A Note on Effects of Generalized and Extended Uncertainty Principles on Jüttner Gas

In recent years, the implications of the generalized (GUP) and extended (EUP) uncertainty principles on Maxwell–Boltzmann distribution have been widely investigated. However, at high energy regimes, the validity of Maxwell–Boltzmann statistics is under debate and instead, the Jüttner distribution is proposed as the distribution function in relativistic limit. Motivated by these considerations, in the present work, our aim is to study the effects of GUP and EUP on a system that obeys the Jüttner distribution. To achieve this goal, we address a method to get the distribution function by starting from the partition function and its relation with thermal energy which finally helps us in finding the corresponding energy density states.

Boltzmann Statistics

When a system is held at a fixed temperature, its higher-energy states are less probable than its lower energy states by an amount that depends on how the energy compares to the temperature. The formula that quantifies this idea is called the Boltzmann distribution. This chapter derives the Boltzmann distribution and shows how to use it to predict the thermal behavior of any system whose microscopic states we can enumerate. The examples go beyond the three simple model systems studied already in Chapters 2 and 3 to include detailed properties of gases, stellar spectra, and paramagnetic materials.

Transparency and stability of low density stellar plasma related to Boltzmann statistics and to thermal bremsstrahlung

The stability of low density stellar plasma is analyzed for a star with a spherical symmetry which is in equilibrium between the gravitational attractive forces and the repulsive pressure forces of an ideal electron gas where the analysis is developed by the use of Boltzmann statistics. Fundamental results are obtained for the radius and total mass of such star and its gravitational forces are large due to the extreme large volume. The absorption and emission of radiation for extremely low density star plasmas is very small over the entire electro-magnetic spectrum.

Statistical Mechanics, Diffusion, and Self-Assembly

Equations for random wandering of particles are derived, and the phenomenon of entropic ordering is explained. Boltzmann’s particle-based approach to diffusion is compared to Maxwell’s continuum hypothesis, and van ’t Hoff’s formula for osmosis is obtained. Other topics include diffusivity, the distribution of energy, and the applications of Maxwell–Boltzmann statistics.