ABS>An attempt is made to correlate the classical Boltzmann, the Bose- Einstein, and the Fermi statistics. This is made pessible by modifying the presentation of the firstmentioned statistics. At the same time, a satisfactory reply is given to the criticism by Loschmidt and others of the inconsistency between the classical Boltzmann theory and the general laws of mechanics. It is pointed out that the customary usage of the term particle for something without identity for the image points in statistical thermodynamics is very misleading. It is suggested that such particles, i.e., image pcints, should be considered as singularities in a field. (auth)
ABS>An attempt is made to correlate the classical Boltzmann, the Bose- Einstein, and the Fermi statistics. This is made pessible by modifying the presentation of the firstmentioned statistics. At the same time, a satisfactory reply is given to the criticism by Loschmidt and others of the inconsistency between the classical Boltzmann theory and the general laws of mechanics. It is pointed out that the customary usage of the term particle for something without identity for the image points in statistical thermodynamics is very misleading. It is suggested that such particles, i.e., image pcints, should be considered as singularities in a field. (auth)
Thermodynamics is used to describe the bulk properties of matter in or near equilibrium. Many scientists, notably Boyle, Carnot, Clausius, Gay-Lussac, Gibbs, Joule, Kelvin, and Rumford, contributed to the development of the field over three centuries. Quantities such as mass, pressure, energy, and so on are readily defined and measured. Classical statistical mechanics attempts to understand thermodynamics by the application of classical mechanics to the microscopic particles making up the system. Great progress in this field was made by physicists like Boltzmann and Maxwell. Temperature, entropy, particle number, and chemical potential are thus understandable in terms of the microscopic nature of matter. Classical mechanics is inadequate in many circumstances however, and ultimately must be replaced by quantum mechanics. In fact, the ultraviolet catastrophe encountered by the application of classical mechanics and electromagnetism to blackbody radiation was one of the problems that led to the development of quantum theory. The development of quantum statistical mechanics was achieved by a number of twentieth century physicists, most notably Planck, Einstein, Fermi, and Bose. The purpose of this chapter is to give a mini-review of the basic concepts of quantum statistical mechanics as applied to noninteracting systems of particles. This will set the stage for the functional integral representation of the partition function, which is a cornerstone of modern relativistic quantum field theory and the quantum statistical mechanics of interacting particles and fields. Ensembles One normally encounters three types of ensemble in equilibrium statistical mechanics. The microcanonical ensemble is used to describe an isolated system that has a fixed energy E , a fixed particle number N , and a fixed volume V .
Review of quantum statistical mechanicsThermodynamics is used to describe the bulk properties of matter in or near equilibrium.Many scientists, notably Boyle, Carnot, Clausius, Gay-Lussac, Gibbs, Joule, Kelvin, and Rumford, contributed to the development of the field over three centuries.Quantities such as mass, pressure, energy, and so on are readily defined and measured.Classical statistical mechanics attempts to understand thermodynamics by the application of classical mechanics to the microscopic particles making up the system.Great progress in this field was made by physicists like Boltzmann and Maxwell.Temperature, entropy, particle number, and chemical potential are thus understandable in terms of the microscopic nature of matter.Classical mechanics is inadequate in many circumstances however, and ultimately must be replaced by quantum mechanics.In fact, the ultraviolet catastrophe encountered by the application of classical mechanics and electromagnetism to blackbody radiation was one of the problems that led to the development of quantum theory.The development of quantum statistical mechanics was achieved by a number of twentieth century physicists, most notably Planck, Einstein, Fermi, and Bose.The purpose of this chapter is to give a mini-review of the basic concepts of quantum statistical mechanics as applied to noninteracting systems of particles.This will set the stage for the functional integral representation of the partition function, which is a cornerstone of modern relativistic quantum field theory and the quantum statistical mechanics of interacting particles and fields. EnsemblesOne normally encounters three types of ensemble in equilibrium statistical mechanics.The microcanonical ensemble is used to describe an isolated system that has a fixed energy E, a fixed particle number N , and a fixed 1
In the previous chapters, the focus was on systems of particles following the law of classical mechanics. Of course, this was mainly done within the framework of classical mechanics but a quantum description of individual (discretised) states was also used, while assuming distinguishability of the particles (for instance, particles located at the nodes of a lattice (spin system, quantum harmonic oscillators,) ). The present chapter as well as Chapters 7 and 8 are dedicated to the study of statistical physics in a fully quantum framework. The first obstacle encountered by a quantum approach is the impossibility to define a quantum phase space (in which the ergodic hypothesis would hold for instance). This problem is inherent to quantum mechanics: because of Heisenberg’s uncertainty principle, one cannot define both the momentum and the position of a particle. Besides, statistical physics considers systems made of a large number of particles, a priori in mutual interaction. In classical physics, model situations for which the interactions between the constituents of the system are neglected can be studied. In quantum mechanics, the indistinguishability of identical particles makes the study of the problem in the absence of interaction already difficult: even for non-interacting particles, the symmetrisation (or antisymmetrisation) of the wave function of https://www.w3.org/1998/Math/MathML" display="inline"> N https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003272427/5ca4d37a-9786-43a3-8b38-8fa9eb3d1a19/content/math_25.tif "/> identical particles induces correlations which have non-trivial thermodynamical effects such as the existence of a “Fermi sea” for fermions (see Section 8.1 ) or the Bose–Einstein condensation phenomenon for bosons (see Section 7.4 ). The present chapter addresses these two points consecutively.
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
Abstract It is well known that a new form of statistical mechanics has been recently developed by Einstein for an ideal gas of structureless mass-points. This starts from a discussion by Bose of the laws of temperature radiation based on the light quantum hypothesis, and has been further analysed by Schrödinger. Yet another new form has been proposed independently by Fermi and Dirac. The latter based his theory on a discussion of lightly coupled systems with the help of Schrödinger’s equation. Combined with Heisenberg’s work on the many-body problem, Dirac’s work forces us to conclude at least that the classical form of statistical mechanics must be changed. It indicates that the true form, which satisfies the laws of the new mechanics, is almost certainly that of Fermi and Dirac, which is the natural generalization of Pauli’s principle of exclusion for electronic orbits in an atom. The work of Heisenberg and Dirac already quoted has shown that Pauli’s principle and its extension are satisfied in the new mechanics by a complete self-consistent solution of the equations of motion. So far as I am aware, the discussions of the new forms have as yet dealt only with the statistics of a gas of structureless mass-points (and, of course, temperature radiation). There has, moreover, been as yet no attempt to define the entropy (and the absolute temperature) in strict analogy with rational thermodynamics by means of the equation dQ = TdS. If another definition is preferred, then this equation must be deduced from it. It has, therefore, seemed worth while to reopen the discussion by examining ab initio a quite general form of statistical mechanics of which the classical form and instein’s and Fermi-Dirac’s are special cases. This is rendered possible by using the powerful method of complex integration already applied to the classical form. The sequence of the argument is then to take the general form, which covers a very large range of ways of assigning possibilities and counting complexions, and construct on that basis exact integral expressions for the number of complexions possible to the assembly and for the average number of systems of the assembly in their various quantum states. We then derive from these the average energies and external reactions and so the form of dQ, deduce from dQ the existence of S and so define S and T. This can be done in the general form for assemblies of ideal systems just as general as can be handled in the classical way—ideal gases of molecules of any structure and crystals and radiation. Such assemblies are in all cases thermodynamical systems.