scholarly journals The Bose-Einstein statistics: Remarks on Debye, Natanson, and Ehrenfest contributions and the emergence of indistinguishability principle for quantum particles

2020 ◽  
Vol 19 ◽  
pp. 423-441
Author(s):  
Józef Spałek
Keyword(s):  
Black Body ◽  
Black Body Radiation ◽  
Quantum Particles ◽  
Particle Wave Function ◽  
Planck’S Law ◽  
The Many ◽  
Bose Einstein ◽  
Grand Canonical ◽  
Fundamental Physical

The principal mathematical idea behind the statistical properties of black-body radiation (photons) was introduced already by L. Boltzmann (1877/2015) and used by M. Planck (1900; 1906) to derive the frequency distribution of radiation (Planck’s law) when its discrete (quantum) structure was additionally added to the reasoning. The fundamental physical idea – the principle of indistinguishability of the quanta (photons) – had been somewhat hidden behind the formalism and evolved slowly. Here the role of P. Debye (1910), H. Kamerlingh Onnes and P. Ehrenfest (1914) is briefly elaborated and the crucial role of W. Natanson (1911a; 1911b; 1913) is emphasized. The reintroduction of this Natanson’s statistics by S. N. Bose (1924/2009) for light quanta (called photons since the late 1920s), and its subsequent generalization to material particles by A. Einstein (1924; 1925) is regarded as the most direct and transparent, but involves the concept of grand canonical ensemble of J. W. Gibbs (1902/1981), which in a way obscures the indistinguishability of the particles involved. It was ingeniously reintroduced by P. A. M. Dirac (1926) via postulating (imposing) the transposition symmetry onto the many-particle wave function. The above statements are discussed in this paper, including the recent idea of the author (Spałek 2020) of transformation (transmutation) – under specific conditions – of the indistinguishable particles into the corresponding to them distinguishable quantum particles. The last remark may serve as a form of the author’s post scriptum to the indistinguishability principle.

scholarly journals Bose-Einstein statistics and degeneracy

1941 ◽  
Vol 178 (973) ◽  
pp. 135-152 ◽  
Keyword(s):  
Thermodynamic Properties ◽  
Black Body ◽  
Black Body Radiation ◽  
Degenerate Case ◽  
Bose Einstein

The thermodynamic properties of degenerate and non-degenerate Bose-Einstein gas in the completely non-relativistic and the completely relativistic cases are derived. The relativistic degenerate case of Bose-Einstein statistics corresponds to black-body radiation. The properties and the possibility of the existence of non-degenerate radiation are discussed.

Are smaller microplastics missing?

2020 ◽  
Author(s):  
Kunihiro Aoki ◽  
Ryo Furue
Keyword(s):  
Size Distribution ◽  
Gradual Increase ◽  
Black Body ◽  
Boltzmann Distribution ◽  
Black Body Radiation ◽  
Rapid Decrease ◽  
Size Distributions ◽  
Biological Uptake ◽  
Data Source ◽  
Planck’S Law

Abstract The size distribution of marine microplastics (< 5 mm) provides a fundamental data source for understanding the dispersal, break down, and biotic impacts of the microplastics in the ocean. The observed size distribution generally shows, from large to small sizes, a gradual increase followed by a rapid decrease. This decrease has led to the hypothesis that the smallest fragments are selectively removed by sinking or biological uptake. Here we propose a new model of size distribution without any removal of material from the system. The model uses an analogy with black-body radiation and the resultant size distribution is analogous to Planck's law. In this model, the original large plastic piece is broken into smaller pieces once by the application of “energy” or work by waves or other processes, under two assumptions, one that fragmentation into smaller pieces requires larger energy and the other that the probability distribution of the “energy” follows the Boltzmann distribution. Our formula well reproduces observed size distributions over wide size ranges from micro- (< 5 mm) to mesoplastics ( > 5 mm). According to this model, the smallest fragments are fewer because large “energy” required to produce such small fragments occurs more rarely.

scholarly journals Novelty Surface Coatings for Far Infrared Spectrum Black Body Radiators

