Field initialization for relativistic species¶
As explained in PIC algorithms, if a net charge is present at the beginning of the simulation, the initial electromagnetic fields are computed. For static charge distributions, the solution of Poisson’s equation will be necessary to find the initial electrostatic field. If the initial charge has a non-zero initial speed, in general the electric and magnetic field should be computed solving the full set of Maxwell’s equations or equivalently the potentials equations. In some physical setups of interest, one or more relativistic species are injected in a plasma. In these cases, the computation of the initial electromagnetic fields can be reduced to the solution of a modified version of Poisson’s equation.
The relativistic Poisson’s equation¶
From the continuity equation
Thus, if a simulation starts with
In the case of a static charge distribution, i.e.
and then be integrated to find the initial electric field:
In general when the initial current
However, if a species is already relativistic when the simulation starts, e.g. a relativistic electron bunch, its initial electromagnetic fields can be computed through a simplified procedure, described in [Vay2008], [Londrillo2014], [Massimo2016] and [Marocchino2018].
An important assumption of this calculation is that the species is highly relativistic, moving in the positive
where
In the relativistic species rest frame
where the Laplacian operator is computed in the reference frame
The vector potential in the species rest frame can be set to zero:
Lorentz transformation of the four-vector
allows to transform the derivatives in Eq. (68) as
The partial derivative along the
Equation (68) can thus be rewritten as
here informally referred to as the relativistic Poisson’s equation. In Smilei, as for Eq. (68), the solution of the relativistic Poisson’s equation is performed through the conjugate gradient method.
Once the potential
From all these relations, the electromagnetic field can be computed as usual, through the definitions of potentials
or in more compact form:
From the previous equations, it can be inferred that, in a 1D cartesian geometry, the fields computed through this procedure equal those obtained through the standard Poisson’s problem.
This can also be inferred from the relativistic transformations of fields, which conserve the
Recommendations for relativistic species field initialization¶
In Smilei, each species can independently benefit from this field initialization procedure. Its field will be initialized when the species will start to move, in order not to interfere with the other species’ dynamics. The initialized fields will be superimposed to the electromagnetic fields already present in the simulation. To have physically meaningful results, we recommend to place a species which requires this method of field initialization far from other species, otherwise the latter could experience instantaneous unphysical forces by the relativistic species’ fields.
Remember that the transverse field of a moving charge with relativistic factor
A relativistic mean velocity in the