Docs: PIC Model

Start writing the PIC model section.
Introduces our implemented system of equations for the reader.

Author: ax3l

docs/source/models/pic.rst

@@ -7,9 +7,81 @@ The Particle-in-Cell Algorithm
 
 For now, please refer to the textbooks [BirdsallLangdon]_, [HockneyEastwood]_, our :ref:`latest paper on PIConGPU <usage-reference>` and [Huebl2014]_ (chapters 2.3, 3.1 and 3.4).
 
+System of Equations
+-------------------
+
+.. math::
+
+   \nabla \cdot \mathbf{E} &= \frac{1}{\varepsilon_0}\sum_s \rho_s
+   
+   \nabla \cdot \mathbf{B} &= 0
+   
+   \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}} {\partial t}
+   
+   \nabla \times \mathbf{B} &= \mu_0\left(\sum_s \mathbf{J}_s + \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t} \right)
+   
+for multiple particle species :math:`s`.
+:math:`\mathbf{E}(t)` represents the electic, :math:`\mathbf{B}(t)` the magnetic, :math:`\rho_s` the charge density and :math:`\mathbf{J}_s(t)` the current density field.
+
+Except for normalization of constants, PIConGPU implements the governing equations in SI units.
+
+Relativistic Plasma Physics
+---------------------------
+
+The 3D3V particle-in-cell method is used to describe many-body systems such as a plasmas.
+It approximates the Vlasov--Maxwell--Equation
+
+.. math::
+   :label: VlasovMaxwell
+
+   \partial_t f_s(\mathbf{x},\mathbf{v},t) + \mathbf{v} \cdot \nabla_x f_s(\mathbf{x},\mathbf{v},t) + \frac{q_s}{m_s} \left[ \mathbf{E}(\mathbf{x},t)  + \mathbf{v} \times \mathbf{B}(\mathbf{x},t) \right] \cdot \nabla_v f_s(\mathbf{x},\mathbf{v},t) = 0
+
+with :math:`f_s` as the distribution function of a particle species :math:`s`, :math:`\mathbf{x},\mathbf{v},t` as position, velocity and time and :math:`\frac{q_s}{m_s}` the charge to mass-ratio of a species.
+The momentum is related to the velocity by :math:`\mathbf{p} = \gamma m_s \mathbf{v}`.
+
+The equations of motion are given by the Lorentz force as
+
+.. math::
+
+  \frac{\mathrm{d}}{\mathrm{d}t} \mathbf{V_s}(t) &= \frac{q_s}{m_s}  \left[ \mathbf{E}(\mathbf{X_s}(t),t) + \mathbf{V_s}(t) \times \mathbf{B}(\mathbf{X_s}(t),t) \right]\\
+ \frac{\mathrm{d}}{\mathrm{d}t} \mathbf{X_s}(t) &= \mathbf{V_s}(t) .
+
+.. attention::
+
+   TODO: write proper relativistic form
+
+where :math:`\mathbf{X_s}, \mathbf{V_s}` describe the ensemble of particle species' position and velocity.
+
+.. note::
+
+   Particles in a particle species can have different charge states in PIConGPU.
+   In the general case, :math:`\frac{q_s}{m_s}` is not required to be constant per particle species.
+
+Electro-Magnetic PIC Method
+---------------------------
+
+**Fields** such as :math:`\mathbf{E}(t), \mathbf{B}(t)` and :math:`\mathbf{J}(t)` are discretized on a regular mesh in Eulerian frame of reference (see [EulerLagrangeFrameOfReference]_).
+
+The distribution function :math:`f_s(\mathbf{x},\mathbf{v},t)` for **particles** is described in Lagrangian frame of reference.
+It is sampled with *markers*, sometimes referred to as *macro-particles*.
+These markers carry a spatial shape of order :math:`n` and a delta-distribution in momentum space.
+In most cases, these shapes are implemented as B-splines and are pre-integrated to *assignment functions* :math:`S` of the form:
+
+.. math::
+
+   S^0(x) = \big\{ \substack{1 \qquad \text{if}~0 \le x \lt 1\\ 0 \qquad \text{else}}
+
+   S^n(x) = \left(S^{n-1} * S^0\right)(x) = \int_{x-1}^x S^{n-1}(\xi) d\xi
+
+PIConGPU implements these up to order :math:`n=4`.
+
 References
 ----------
 
+.. [EulerLagrangeFrameOfReference]
+        Eulerian and Lagrangian specification of the flow field.
+        https://en.wikipedia.org/wiki/Lagrangian_and_Eulerian_specification_of_the_flow_field
+
 .. [BirdsallLangdon]
         C.K. Birdsall, A.B. Langdon.
         *Plasma Physics via Computer Simulation*,