scholarly journals A Consistent BGK Model with Velocity-Dependent Collision Frequency for Gas Mixtures

2021 ◽  
Vol 184 (3) ◽  
Author(s):  
J. Haack ◽  
C. Hauck ◽  
C. Klingenberg ◽  
M. Pirner ◽  
S. Warnecke
Keyword(s):  
Total Energy ◽  
Unique Solution ◽  
Minimization Problem ◽  
Collision Frequency ◽  
Total Momentum ◽  
Gas Mixture ◽  
Bgk Model ◽  
H Theorem ◽  
Weighted Entropy ◽  
Number Of Particles

AbstractWe derive a multi-species BGK model with velocity-dependent collision frequency for a non-reactive, multi-component gas mixture. The model is derived by minimizing a weighted entropy under the constraint that the number of particles of each species, total momentum, and total energy are conserved. We prove that this minimization problem admits a unique solution for very general collision frequencies. Moreover, we prove that the model satisfies an H-Theorem and characterize the form of equilibrium.

A kinetic ellipsoidal BGK model for a binary gas mixture

2011 ◽  
Vol 96 (6) ◽  
pp. 64002 ◽  
Author(s):  
M. Groppi ◽  
S. Monica ◽  
G. Spiga
Keyword(s):  
Gas Mixture ◽  
Bgk Model ◽  
Binary Gas Mixture

scholarly journals The computation of Fermi-Dirac functions

1938 ◽  
Vol 237 (773) ◽  
pp. 67-104 ◽  
Keyword(s):  
Energy Distribution ◽  
Energy Range ◽  
Total Energy ◽  
Variational Equation ◽  
Statistical Treatment ◽  
Unit Energy ◽  
Boltzmann Statistics ◽  
Number Of Particles

The quantitative application of Fermi-Dirac statistics involves the evaluation of certain integrals which have not previously been tabulated. In this paper, tables are given of the values of the basic integrals most frequently required , with a view to placing Fermi-Dirrac statistics on as firm a numerical basis as is Maxwell-Boltzmann statistics. T e expression for the energy distribution of particles subject to Fermi-Dirrac statistics may be written in the form dN He) de e<*+Pe -)-1 ’ wherev(e) is the number of states per unit energy range, and dN is the number of particles in the energy range e to e--de. In the statistical treatment, the parameters ot and fi, which are usually introduced as undetermined multipliers in a variational equation, are to be determined from two equations expressing conditions imposed by the total number of particles, and the total energy of the system. By linking up the statistical and thermodynamical treatments, interpretation can be given to a and b this is expressed by P**:l IkT, a = -C lk T ,

scholarly journals Optimization-Based Construction of Quadrilateral Table Cartograms

2020 ◽  
Vol 9 (1) ◽  
pp. 43
Author(s):  
Ryo Inoue ◽  
Mao Li
Keyword(s):  
Unique Solution ◽  
Minimization Problem ◽  
Optimization Problem ◽  
Construction Method ◽  
Shape Deformation ◽  
Output Table ◽  
Positive Weights ◽  
Original Table ◽  
New Construction ◽  
Complex Settings

A quadrilateral table cartogram is a rectangle-shaped figure that visualizes table-form data; quadrilateral cells in a table cartogram are transformed to express the magnitude of positive weights by their areas, while maintaining the adjacency of cells in the original table. However, the previous construction method is difficult to implement because it consists of multiple operations that do not have a unique solution and require complex settings to obtain the desired outputs. In this article, we propose a new construction for quadrilateral table cartograms by recasting the construction as an optimization problem. The proposed method is formulated as a simple minimization problem to achieve mathematical clarity. It can generate quadrilateral table cartograms with smaller deformation of rows and columns, thereby aiding readers to recognize the correspondence between table cartograms and original tables. In addition, we also propose a means of sorting rows and/or columns prior to the construction of table cartograms to reduce excess shape deformation. Applications of the proposed method confirm its capability to output table cartograms that clearly visualize the characteristics of datasets.

scholarly journals Statistical Mechanics and Equilibrium Sequences of Ellipticals

1987 ◽  
Vol 127 ◽  
pp. 505-506 ◽  
Author(s):  
M. Stiavelli ◽  
G. Bertin
Keyword(s):  
Statistical Mechanics ◽  
Total Energy ◽  
Elliptical Galaxies ◽  
Number Of Particles

Elliptical galaxies are expected to have undergone incomplete violent relaxation. Here incomplete relaxation is regarded as a process producing a metastable, long-lived state which is dynamically stabilized by the approximate conservation of one global quantity in addition to the total energy and number of particles.

Kramers' problem for a variable collision frequency model

2001 ◽  
Vol 12 (2) ◽  
pp. 179-191 ◽  
Author(s):  
C. E. SIEWERT
Keyword(s):  
Gas Dynamics ◽  
Rarefied Gas ◽  
Collision Frequency ◽  
Discrete Ordinates ◽  
Rigid Sphere ◽  
Sphere Model ◽  
Bgk Model ◽  
Frequency Model ◽  
Change Of Variables ◽  
Kramers Problem

The often-studied problem known as Kramers' problem, in the general area of rarefied-gas dynamics, is investigated in terms of a linearized, variable collision frequency model of the Boltzmann equation. A convenient change of variables is used to reduce the general case considered to a canonical form that is well suited for analysis by analytical and/or numerical methods. While the general formulation developed is valid for an unspecified collision frequency, a recently developed version of the discrete-ordinates method is used to compute the viscous-slip coefficient and the velocity defect in the Knudsen layer for three specific cases: the classical BGK model, the Williams model (the collision frequency is proportional to the magnitude of the velocity) and the rigid-sphere model.

scholarly journals Steepest Entropy Ascent Models of the Boltzmann Equation: Comparisons With Hard-Sphere Dynamics and Relaxation-Time Models for Homogeneous Relaxation From Highly Non-Equilibrium States

2013 ◽  
Author(s):  
Gian Paolo Beretta ◽  
Nicolas G. Hadjiconstantinou
Keyword(s):  
Relaxation Time ◽  
Boltzmann Equation ◽  
Power Law ◽  
Collision Frequency ◽  
Equilibrium States ◽  
Space Distribution ◽  
Bgk Model ◽  
The Boltzmann Equation ◽  
Far From Equilibrium ◽  
Non Equilibrium

We present a family of steepest entropy ascent (SEA) models of the Boltzmann equation. The models preserve the usual collision invariants (mass, momentum, energy), as well as the non-negativity of the phase-space distribution, and have a strong built-in thermodynamic consistency, i.e., they entail a general H-theorem valid even very far from equilibrium. This family of models features a molecular-speed-dependent collision frequency; each variant can be shown to approach a corresponding BGK model with the same variable collision frequency in the limit of small deviation from equilibrium. This includes power-law dependence on the molecular speed for which the BGK model is known to have a Prandtl number that can be adjusted via the power-law exponent. We compare numerical solutions of the constant and velocity-dependent collision frequency variants of the SEA model with the standard relaxation-time model and a Monte Carlo simulation of the original Boltzmann collision operator for hard spheres for homogeneous relaxation from near-equilibrium and highly non-equilibrium states. Good agreement is found between all models in the near-equilibrium regime. However, for initial states that are far from equilibrium, large differences are found; this suggests that the maximum entropy production statistical ansatz is not equivalent to Boltzmann collisional dynamics and needs to be modified or augmented via additional constraints or structure.

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