Globally hyperbolic moment model of arbitrary order for the three-dimensional special relativistic Boltzmann equation with the Anderson-Witting collision

2021 ◽  
Author(s):  
Yangyu Kuang ◽  
Huazhong Tang
Keyword(s):  
Boltzmann Equation ◽  
Arbitrary Order ◽  
Three Dimensional ◽  
Globally Hyperbolic ◽  
Relativistic Boltzmann Equation

scholarly journals On the Determinant Problem for the Relativistic Boltzmann Equation

2021 ◽  
Author(s):  
James Chapman ◽  
Jin Woo Jang ◽  
Robert M. Strain
Keyword(s):  
Boltzmann Equation ◽  
Upper Bound ◽  
Numerical Study ◽  
Jacobian Determinant ◽  
Large Role ◽  
Relativistic Boltzmann Equation ◽  
Relativistic Analog ◽  
Relativistic Collision ◽  
Pointwise Limit ◽  
Open Question

AbstractThis article considers a long-outstanding open question regarding the Jacobian determinant for the relativistic Boltzmann equation in the center-of-momentum coordinates. For the Newtonian Boltzmann equation, the center-of-momentum coordinates have played a large role in the study of the Newtonian non-cutoff Boltzmann equation, in particular we mention the widely used cancellation lemma [1]. In this article we calculate specifically the very complicated Jacobian determinant, in ten variables, for the relativistic collision map from the momentum p to the post collisional momentum $$p'$$ p ′ ; specifically we calculate the determinant for $$p\mapsto u = \theta p'+\left( 1-\theta \right) p$$ p ↦ u = θ p ′ + 1 - θ p for $$\theta \in [0,1]$$ θ ∈ [ 0 , 1 ] . Afterwards we give an upper-bound for this determinant that has no singularity in both p and q variables. Next we give an example where we prove that the Jacobian goes to zero in a specific pointwise limit. We further explain the results of our numerical study which shows that the Jacobian determinant has a very large number of distinct points at which it is machine zero. This generalizes the work of Glassey-Strauss (1991) [8] and Guo-Strain (2012) [12]. These conclusions make it difficult to envision a direct relativistic analog of the Newtonian cancellation lemma in the center-of-momentum coordinates.

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.

EVOLUTION OF CENTRAL MOMENTS FOR A GENERAL-RELATIVISTIC BOLTZMANN EQUATION: THE CLOSURE BY ENTROPY MAXIMIZATION

2002 ◽  
Vol 14 (05) ◽  
pp. 469-510 ◽  
Author(s):  
ZBIGNIEW BANACH ◽  
WIESLAW LARECKI
Keyword(s):  
Vector Field ◽  
Boltzmann Equation ◽  
Velocity Vector ◽  
Hyperbolic Systems ◽  
Velocity Vector Field ◽  
Field Equations ◽  
Entropy Maximization ◽  
Central Moments ◽  
Relativistic Boltzmann Equation ◽  
Lorentzian Metric

Beginning from the relativistic Boltzmann equation in a curved space-time, and assuming that there exists a fiducial congruence of timelike world lines with four-velocity vector field u, it is the aim of this paper to present a systematic derivation of a hierarchy of closed systems of moment equations. These systems are found by using the closure by entropy maximization. Our concepts are primarily applied to the formalism of central moments because if an alternative and more familiar theory of covariant moments is taken into account, then the method of maximum entropy is ill-defined in a neighborhood of equilibrium states. The central moments are not covariant in the following sense: two observers looking at the same relativistic gas will, in general, extract two different sets of central moments, not related to each other by a tensorial linear transformation. After a brief review of the formalism of trace-free symmetric spacelike tensors, the differential equations for irreducible central moments are obtained and compared with those of Ellis et al. [Ann. Phys. (NY)150 (1983) 455]. We derive some auxiliary algebraic identities which involve the set of central moments and the corresponding set of Lagrange multipliers; these identities enable us to show that there is an additional balance law interpreted as the equation of balance of entropy. The above results are valid for an arbitrary choice of the Lorentzian metric g and the four-velocity vector field u. Later, the definition of u as in the well-known theory of Arnowitt, Deser, and Misner is proposed in order to construct a hierarchy of symmetric hyperbolic systems of field equations. Also, the Eckart and Landau–Lifshitz definitions of u are discussed. Specifically, it is demonstrated that they lead, in general, to the systems of nonconservative equations.

Sign in / Sign up

Export Citation Format

Share Document