Preliminary Study of Uncertainty-Driven Plasma Diffusion II

Abstract

We have constructed a semiclassical collisional diffusion model. In this model, a field particle is treated as either a point charge or a spatially distributed charge. The test particle is treated as a distributed point charge with Gaussian distribution. It was shown that the collisional changes in velocity in our model is of the same order as the classical theory for a typical proton in a fusion plasma of T = 10 keV and n = 10 20 m −3 . It was also shown that the spatial extent of the distribution, or the quantum-mechanical uncertainty in position, for the test particle obtained in our model grows in time, and becomes of the order of the average interparticle separation Δ ≡ n −1/3 during a time interval τ r ∼ × 10 7 Δ /g th , where g th = √ 2T/m is the thermal speed, with m being the mass of the particle under consideration. The time interval is 3-4 order of magnitudes smaller than the collision time. This suggests that particle transport cannot be understood in the framework of classical mechanics, and that the quantum-mechanical distribution of individual particles in plasmas may cause the anomalous diffusion.

Numerical Analysis of Quantum-Mechanical Non-Uniform <i>E × B </i>Drift

“…Here the superscript n represents time-label, I and H are unit matrix and numerical Hamiltonian matrix [6,9]. This time integrator is not only unconditionally stable but also norm-conserving scheme for discretized wavefunction {ψ}, the latter of which leads to the strict particle conservation, irrespective of Δt, Δx and Δy, since the matrix H in Eq.…”

“…[5][6][7] that (i) for distant encounters in typical fusion plasmas of a temperature T = 10 keV and n = 10 20 m −3 , the average potential energy U ∼ 30 meV is as small as the uncertainty in energy ΔE ∼ 40 meV, and (ii) for a magnetic field B ∼ 3 T, the spatial size of the wavefunction in the plane perpendicular to the magnetic field is as large as the magnetic length…”

Quantum mechanical expansion of a magnetized particle in the presence of a field particle for magnetic fusion plasmas

“…[7][8][9], the authors have also shown that for distant encounters in typical fusion plasmas of a temperature ~10 T keV and 20 3 10 m n   , the average potential energy ~30 U meV is as small as the uncertainty in energy~40 E  meV, and for a magnetic field ~3 B T, the spatial size of the wavefunction in the plane perpendicular to the magnetic field is as large as the magnetic length .…”

Numerical Analysis of Quantum Mechanical ∇B Drift

“…We have solved the two-dimensional time-dependent Schödinger equation for a particle with and without the interparticle potential in a fusion plasma [1], and similar analysis was made with a semi-classical diffusion model [2]. It was shown in such analyses, especially in Ref.…”

“…[1], that spatial extent of a free particle grows monotonically in time. Such expansion leads to a spatial extent or size of a proton of the order of the average interparticle separation Δ ≡ n −1/3 ∼ 2 × 10 −7 m in a time interval of 10 6 × Δ /v th ∼ 10 −7 sec for a plasma with a density n ∼ 10 20 m −3 and a temperature T = mv 2 th /2 ∼ 10 keV. It was also shown that, under a Coulomb potential, the wavefunction of a charged particle first shrink and expand in time.…”