This chapter discusses the main aspects of the Boltzmann distribution which played an important role for the discovery of Planck’s law of the spectral radiance of the thermal black-body experiment and Einstein’s equipartition as a derivation of Planck’s law. This chapter introduces the concepts of microstate, microstate, statistical weight of the distribution in order to obtain the Boltzmann distribution of distinguished particles. In the next section, it is discussed the statistics of undistinguishable particles of fermions (electrons, protons, neutrons) known as Fermi-Dirac statistics which began with Fermi’s approach of monoatomic molecules of an ideal gas where he applied the concept of quantum numbers associated with each molecule and Pauli’s exclusion principle. Afterwards, it is shown how Dirac amplified Fermi statistical distribution to many-particle systems. Dirac also used Pauli’s exclusion principle in order to arrive at the concept of antisymmetrical eigenfunctions and their determinantal form which later became known as Slater’s determinant. (151 words).
This chapter discusses the main aspects of the Boltzmann distribution which played an important role for the discovery of Planck’s law of the spectral radiance of the thermal black-body experiment and Einstein’s equipartition as a derivation of Planck’s law. This chapter introduces the concepts of microstate, microstate, statistical weight of the distribution in order to obtain the Boltzmann distribution of distinguished particles. In the next section, it is discussed the statistics of undistinguishable particles of fermions (electrons, protons, neutrons) known as Fermi-Dirac statistics which began with Fermi’s approach of monoatomic molecules of an ideal gas where he applied the concept of quantum numbers associated with each molecule and Pauli’s exclusion principle. Afterwards, it is shown how Dirac amplified Fermi statistical distribution to many-particle systems. Dirac also used Pauli’s exclusion principle in order to arrive at the concept of antisymmetrical eigenfunctions and their determinantal form which later became known as Slater’s determinant. (151 words).
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.
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 Planck’s original introduction of the quantum idea was based on a statistical mechanical treatment of black body radiation. S. Bose, a physicist working in India, found a new way of deriving Planck’s law. Einstein applied Bose’s method to a gas of massive particles. This led to the prediction of a new physical effect (Bose-Einstein condensation). With the introduction of the exclusion principle by Pauli, Fermi, and Dirac independently introduced a new kind of statistics. Considerations by Einstein and Schrödinger introduced a preliminary form of second quantization.
In a previous work [Phys. Rev. E 48, 4263 (1993)] we have derived a nonlinear one-dimensional kinetic equation for the distribution function of particles obeying an exclusion principle. In the present work, on the same grounds, we extend this kinetics to D-dimensional continuous or discrete space, in order to study the distribution function of particles obeying a generalized exclusion-inclusion Pauli principle (EIP). This exclusion or inclusion principle is introduced into the classical transition rates by means of an inhibition or an enhancement factor, which contains a parameter \ensuremath{\kappa}, whose values range between -1 and +1 and can balance the effect of the full or partial validity of EIP. After deriving the kinetic equation we obtain a general expression of the stationary distribution function, depending on the value we give to the parameter \ensuremath{\kappa}. When we limit ourselves to Brownian particles, we derive exactly for \ensuremath{\kappa}=-1 the Fermi-Dirac (FD) distribution, for \ensuremath{\kappa}=0 the Maxwell-Boltzmann distribution, and for \ensuremath{\kappa}=1 the Bose-Einstein (BE) distribution. When \ensuremath{\kappa} assumes an intermediate value, except zero, between the extreme values -1 and +1, we obtain statistical distributions different from the FD and BE ones. We attribute to the parameter \ensuremath{\kappa} the meaning of the degree of indistinguishability of identical particles, the degree of antisymmetrization, or the symmetrization of the wave function of the particle system.
By exploiting Einstein’s rate-equation approach [A. Einstein, Phys. Z. 18, 121 (1917)] to Planck’s law of blackbody radiation [M. Planck, Ann. Phys. 309, 553 (1901)], we obtain a simple relation between the population densities of the two energy levels of the atomic oscillators in the walls of the black body, as assumed by Einstein in his paper from 1917, and the occupation numbers in a photonic excited and ground state. This relation establishes a quantum principle for photons and, more generally, all bosons, which has the same physical relevance as Pauli’s exclusion principle [W. Pauli, Z. Phys. 31, 765 (1925)], which is the quantum principle for fermions. We demonstrate the equivalence of these two quantum principles by inserting either of them into the Boltzmann distribution, thereby transforming the Boltzmann distribution into either the Fermi-Dirac [E. Fermi, Rendiconti Lincei 3, 145 (1926)] and Bose-Einstein [Bose, Z. Phys. 26, 178 (1924)] distribution.