2021 ◽  
Vol 25 (4) ◽  
pp. 77-82
Author(s):  
Andrzej Ligienza ◽  
Grzegorz Bieszczad ◽  
Tomasz Sosnowski ◽  
Bartosz Bartosewicz ◽  
Krzysztof Firmanty
Keyword(s):  
Thermal Imaging ◽  
Far Infrared ◽  
Black Body ◽  
Black Body Radiation ◽  
Surface Coatings ◽  
Reference Surface ◽  
Emitter Surface ◽  
Radiation Sources ◽  
Physical Approximation ◽  
Planck’S Law

Black body radiation sources are commonly used devices in areas related to thermal imaging and radiometry. They are the closest physical approximation of theoretical black body emitter derived from the Planck’s law. Majority of such devices are costly with restricted information about their production technology, including their emitter surface. A few relatively easily accessible coatings with potential application in such devices have been chosen and their emissivity measured. The paper presents measurements that provides information necessary to determine whether there are coatings viable for black body emitter or reference surface.

Thermodynamics

2017 ◽  
Author(s):  
Nicholas Manton ◽  
Nicholas Mee
Keyword(s):  
Chemical Potential ◽  
Black Body ◽  
Black Body Radiation ◽  
Spin Systems ◽  
Atomic Scale ◽  
Einstein Condensation ◽  
Maxwell Distribution ◽  
Bose Einstein Condensation ◽  
Thermodynamic Variables ◽  
Bose Einstein

This chapter is about thermodynamics, or statistical mechanics, which explains macroscopic features of the world in terms of the motion of vast numbers of particles on the atomic scale. It discusses how macroscopic variables such as temperature and entropy were originally introduced, before presenting modern definitions of temperatures and entropy and the Laws of Thermodynamics. Alternative thermodynamic variables, including enthalpy and the Gibbs free energy, are defined and the Gibbs distribution is explained. The Maxwell distribution is derived. The chemical potential is introduced and the pressure and heat capacity of an electron gas is calculated, The Fermi–Dirac and Bose–Einstein functions are derived. Bose–Einstein condensation is explained. Black body radiation is discussed and the Planck formula is derived. Lasers are explained. Spin systems are used to model magnetization. Phase transitions are briefly discussed. Hawking radiation and the thermodynamics of black holes is explained.

An Introduction to Statistical Mechanics and Thermodynamics

2019 ◽  
Author(s):  
Robert H. Swendsen
Keyword(s):  
Statistical Mechanics ◽  
Quantum Statistical Mechanics ◽  
Black Body ◽  
Black Body Radiation ◽  
Ideal Gas ◽  
Band Theory ◽  
Multiple Systems ◽  
Thermodynamic Potentials ◽  
Legendre Transforms ◽  
Bose Einstein

This is a textbook on statistical mechanics and thermodynamics. It begins with the molecular nature of matter and the fact that we want to describe systems containing many (1020) particles. The first part of the book derives the entropy of the classical ideal gas using only classical statistical mechanics and Boltzmann’s analysis of multiple systems. The properties of this entropy are then expressed as postulates of thermodynamics in the second part of the book. From these postulates, the structure of thermodynamics is developed. Special features are systematic methods for deriving thermodynamic identities using Jacobians, the use of Legendre transforms as a basis for thermodynamic potentials, the introduction of Massieu functions to investigate negative temperatures, and an analysis of the consequences of the Nernst postulate. The third part of the book introduces the canonical and grand canonical ensembles, which are shown to facilitate calculations for many models. An explanation of irreversible phenomena that is consistent with time-reversal invariance in a closed system is presented. The fourth part of the book is devoted to quantum statistical mechanics, including black-body radiation, the harmonic solid, Bose–Einstein and Fermi–Dirac statistics, and an introduction to band theory, including metals, insulators, and semiconductors. The final chapter gives a brief introduction to the theory of phase transitions. Throughout the book, there is a strong emphasis on computational methods to make abstract concepts more concrete.

Sign in / Sign up

Export Citation Format

Share Document