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Quantum statistical mechanics

James P. Sethna

Published Jan 26, 2021• 0 citations
Publisher

Abstract

Abstract Quantum statistical mechanics governs metals, semiconductors, and neutron stars. Statistical mechanics spawned Planck’s invention of the quantum, and explains Bose condensation, superfluids, and superconductors. This chapter briefly describes these systems using mixed states, or more formally density matrices, and introducing the properties of bosons and fermions. We discuss in unusual detail how useful descriptions of metals and superfluids can be derived by ignoring the seemingly important interactions between their constituent electrons and atoms. Exercises explore how gregarious bosons lead to superfluids and lasers, how unsociable fermions explain transitions between white dwarfs, neutron stars, and black holes, how one calculates materials properties in semiconductors, insulators, and metals, and how statistical mechanics can explain the collapse of the quantum wavefunction during measurement.

DOI
10.1093/oso/9780198865247.003.0007

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Abstract Quantum statistical mechanics governs metals, semiconductors, and neutron stars. Statistical mechanics spawned Planck’s invention of the quantum, and explains Bose condensation, superfluids, and superconductors. This chapter briefly describes these systems using mixed states, or more formally density matrices, and introducing the properties of bosons and fermions. We discuss in unusual detail how useful descriptions of metals and superfluids can be derived by ignoring the seemingly important interactions between their constituent electrons and atoms. Exercises explore how gregarious bosons lead to superfluids and lasers, how unsociable fermions explain transitions between white dwarfs, neutron stars, and black holes, how one calculates materials properties in semiconductors, insulators, and metals, and how statistical mechanics can explain the collapse of the quantum wavefunction during measurement.
Overview Classical Statistical Mechanics (CSM) ignores the quantum nature of particles. This introduces three errors in the calculation of the partition function. First is the assumption of distinguishability of individual particles not possible in the microscopic world. Second, CSM neglects the discrete nature of system’s energy levels. Third, CSM fails to account for the spin of the particles which divides all particles into two types: boson and fermion, each following its own law of statistical distribution among energy levels. While CSM is impressively successful in describing observed properties of many naturally occurring large scale phenomena (particularly in liquids and biological systems), it fails to describe many phenomena where quantum nature of particles and quantum statistics play important role in determining properties such as electrical and thermal conductivity (particularly in solid materials) and superfluidity in liquid helium. It is not just at low temperatures where quantum nature is important, but also when the particle is intrinsically quantum, such as an electron. Quantum Statistical Mechanics (QSM) is also needed to understand biochemical reactions, like enzyme kinetics and optical properties of DNA. In this chapter, we discuss elementary aspects of quantum Statistical Mechanics, with detailed treatment of ideal Bose gas, leading to Bose-Einstein condensation.
Abstract This chapter begins by extending the Boltzmann distribution to the case of a system that exchanges particles with its environment. This idea finds direct applications ranging from hemoglobin adsorption to ionization of atoms in stars. But the main applications are to dense “gases” in which the quantum behavior of identical particles comes into play. Examples include conduction electrons in metals and semiconductors; white dwarf and neutron stars; photon gases and thermal radiation from incandescent objects; neutrino and electron-positron production in the early universe; quasiparticles associated with vibrations and magnetic excitations in solids; and Bose-Einstein condensation of ultracold clouds of atoms.
Under conditions of high temperature and low density, we shall ˆnd that it does not matter whether classical or quantum mechanical descriptions are used for a system of particles. At high densities and low temperatures this is no longer true, because of the overlap of the particles’ wave functions and quantum mechanics must be used, giving rise to quantum statistics. Under high-density, low-temperature conditions, the properties of a system depend in a crucial way on whether the particles that make up the system are fermions or bosons. Many fascinating phenomena occur in condensed matter as the temperature is lowered. Examples are ferromagnetism, superconductivity, and super uidity. ese important new properties appear fairly abruptly at phase transitions. Progress in the microscopic understanding and description of the behavior of these systems involves quantum mechanics and statistical physics ideas. Applications of quantum statistics are not conˆned to terrestrial systems and include astrophysical phenomena such as the microwave background radiation from the Big Bang and the mass-radius relationships for white dwarf stars and neutron stars. An important concept in thermal physics is that of entropy, which, as we shall see, increases with time as systems become more disordered. e increase of the entropy of the universe with the passage of time provides what is termed time’s arrow. An interesting and unanswered question arises as to why the entropy of the universe was so low at the beginning of time.
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 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
The concepts of statistical physics have been used to study fermionic systems rapidly after the birth of the “new" quamtum mechanics in 1925. This proved to be extremely fruitful and led to major advances in atomic and solid state physics. Fermi-Dirac statistics was introduced in 1926 by Enrico Fermi and Paul Dirac. It was used by Ralph Fowler in 1926 to describe the collapsed state of a star (a so-called white dwarf ) and by Arnold Sommerfeld in 1928 to describe conduction electrons in solids. This chapter first presents some general results and then discusses a choice of specific systems.