China
Africa
Explore chapters and articles related to this topic
Time-Domain Solutions
Published in Dikshitulu K. Kalluri, Principles of Electromagnetic Waves and Materials, 2017
The simplest approximation of the Boltzmann equation is to ignore the collision term on the RHS of (13.320) all together: ∂f∂t+v⋅∇rf+qmE+v×B⋅∇vf=0. Equation (13.326) is called Boltzmann–Vlasov equation.
Uses of Intense Microwaves in Tokamaks
Published in R A Cairns, A D R Phelps, P Osborne, Generation and Application of High Power Microwaves, 2020
The distribution function of electrons or ions, in the so-called collisionless limit, obeys the Vlasov equation. The Vlasov Equation is a continuity equation for the density f in the six dimensional space that includes both the three dimensional configuration space x and the three dimensional velocity space v. This continuity equation presents one constraint on the motions of particles in any program of phase space engineering.
Dynamics of particles in cold electrons plasma: fractional actionlike variational approach versus fractal spaces approach
Published in Waves in Random and Complex Media, 2021
Rami Ahmad El-Nabulsi, Alireza Khalili Golmankhaneh
In the next section, we will show that this term may have new impacts in plasma theory. When , Equation (6) is reduced to the standard Boltzmann/Vlasov equation which describes time evolution of the distribution function of plasma consisting of charged particles subject to Coulomb long-range interaction. In order to join the fractional Vlasov equation with Maxwell's equations for the fields, we first write equation (6) as The Vlasov–Maxwell equation in general describes collisionless magnetized charged particles in an electromagnetic field which are generated by an external applied field or by the particles themselves. We can now use a self-consistent collective field created by charged plasma particles using distribution functions for electrons and positive plasma ions. Accordingly, we can replace the Lorentz force into Equation (7) where and are collective self-consistent electromagnetic field created in the point at time moment τ by plasma particles of charge q. The fractional Vlasov–Maxwell system of equations is now (i is for ions): where for electrons and for positive plasma ions, e is the electron charge, c is the speed of light, and are respectively the charge and current densities, and are the permittivity and permeability of free space, and finally . At this stage, we simply note that if we perform the reversal transformation: , then Equation (8) is invariant. In the next section, we will derive the consequential fractional fluid equations in unmagnetized plasma. The system of equations (8) and (9) is defined as the fractional Vlasov–Maxwell equation (FVME) and may be used to model collisionless plasmas at relativistic velocities. It is notable that the Maxwell–Vlasov equation plays a leading role in studying the transport processes in several fields of sciences including semiconductors and microelectronic structures where quantum, and/or kinetic effects are important.