scholarly journals On hydrodynamics in the presense of strong external potential field

2021 ◽  
Vol 29 (1) ◽  
pp. 21-28
Author(s):  
A. I. Sokolovsky ◽  
S. A. Sokolovsky
Keyword(s):  
Distribution Function ◽  
Kinetic Equation ◽  
Potential Field ◽  
Chemical Potential ◽  
Local Equilibrium ◽  
Practical Importance ◽  
External Potential ◽  
Equilibrium Distribution Function ◽  
Kinetic Coefficients ◽  
Functional Hypothesis

On the base of the Boltzmann kinetic equation, hydrodynamics of a dilute gas in the presence of the strong external potential field is investigated. First of all, a gravitational field is meant, because the consistent development of hydrodynamics in this environment is of great practical importance. In the present paper it is assumed that it is possible to neglect the influence of the field on the particle collisions. The study is based on the Chapman–Enskog method in a Bogolyubov’s formulation, which uses the idea of the functional hypothesis. Consideration is limited to steady gas states, which are subjected to a simpler experimental study. Chemical potential μ0 of the gas at the point where the external field has zero value and its temperature T are selected as the reduced description parameters of the system. In equilibrium, in the presence of the field, these values do not depend on the coordinates. It is assumed that in thehydrodynamic states T and μ0 are weakly dependent on the coordinates and therefore their gradients, considered on the scale of the free path length of the gas, are small. The kinetic equation, accounting for the functional hypothesis, gives an integro-differential equation for a gas distribution function at the hydrodynamic stage of evolution. This equation is solved in perturbation theory in gradients of T and μ0. The main approximation is analyzed for possibility of the system to be in a local equilibrium by means of comparing it with an equilibrium distribution function. Next, the distribution function is calculated in the first approximation in gradients and it is expressed in terms of solutions Ap , Bp of some first kind integral Fredholm equations. An approach to the approximate solution of these equations is discussed. The found distribution function is used to calculate the fluxes of the number of gas particles and their energy in the first order in gradients T and μ0 . Kinetic coefficients, which describe the structure of these fluxes, are introduced. Matrix elements of the operator of the linearized collision integral (integral brackets) are used for their research. It is a question of validity of the principle of symmetry of kinetic coefficients and definition of their signs.

A LATTICE BGK MODEL FOR SIMULATING SOLITARY WAVES OF THE COMBINED KdV–MKDV EQUATION

2011 ◽  
Vol 25 (04) ◽  
pp. 589-597 ◽  
Author(s):  
CHANGFENG MA
Keyword(s):  
Distribution Function ◽  
Solitary Waves ◽  
Equilibrium Distribution ◽  
Local Equilibrium ◽  
Mkdv Equation ◽  
Equilibrium Distribution Function ◽  
Bgk Model ◽  
Theoretical Results

A lattice BGK model for simulating solitary waves of the combined KdV–MKDV equation, ut+αuux-βu2ux+δuxxx = 0, is established. The tunable parameters in Chapman–Enskog expansion of the local equilibrium distribution function are determined by the coefficient of the combined KdV–MKDV equation. Simulating results fit close in with the theoretical results.

scholarly journals Sterile Neutrinos as Dark Matter: Alternative Production Mechanisms in the Early Universe

Universe ◽  
2021 ◽  
Vol 7 (8) ◽  
pp. 264
Author(s):  
Daniel Boyanovsky
Keyword(s):  
Dark Matter ◽  
Distribution Function ◽  
Kinetic Equation ◽  
Standard Model ◽  
Early Universe ◽  
Quantum Kinetic Equation ◽  
Sterile Neutrinos ◽  
The Standard Model ◽  
Production Mechanisms ◽  
Freeze Out

We study various production mechanisms of sterile neutrinos in the early universe beyond and within the standard model. We obtain the quantum kinetic equations for production and the distribution function of sterile-like neutrinos at freeze-out, from which we obtain free streaming lengths, equations of state and coarse grained phase space densities. In a simple extension beyond the standard model, in which neutrinos are Yukawa coupled to a Higgs-like scalar, we derive and solve the quantum kinetic equation for sterile production and analyze the freeze-out conditions and clustering properties of this dark matter constituent. We argue that in the mass basis, standard model processes that produce active neutrinos also yield sterile-like neutrinos, leading to various possible production channels. Hence, the final distribution function of sterile-like neutrinos is a result of the various kinematically allowed production processes in the early universe. As an explicit example, we consider production of light sterile neutrinos from pion decay after the QCD phase transition, obtaining the quantum kinetic equation and the distribution function at freeze-out. A sterile-like neutrino with a mass in the keV range produced by this process is a suitable warm dark matter candidate with a free-streaming length of the order of few kpc consistent with cores in dwarf galaxies.

