Our work set up a set of two-fluid, collision-free Vlasov-Maxwell equations to obtain self-similar differential equations which were used to simulate the excitation and propagation of nonlinear brain EEG waves.
(20) Such a non-classical system should not still be treated by employing the Vlasov-Maxwell equations. In this case, the Wigner-Poisson or Wigner-Maxwell equations come to the stage by incorporating the quantum term into account in the Vlasov equations under electrostatic or electromagnetic conditions, respectively.
This paper focuses on investigating the role played by the quantum term in the brain consciousness by generalizing PBD's Vlasov-Maxwell equations with the extra quantum term, thus forming the Wigner-Maxwell equations.
A set of electrostatic Wigner-Poisson equations are given in the absence of an external magnetic field, [B.sub.0], where an additional quantum term comes into being relative to the classical Vlasov-Maxwell equations under electrostatic conditions.
Relative to the classical Vlasov-Maxwell equations under electrostatic conditions, this set of equations includes an additional quantum term.
The nonlinear gyrokinetic theory of the Vlasov-Maxwell equations
can be carried out by appealing to Lagrangian and Hamiltonian methods , , .
Individual topics include hydrodynamic limits of kinetic models, collisionless plasmas and the Vlasov Maxwell system, irreversible behaviors in Vlasov equation and many-body Hamiltonian dynamics in terms of Landau damping, chaos and granularity, guiding center theory, variational formulation of exact and reduced Vlasov-Maxwell equations
, general gyrokinetic theory (including an article with applications in magnetic confinement research in plasma plastics) kinetic to fluid descriptions in plasmas, nonlocal closures in long mean free path regimes, modeling quantum plasmas and inelastic kinetic theory in terms of the granular gas.
(17) Such a system can be described by a set of Vlasov-Maxwell equations to evaluate brain functions by merely taking ion dynamics into consideration (18) in the nervous extracellular space where the interstitial fluid is in contact with the cerebrospinal fluid from the ventricular surfaces; surrounded by the extracellular space there exists larger intracellular neuronal compartment which occupies about 85% of the brain volume.
The layout of the paper is as follows: Section 2 estimates brain plasma parameters and set up a set of two-fluid Vlasov-Maxwell equations; Section 3 derives a set of nonlinear, self-similar differential equations for the excitation and propagation of EEG waves; Section 4 exposes the data-fit modeling of the solitary EEG waves under different conditions.
BRAIN PLASMA MODEL AND TWO-SPECIES VLASOV-MAXWELL EQUATIONS
The two types of particles are therefore described by different collision-free Vlasov-Maxwell equations. Reduced from the collisional Boltzmann equation, the mandatory set of the Vlasov-Maxwell equations for ions and electrons are expressed jointly as follows: (15,35)
We derived a set of two-fluid, self-similar, nonlinear solitary wave equations from PBD's Vlasov-Maxwell equations. This model treats brain aqua-ions and electrons as two different fluids which are coupled with each other in the presence of the internal electric and magnetic fields and the external geomagnetic field.