CRC Press eBooks

The Quantum Distribution Functions

Michael J. R. Hoch

Published Mar 30, 2021• 0 citations

Abstract

In nature, two different classes of particles are found, called fermi particles, or fermions, and bose particles, or bosons, respectively. The MB statistics is therefore an approximation to the true particle statistics and is applicable only in situations when most states are empty with some states occupied by a single particle and very few states, in the case of bosons, occupied by more than one particle. At high temperatures, the chemical potential decreases and the distribution changes its form dramatically. In discussing the quantum distribution functions, the authors have for convenience chosen the lowest state to have energy zero. The quantum distribution functions derived in this chapter are extremely useful in many areas of physics whenever what are called degenerate Fermi or Bose fluids are considered.

DOI: 10.1201/9781003028604-19

Related papers

Quantum ideal gases

B. Widom

Bose–Einstein and Fermi–Dirac statistics versus Boltzmann statistics It was remarked in Chapter 2, §2.1, that there are circumstances in which the “Boltzmann statistics” for an ideal gas – the approximation in Eq. (2.6), in which the total partition function is decomposed into a product of single-molecule partition functions, with division by N 1 ! N 2 ! … to correct for particle identity – is inaccurate even if the gas is still ideal (non-interacting particles). That happens when the number of single-particle states within the energy kT of the ground state is no longer very much greater than the number of particles. (Recall Fig. 2.1 and the accompanying discussion.) We noted that when the Boltzmann statistics is no longer an adequate approximation one must distinguish the particles as fermions, no two of which can be in the same single-particle state (counting states of different spin as different), or as bosons, for which there is no restriction on multiple occupancy of the singleparticle states. Fermions are particles of half-odd-integer spin quantum number; they could be composite particles consisting of an odd number of spin-½ particles. Protons, neutrons, and electrons, all of which are of spin ½, are fermions. Bosons are particles of integer spin; they could be spinless, or of spin 1, etc., or could be composites consisting of an even number of fermions. The nuclei of helium atoms of mass number 4 consist of two protons and two neutrons, and the neutral atom has two electrons outside the nucleus; 4 He atoms are bosons.

Cambridge University Press eBooksApr 18, 20020 citations

Fermi and Bose particles

Roger M. Bowley, Mariana Sánchez

This chapter reviews the techniques of the grand canonical ensemble to give an exact treatment for both Bose and Fermi particles. It looks at the results of Maxwell-Boltzmann statistics for the average number of particles in sufficiently high temperatures and in a single-particle state for both fermions and bosons. It also examines the exclusion principle in Fermi particles, which means that there cannot be two particles in the same single-particle state, while there is no such restriction for Bose particles. The chapter clarifies that the resulting formulae for the average number of particles in a single-particle state are different at low temperatures. It describes Fermi particles at low temperatures that have a large kinetic energy, an energy which can be much larger than the interaction between particles.

Oxford University Press eBooksOct 21, 19990 citations

Quantum Statistics

Andrew F. Rex, C.B.P. Finn

This chapter aims to develop the Boltzmann factor by considering a system attached to a reservoir, assuming that the system and reservoir were fixed in size but could exchange energy. When dealing with quantum statistics, it will be convenient to consider systems in which both particles and energy can be exchanged with the reservoir. An alternate derivation of the Stefan–Boltzmann law involves integrating the spectral energy density or Planck law over all wavelengths. The chapter presents a theoretical model for estimating the maximum temperature for a particular Bose–Einstein condensate (BEC). A BEC can exhibit unusual properties, such as quantized vortices. The vortices have been used to model the behavior of black holes. Another interesting effect observed is interference, attributed to the system’s wave-particle duality. For classical systems, there is no need to consider whether particles are fermions or bosons, because their wave functions are considered non-overlapping. Therefore, classical particles are described by a type of statistics, called Maxwell–Boltzmann (MB) statistics.

CRC Press eBooksFeb 22, 20240 citations

Bose–Einstein Statistics

Robert H. Swendsen

The properties of the ideal Bose gas are calculated from the integral equations for the energy and the number of particles as a function of the temperature and chemical potential. It is shown that the integral equations break down below the Einstein temperature that corresponds to the transition to the low-temperature state. The lowest single-particle energy level must be treated explicitly to get the proper equations. With the inclusion of the lowest single-particle energy level, the low-temperature behavior is calculated. The occupation of the lowest level becomes comparable to the total number of particles in the system below the Einstein temperature, and equal to the total number of particles at zero temperature. A numerical solution to the properties of the Bose gas is discussed, and the detailed calculations are assigned to the problems at the end of the chapter.

Oxford University Press eBooksDec 10, 20191 citation

Quantum ideal gases: Fermi-Dirac and Bose-Einstein statistics

Mark E. Tuckerman

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

Aug 1, 20230 citations

The Spin-Statistics Theorem and Identical Particle Distribution Functions

Dwight E. Neuenschwander

In the restaurants of some countries, the maître d ’ will seat your party at a table already occupied by strangers if the table has sufficient empty chairs. In other cultures, new diners expect to be seated at empty tables even when occupied tables are surrounded by sur-plus seats. Elementary particles, and composites made of them, fall analogously into two categories: bosons and fermions. Bosons readily share the same state, analogous to the first group of diners. Fermions prefer solitude, like the second group of diners. This article offers a simple argument for their respective distribution functions that describe how a system of identical particles populates the states available to them. Elegant Connections in Physics Since the\tmost\tstable\tconfiguration\tof a system\t typically\t finds\t it\t in\t the\t state\t of lowest energy,\t why\tdon’t\tall\tthe\telectrons in an\tatom\treside\tin\tthe\t1s orbital?\t If\tthey did,\t there\twould\tbe\tno\tchemistry,\t and\tthus no biology\tand\tno\tlife—nor\twould\tmetals

Jan 1, 20130 citations