scholarly journals A construction of the general relativistic Boltzmann equation

1984 ◽  
Vol 7 (3) ◽  
pp. 591-597 ◽  
Author(s):  
P. Dolan ◽  
A. C. Zenios
Keyword(s):  
Distribution Function ◽  
Boltzmann Equation ◽  
Kinetic Equation ◽  
Simple Model ◽  
State Space ◽  
Differential Forms ◽  
Relativistic Boltzmann Equation ◽  
Dimensional State Space ◽  
General Relativistic ◽  
Invariant Differential

Our work depends essentially on the notion of a one-particle seven-dimensional state-space. In constructing a general relativistic theory we assume that all measurable quantities arise from invariant differential forms. In this paper, we study only the case when instantaneous, binary, elastic collisions occur between the particles of the gas. With a simple model for colliding particles and their collisions, we derive the kinetic equation, which gives the change of the distribution function along flows in state-space.

scholarly journals Validity of the molecular-dynamics-lattice-gas global equilibrium distribution function

2019 ◽  
Vol 30 (10) ◽  
pp. 1941007 ◽  
Author(s):  
M. Reza Parsa ◽  
Aleksandra Pachalieva ◽  
Alexander J. Wagner
Keyword(s):  
Molecular Dynamics ◽  
Distribution Function ◽  
Lattice Gas ◽  
Equilibrium Distribution ◽  
Coarse Graining ◽  
High Volume ◽  
Equilibrium Distribution Function ◽  
Theory Approach ◽  
Lattice Boltzmann Methods ◽  
Definition Of

The molecular-dynamics-lattice-gas (MDLG) method establishes a direct link between a lattice-gas method and the coarse-graining of a molecular dynamics (MD) approach. Due to its connection to MD, the MDLG rigorously recovers the hydrodynamics and allows to validate the behavior of the lattice-gas or lattice-Boltzmann methods directly without using the standard kinetic theory approach. In this paper, we show that the analytical definition of the equilibrium distribution function remains valid even for very high volume fractions.

scholarly journals Mass-Conserved Wall Treatment of the Non-Equilibrium Extrapolation Boundary Condition in Lattice Boltzmann Method

Energies ◽  
2018 ◽  
Vol 11 (10) ◽  
pp. 2585
Author(s):  
Zhe Feng ◽  
Hee-Chang Lim
Keyword(s):  
Boundary Condition ◽  
Distribution Function ◽  
Lattice Boltzmann ◽  
Treatment Method ◽  
Extrapolation Method ◽  
Equilibrium Distribution Function ◽  
Boundary Node ◽  
External Force Field ◽  
Mass Leakage ◽  
Non Equilibrium

In lattice Boltzmann simulations, the widely used non-equilibrium extrapolation method for velocity and pressure boundary conditions can cause a constant mass leakage under certain circumstances, particularly when an external force field is imposed on the fluid domain. The non-equilibrium distribution function at the boundary uses a first-order extrapolation method on the corresponding data of adjacent fluid nodes. In addition, based on this extrapolation method, the macroscopic velocity and density at the boundary nodes are obtained. Therefore, the corresponding equilibrium component of the distribution function can be calculated explicitly. Regarding the no-slip wall boundary condition, we found that the mass leakage primarily results from the extrapolation scheme for the density term in the equilibrium component of the distribution function at the boundary node. In this study, a mass-conserved wall treatment method is developed to correct the existing density term for guaranteeing the conservation of mass. Several benchmark test cases were simulated and compared to prove the justification of the newly developed mass-conserved boundary condition, and the results show a good agreement with those in the existing literature.

Bootstrap-current computations for a stellarator-reactor plasma

1990 ◽  
Vol 44 (3) ◽  
pp. 431-453 ◽  
Author(s):  
W. D. D'Haeseleer ◽  
W. N. G. Hitchon ◽  
C. D. Beidler ◽  
J. L. Shohet
Keyword(s):  
Distribution Function ◽  
Kinetic Equation ◽  
Physical Interpretation ◽  
Collision Operator ◽  
Numerical Code ◽  
Parallel Flows ◽  
Bootstrap Current ◽  
The Difference ◽  
Rotational Transform ◽  
The Impact

Numerical results for the bootstrap current in a stellarator-reactor plasma are presented. The distribution function f is computed numerically from a kinetic equation that is averaged over the helical ripple. The parallel flows and the current are obtained as v‖ moments of f. The physics issues embedded in the code are discussed concisely, concentrating on the justification as to why the bootstrap current can be estimated from an averaged scheme. Results are presented for typical stellarator-reactor parameters. The numerical code FLOCS predicts that the momentum-restoring terms in the collision operator have no significant impact on the value of the bootstrap current (the difference being about 10%). The results obtained are related to the equilibrium flows, and a physical interpretation based on the kinetic picture is presented. Finally, an estimate for the impact of J‖ on the rotational transform is given.

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