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.
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.
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
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.
Classical statistical mechanics tries to derive the macroscopic properties of matter starting from the mechanical laws that rule the behaviour of single particles. To describe equilibrium states, only observables accounting for correlations among states are considered, and only systems consisting of a large number of particles are taken into account. The observables are described by continuous functions on phase space, and the states are represented by linear assignments of a number to each observable. Within the framework of the canonical ensemble, one deals with mechanical systems in thermal equilibrium with a thermal reservoir, and the equilibrium state is reached as a result of the interaction with the external world. The external world may be really external, or equally well, the unrecognized internal degrees of freedom, like in the calculation of viscosity or thermal conductivity. If really only the external world is the cause we may expect some surface dependence while an internal unrecognized degree of freedom would have a volume effect, unless the interactions are long range. The microcanonical ensemble is instead introduced to study isolated mechanical systems, and the equilibrium is viewed as a temporal average, rather than as a limit. Attention is then focused on partition functions, the theorem of equipartition of energy and an elementary theory of specific heats. In the second part, the Planck derivation of the law of black-body radiation is analysed, presenting in chronological order the Kirchhoff theorem, the Stefan law, the Wien displacement law, the Rayleigh–Jeans formula and the Planck hypothesis. Further topics discussed are the Einstein and Debye quantum models for specific heats of solids. These topics prepare the ground for the introduction of quantum statistical mechanics. The third part is, in fact, devoted to the analysis of identical particles in quantum mechanics.
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.
The topic of Bose–Einstein condensation in a uniform, non-interacting gas of bosons is treated in most textbooks on statistical mechanics. In the present chapter we discuss the properties of a non-interacting Bose gas in a trap. We shall calculate equilibrium properties of systems in a semi-classical approximation, in which the energy spectrum is treated as a continuum. For this approach to be valid the temperature must be large compared with Δ∊/ k , where Δ∊ denotes the separation between neighbouring energy levels. As is well known, at temperatures below the Bose–Einstein condensation temperature, the lowest-energy state is not properly accounted for if one simply replaces sums by integrals, and it must be included explicitly. The statistical distribution function is discussed in Sec. 2.1, as is the single-particle density of states, which is a key ingredient in the calculations of thermodynamic properties. Calculations of the transition temperature and the fraction of particles in the condensate are described in Sec. 2.2. In Sec. 2.3 the semi-classical distribution function is introduced. From this we obtain the density profile and the velocity distribution of particles, and use these to determine the shape of an anisotropic cloud after free expansion. Thermodynamic properties of Bose gases are calculated as functions of the temperature in Sec. 2.4. Finally, Sec. 2.5 treats corrections to the transition temperature due to a finite particle number.
The topic of Bose–Einstein condensation in a uniform, non-interacting gas of bosons is treated in most textbooks on statistical mechanics. In the present chapter we discuss the properties of a non-interacting Bose gas in a trap. We shall calculate equilibrium properties of systems in a semi-classical approximation, in which the energy spectrum is treated as a continuum. For this approach to be valid the temperature must be large compared with Δ∈/ k , where Δ∈ denotes the separation between neighbouring energy levels. As is well known, at temperatures below the Bose–Einstein condensation temperature, the lowest energy state is not properly accounted for if one simply replaces sums by integrals, and it must be included explicitly. The statistical distribution function is discussed in Sec. 2.1, as is the single-particle density of states, which is a key ingredient in the calculations of thermo dynamic properties. Calculations of the transition temperature and the fraction of particles in the condensate are described in Sec. 2.2. In Sec. 2.3 the semi-classical distribution function is introduced, and from this we determine the density profile and the velocity distribution of particles. Thermodynamic properties of Bose gases are calculated as functions of the temperature in Sec. 2.4. The final two sections are devoted to effects not captured by the simplest version of the semi-classical approximation: corrections to the transition temperature due to a finite particle number (Sec. 2.5), and thermodynamic properties of gases in lower dimensions (Sec. 2.6).