VectorSpin

This package is designed to solve Vlasov system with spin effects.

This code is used in the paper (Crouseilles et al., 2023).

This is the vectorial version of the SpinGEMPIC.jl code that has been used for the paper (Crouseilles et al., 2021)

Scalar spin laser plasma model

Particle distribution function $f(x, p, {\mathbf s}, t)$,

• $x\in [0,L]$,
• $p \in \mathbb{R}$ are scalars,
• ${\mathbf s}=(s_1,s_2,s_3) \in \mathbb{R}^3$,
• ${\mathbf E} = (E_x, {\mathbf E}_\perp) = (E_x, E_y, E_z)$,
• ${\mathbf A} = (A_x, {\mathbf A}_\perp) = (0, A_y, A_z)$,
• ${\mathbf B} =\nabla\times{\mathbf A} = (0,- \partial_xA_z, \partial_xA_y)$.

The scalar spin Vlasov-Maxwell system is:

\[\left\{ \begin{aligned} &\frac{\partial f}{\partial t} + p \frac{\partial f}{\partial x} + [ E_x - \mathfrak{h} s_2 \frac{\partial^2 A_z}{\partial x^2} + \mathfrak{h} s_3 \frac{\partial^2 A_y}{\partial x^2} - {\mathbf A}_\perp \cdot \frac{\partial {\mathbf A}_\perp}{\partial x} ]\frac{\partial f}{\partial p} \\ & \hspace{3cm}+ [s_3 \frac{\partial A_z}{\partial x} + s_2 \frac{\partial A_y}{\partial x}, -s_1 \frac{\partial A_y}{\partial x}, -s_1 \frac{\partial A_z}{\partial x} ] \cdot \frac{\partial f}{\partial {\mathbf s}} = 0,\\ &\frac{\partial E_x}{\partial t} = -\int_{\mathbb{R}^4} p f \mathrm{d}{p}\mathrm{d}\mathrm{\mathbf s},\\ &\frac{\partial E_y}{\partial t} = - \frac{\partial^2 A_y}{\partial x^2} + A_y \int_{\mathbb{R}^4} f \mathrm{d}{p}\mathrm{d}\mathrm{\mathbf s} + \mathfrak{h}\int_{\mathbb{R}^4} s_3 \frac{\partial f}{\partial x}\mathrm{d}{p}\mathrm{d}\mathrm{\mathbf s},\\ &\frac{\partial E_z}{\partial t} = - \frac{\partial^2 A_z}{\partial x^2} + A_z \int_{\mathbb{R}^4} f \mathrm{d}{p}\mathrm{d}\mathrm{\mathbf s} - \mathfrak{h}\int_{\mathbb{R}^4} s_2 \frac{\partial f}{\partial x}\mathrm{d}{p}\mathrm{d}\mathrm{\mathbf s},\\ & \frac{\partial {\mathbf A}_\perp}{\partial t} = - {\mathbf E}_\perp,\\ &\frac{\partial E_x}{\partial x} = \int_{\mathbb{R}^4} f \mathrm{d}{p}\mathrm{d}\mathrm{\mathbf s} - 1. \ \text{(Poisson equation)} \end{aligned} \right.\]

The system numerically solve is the vector model:

\[f(t, x,p,{\mathbf{s}})=\frac{1}{4\pi}(f_0(t, x,p)+3s_1f_1(t, x,p)+3s_2f_2(t, x,p)+3s_3f_3(t, x,p)).\]

\[\left\{ \begin{aligned} &\frac{\partial f_0}{\partial t} + p \frac{\partial f_0}{\partial x} + \left(E_x - {\mathbf A}_\perp \cdot \frac{\partial {\mathbf A}_\perp}{\partial x} \right) \frac{\partial f_0}{\partial p} - \mathfrak{h}\frac{\partial^2 A_z}{\partial x^2}\frac{\partial f_2}{\partial p} + \mathfrak{h}\frac{\partial^2 A_y}{\partial x^2} \frac{\partial f_3}{\partial p} = 0,\\ &\frac{\partial f_1}{\partial t} + p \frac{\partial f_1}{\partial x} + \left(E_x - {\mathbf A}_\perp \cdot \frac{\partial {\mathbf A}_\perp}{\partial x} \right) \frac{\partial f_1}{\partial p} - \frac{\partial A_z }{\partial x} f_3 - \frac{\partial A_y }{\partial x} f_2 = 0,\\ & \frac{\partial f_2}{\partial t} + p \frac{\partial f_2}{\partial x} + \left(E_x - {\mathbf A}_\perp \cdot \frac{\partial {\mathbf A}_\perp}{\partial x} \right) \frac{\partial f_2}{\partial p} - {\frac{\mathfrak{h}}{3}} \frac{\partial^2 A_z}{\partial x^2}\frac{\partial f_0}{\partial p} + \frac{\partial A_y }{\partial x} f_1 = 0,\\ & \frac{\partial f_3}{\partial t} + p \frac{\partial f_3}{\partial x} + \left(E_x - {\mathbf A}_\perp \cdot \frac{\partial {\mathbf A}_\perp}{\partial x} \right) \frac{\partial f_3}{\partial p} + {\frac{\mathfrak{h}}{3}} \frac{\partial^2 A_y}{\partial x^2}\frac{\partial f_0}{\partial p} + \frac{\partial A_z }{\partial x} f_1 = 0,\\ &\frac{\partial E_x}{\partial t} = -\int_{\mathbb{R}} p f_0 \mathrm{d}\mathrm{p},\\ &\frac{\partial E_y}{\partial t} = - \frac{\partial^2 A_y}{\partial x^2} + A_y \int_{\mathbb{R}} f_0 \mathrm{d}\mathrm{p} + \mathfrak{h}\int_{\mathbb{R}} \frac{\partial f_3}{\partial x}\mathrm{d}\mathrm{p},\\ &\frac{\partial E_z}{\partial t} = - \frac{\partial^2 A_z}{\partial x^2} + A_z \int_{\mathbb{R}} f_0 \mathrm{d}\mathrm{p} -\mathfrak{h} \int_{\mathbb{R}} \frac{\partial f_2}{\partial x}\mathrm{d}\mathrm{p},\\ & \frac{\partial {\mathbf A}_\perp}{\partial t} = - {\mathbf E}_\perp,\\ &\frac{\partial E_x}{\partial x} = \int_{\mathbb{R}} f_0 \mathrm{d}\mathrm{p} - 1.\ \text{(Poisson equation)} \end{aligned} \right.\]