Statistical theorem

In this section the statistical theorems or mathematical tools needed to understand the Boltzmann equation in itself, and the mathematical operations performed developing the macroscopic conservation equations starting out from the microscopic Boltzmann equation, are presented. 

The relation between the spin and statistics of particles is given by the spin-statistics theorem. 

Chapter 10 gives two further quantum-mechanical postulates that deal with spin and the spin-statistics theorem. 

Chapter 10 Electron Spin and the Spin-Statistics Theorem... 

Many proofs of varying validity have been offered for the spin-statistics theorem see I. Duck and E. C. G. Sudurshan, Pauli and the Spin-Statistics Theorem, World... 

Scientific, 1997 Am. J. Phys., 66, 284 (1998) Sudurshan and Duck, Pramana-J. Phys., 61, 645 (2003) (available at www.ias.ac.in/pramana/v61/p645/fulltext.pdf). Several experiments have confirmed the validity of the spin-statistics theorem to extremely high accuracy see G. M. Tino, Fortschr. Phys., 48, 537 (2000) (available at arxiv.org/ abs/quant-ph/9907028). 

The spin-statistics theorem has an important consequence for a system of identical fermions. The antisymmetry requiranent means that... 

If Ej[t) is the amplitude of the scattered field at some time, t, then ,(t + T) is the amplitude of the mattered field at some time t+r later. If one is interested in the frequency spectrum in the range 10 to 10 Hertz then one needs to obtain correlations in the amplitude of the scattered wave fi om approximately 10" to 10" seconds. As we will see later, this can be done experimentally, but now we will show that the autocorrelation function has a simple mathematical function for simple, monodisperse polymer solutions. The use of the Wiener-Kinchine theorem is of prime importance in the theory and readers who are not familiar with this statistical theorem will find a simple explanation and derivation in almost any statistical mechanics text. 

In this section the statistical theorems or mathematical tools needed to understand the Boltzmann equation in itself, and the mathematical operations performed developing the macroscopic conservation equations starting out from the microscopic Boltzmann equation, are presented. Introductory it is stressed that a heuristic theory, which resembles the work of Boltzmann [10] and the standard kinetic theory literature, is adopted in this section and the subsequent sections deriving the Boltzmann equation. Irrespective, the notation and concepts presented in Sect. 2.2 are often referred, or even redefined in a less formal wrapping, thus the underlying elements of classical mechanics are prescience of outmost importance understanding the true principles of kinetic theory. 

In this section the elementary definitions and results deduced from the H-theorem are given [10]. In summary, the statistical //-theorem of kinetic theory relates to the Maxwellian velocity distribution function and thermodynamics. Most important, the Boltzmann s //-theorem provides a mechanistic or probabilistic prove for the second law of thermodynamics. In this manner, the //-theorem also relates the thermodynamic entropy quantity to probability concepts. Further details can be found in the standard references [20, 39, 55, 68, 90, 117, 150, 157]. In practice, during the process of developing novel models for the collision term, the //-theorem merely serves as a requirement for the constitutive relations in order to fulfill the second law of thermodynamics (in a similar manner as for the continuum models). 

Particles in nature posses a property called spin. According to Pauli s famous Spin-Statistics-Theorem all particles possessing integer spin values (e.g. photons whose spin is one) are Bosons, whereas particles possessing half-integer spin values (e.g. the electron has spin one halO are Fermions. ... 

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