns the time-evolution of initially trapped Bose gases in the ultra-dilute Gross-Pitaevskii regime. Under some assumptions on the initial data satisfied in particular by a class of states that are expected to approximate the ground state in a trap, we derive a quasi-free approximation that is close to the true solution in L2-norm. In particular, we show that this approximate wave function is described in terms of a Bose-Einstein condensate in a one-particle state following the time-dependent Gross-Pitaevskii equation together with pair excitations with respect to this condensate. As a corollary, we establish a central limit theorem for bounded one-particle observables. The second result is a simple proof of Bose-Einstein condensation for interacting, ultra-dilute gases. More specifically, we consider scalings interpolating between the Gross-Pitaevskii regime and the thermodynamic limit. While condensation had been shown to hold in these regimes before, the advantage of the method used here is its simplicity. Lastly, the third result establishes an upper bound on the ground state energy of a dilute Bose gas in the thermodynamic limit. It captures rigorously the third order correction to the energy per unit volume for sufficiently low densities as predicted in the physics literature first by Wu and subsequently others. Full-text embargoed until: 2026-01-01