The Solvability of Mixed Value Problem for the First and Second Approximations of One-Dimensional Nonlinear System of Moment Equations with Microscopic Boundary Conditions

The paper gives a derivation of a new one-dimensional non-stationary nonlinear system of moment equations, that depend on the flight velocity and the surface temperature of an aircraft. Maxwell microscopic condition is approximated for the distribution function on moving boundary, when one fraction of molecules reflected from the surface specular and another fraction diffusely with Maxwell distribution. Moreover, macroscopic boundary conditions for the moment system of equations depend on evenness or oddness of approximation $${f}_{k}(t,x,c)$$ f k ( t , x , c ) , where $${f}_{k}(t,x,c)$$ f k ( t , x , c ) is partial expansion sum of the molecules distribution function over eigenfunctions of linearized collision operator around local Maxwell distribution. The formulation of initial and boundary value problem for the system of moment equations in the first and second approximations is described. Existence and uniqueness of the solution for the above-mentioned problem using macroscopic boundary conditions in the space of functions $$C\left(\left[0,T\right];{L}^{2}\left[-a,a\right]\right)$$ C 0 , T ; L 2 - a , a are proved.

An experiment design for schools to demonstrate the Maxwell distribution

The article describes a new version of a demonstration experiment for the Maxwell distribution. In the first part students analyse the applicability of the Gaussian distribution to the projection of the particle velocities in the suggested experiment. Further, students observe two-dimensional distribution of particles by the modulus of velocity in a mechanical demonstration model and compare the results with theoretical provisions. Demonstration of the two-dimensional version of the Maxwell distribution for particle interaction allows students to independently derive formulas for the three-dimensional Maxwell distribution for particles in an ideal gas. The use of the suggested demonstration ensures active engagement in fundamentally important physical content.

A Computational Model of Maxwell’s Distribution for Undergraduates

In this paper we employ a numerical approach to perform simulations of Maxwell distribution for several dimensionalities, based on the Central Limit Theorem. We show that by increasing the number of molecules of the gas N, the simulated distributions tend toward the respective theoretical distributions. Also, we observed that by increasing the model temperature n, the distribution shifted toward higher speeds, in agreement with theoretical results. The numerical simulations provide a physical definition of the concept of temperature. The codes used to perform the simulations are quite easy to construct and implement, while the results strikingly satisfy theoretical expectations. Furthermore, the actual approach makes it possible to skip the mathematical details and explain the distribution by just following the algorithm of simulations. We recommend such approach as a demonstrative tool that can be shown in a lecture class thus enriching the teaching quality and improving students’ understanding.

Bayesian Estimation of Weighted Inverse Maxwell Distribution under Different Loss Functions

In this study, the shape parameter of the weighted Inverse Maxwell distribution is estimated by employing Bayesian techniques. To produce posterior distributions, the extended Jeffery's prior and the Erlang prior are utilised. The estimators are derived from the squared error loss function, the entropy loss function, the precautionary loss function, and the Linex loss function. Furthermore, an actual data set is studied to assess the effectiveness of various estimators under distinct loss functions.

The Angular Speed Distribution of Randomly Moving Particles

Spin is a common phenomenon in nature, but the detailed statistical principle behind it is seldom discussed. This article aims to inspire people to think about such problems. In this article, probability theory and the corresponding modules of Mathematica software are used to study the distribution of angular speed generated by a population of equal speed particles moving randomly in space. The probability density expression of this distribution is derived successfully, and then its accuracy is verified by simulation data. Finally, it is confirmed that the angular speed follows the Maxwell distribution with a specified scale parameter. This research may be helpful for understanding why most celestial bodies or micro particles in the universe spin.