Related Experiment Video
Updated: Jun 18, 2026
11:03
An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
Published on: December 4, 2017
Linear transport equations valid for arbitrary collisionality: comparison with the Chapman-Enskog expansion
A Bendib1, K Bendib-Kalache, M-M Gombert
1Laboratoire d'Electronique Quantique, Faculté de Physique, USTHB, El Alia, BP 32, Bab Ezzouar, 16111 Algiers, Algeria.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 13, 2009
Summary
This study extends a method to solve the Boltzmann equation, deriving linear transport equations across all collisionality ranges. The findings reveal crucial roles for projection operators and propose stable transport coefficients.
Area of Science:
- Plasma Physics
- Statistical Mechanics
- Kinetic Theory
Background:
- The Bhatnagar-Gross-Krook (BGK) operator models collisions in kinetic theory.
- Solving the perturbed Boltzmann equation is essential for understanding transport phenomena.
- Previous work proposed a method for solving the perturbed Boltzmann equation with the BGK operator.
Purpose of the Study:
- To derive linear transport equations valid across the entire collisionality range using a previously proposed method.
- To compare closure relations derived via this method with established expansions like Chapman-Enskog.
- To investigate the contribution of projection operators and assess the stability of the derived equations.
Main Methods:
- Application of a novel method for solving the perturbed Boltzmann equation.
- Derivation of closure relations up to the third order in Knudsen number (super-Burnett expansion).
- Analysis of projection operator contributions and stability tests using generalized transport coefficients and Padé approximants.
Main Results:
- Linear transport equations were successfully derived for the entire collisionality range.
- Super-Burnett closure relations matched the Chapman-Enskog expansion results.
- Projection operators were found to have significant contributions, especially in the collisionless regime.
- Stability of super-Burnett equations was linked to the positivity of generalized transport coefficients.
Conclusions:
- The proposed method effectively derives linear transport equations across all collisionality regimes.
- Projection operators play a non-negligible role in super-Burnett equations and are significant in the collisionless limit.
- Padé approximants yield nonlocal transport coefficients with improved stability properties, addressing key challenges in kinetic theory.
Related Concept Videos
01:06Collisions in Multiple Dimensions: Problem Solving
In multiple dimensions, the conservation of momentum applies in each direction independently. Hence, to solve collisions in multiple dimensions, we should write down the momentum conservation in each direction separately. To help understand collisions in multiple dimensions, consider an example.
A small car of mass 1,200 kg traveling east at 60 km/h collides at an intersection with a truck of mass 3,000 kg traveling due north at 40 km/h. The two vehicles are locked together. What is the...
A small car of mass 1,200 kg traveling east at 60 km/h collides at an intersection with a truck of mass 3,000 kg traveling due north at 40 km/h. The two vehicles are locked together. What is the...
01:05Collisions in Multiple Dimensions: Introduction
It is far more common for collisions to occur in two dimensions; that is, the initial velocity vectors are neither parallel nor antiparallel to each other. Let's see what complications arise from this. The first idea is that momentum is a vector. Like all vectors, it can be expressed as a sum of perpendicular components (usually, though not always, an x-component and a y-component, and a z-component if necessary). Thus, when the statement of conservation of momentum is written for a problem,...
01:24Reynolds Transport Theorem
The Reynolds transport theorem provides a framework to relate the time rate of change of an extensive property within a system to that in a control volume, which is crucial for analyzing fluid dynamics. Extensive properties, such as mass, velocity, acceleration, temperature, and momentum, can be expressed in terms of the mass of a fluid portion. These properties are called extensive because they depend on the system's size, while intensive properties are their corresponding values per unit mass.
01:15Elastic Collisions: Case Study
Elastic collision of a system demands conservation of both momentum and kinetic energy. To solve problems involving one-dimensional elastic collisions between two objects, the equations for conservation of momentum and conservation of internal kinetic energy can be used. For the two objects, the sum of momentum before the collision equals the total momentum after the collision. An elastic collision conserves internal kinetic energy, and so the sum of kinetic energies before the collision equals...
01:00Elastic Collisions: Introduction
An elastic collision is one that conserves both internal kinetic energy and momentum. Internal kinetic energy is the sum of the kinetic energies of the objects in a system. Truly elastic collisions can only be achieved with subatomic particles, such as electrons striking nuclei. Macroscopic collisions can be very nearly, but not quite, elastic, as some kinetic energy is always converted into other forms of energy such as heat transfer due to friction and sound. An example of a nearly...
01:21Carrier Transport
The generation of electrical current in semiconductors is fundamentally driven by two mechanisms: drift and diffusion. These processes are essential for the functionality and performance of semiconductor-based devices.
Drift Current:
The drift of charge carriers is started by an external electric field (E). Charged particles, such as electrons and holes, experience an acceleration between collisions with lattice atoms. For electrons, this results in a drift velocity (vd) given by:
Drift Current:
The drift of charge carriers is started by an external electric field (E). Charged particles, such as electrons and holes, experience an acceleration between collisions with lattice atoms. For electrons, this results in a drift velocity (vd) given by: