Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fluid Models from Kinetic Models using a Geometric Averaging Procedure

Ricardo Gallego Torrome

Abstract

We interpret the Lorentz force equation as a geodesic equation associated with a non-linear connection. Using a geometric averaging procedure, we prove that for narrow and smooth one-particle distribution functions whose supports are invariant under the flow of the Lorentz equation, a bunch of charged point particles can be described by a charged cold fluid model in the ultra-relativistic regime. The method used to prove this result does not require additional hypotheses on the higher moments of the distribution. This is accomplished by estimating the expressions that include the differential operators appearing in the charged cold fluid model equation. Under the specified conditions of narrowness and ultra-relativistic dynamics, it turns out that these differential expressions are close to zero, justifying the use of the charged cold fluid model. The method presented in the work can also be applied to justify the use of warm plasmas and other models.

Fluid Models from Kinetic Models using a Geometric Averaging Procedure

Abstract

We interpret the Lorentz force equation as a geodesic equation associated with a non-linear connection. Using a geometric averaging procedure, we prove that for narrow and smooth one-particle distribution functions whose supports are invariant under the flow of the Lorentz equation, a bunch of charged point particles can be described by a charged cold fluid model in the ultra-relativistic regime. The method used to prove this result does not require additional hypotheses on the higher moments of the distribution. This is accomplished by estimating the expressions that include the differential operators appearing in the charged cold fluid model equation. Under the specified conditions of narrowness and ultra-relativistic dynamics, it turns out that these differential expressions are close to zero, justifying the use of the charged cold fluid model. The method presented in the work can also be applied to justify the use of warm plasmas and other models.

Paper Structure

This paper contains 5 sections, 12 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Lorentz Force Equation and Averaged Lorentz Force Equation
  3. Comparison of the Solutions of the Vlasov and Averaged Vlasov Equations
  4. The Charged Cold Fluid Model from the Averaged Vlasov Model
  5. Discussion

Key Result

Proposition 2.2

The Lorentz force equation can be written as the auto-parallel condition where ${x}:{\bf I}\longrightarrow {\bf M}$ is a time-like curve parameterized with respect to the proper time of the Lorentzian metric $\eta$, $\tilde{x}$ is the horizontal lift on N and $^LD$ is the Koszul linear connection determined with connection coefficients given by equation $(2.2)$.

Theorems & Definitions (16)

  • Definition 1.1
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Proposition 3.1
  • Definition 4.1
  • ...and 6 more