I would like to show the effect of a stochastic electromagnetic field on a relativistic charged particle, using a manipulate box.
The field should be randomly varying in time and in space, and be defined as a superposition of plane waves with a specific frequency distribution. The polarisation state of each plane wave, their amplitude and phase constant should be fully random. And finally the frequency distribution should be Lorentz invariant : \begin{equation} df = \frac{1}{\pi^2 c^3} \; \omega^2 \, d\omega. \end{equation}
Here's a MWE code that shows the effect of a very simple constant and uniform EM field :
a = {0, 1, 0}; (* Polarisation vector should be orthogonal to wave orientation *) b = {0, 0, 1}; FieldE[t_, x_, y_, z_] := a FieldB[t_, x_, y_, z_] := b Velocity[t_] := {x'[t], y'[t], z'[t]} Force[t_, q_] := q(FieldE[t, x[t], y[t], z[t]] + Cross[Velocity[t], FieldB[t, x[t], y[t], z[t]]]) Acceleration[t_, q_] := Force[t, q] - (Force[t, q].Velocity[t]) Velocity[t] Motion[q_, v0_, theta_, phi_] := NDSolve[{ x''[t] == Sqrt[1 - Velocity[t].Velocity[t]]{1, 0, 0}.Acceleration[t, q], y''[t] == Sqrt[1 - Velocity[t].Velocity[t]]{0, 1, 0}.Acceleration[t, q], z''[t] == Sqrt[1 - Velocity[t].Velocity[t]]{0, 0, 1}.Acceleration[t, q], x[0] == 0, y[0] == 0, z[0] == 0, x'[0] == v0 Sin[theta]Cos[phi], y'[0] == v0 Sin[theta]Sin[phi], z'[0] == v0 Cos[theta] }, {x, y, z}, {t, 0, 10}, Method -> Automatic, MaxSteps -> Automatic] Trajectory[t_, q_, v0_, theta_, phi_] := ParametricPlot3D[Evaluate[{x[s], y[s], z[s]}/.Motion[q, v0, theta, phi]], {s, 0.001, t} ] Manipulate[ Show[ Trajectory[t, q, v0, theta Pi/180, phi Pi/180], PlotRange -> {{-3, 3}, {-3, 3}, {-3, 3}}, Boxed -> True, Axes -> True, AxesOrigin -> {0, 0, 0}, SphericalRegion -> True, Method -> {"RotationControl" -> "Globe"}, ImageSize -> {700, 700} ], {{t, 0, "Time"}, 0, 10, 0.01}, {{q, Pi, "Coupling constant"}, -10, 10, 0.01}, {{v0, 0.5, "Initial velocity"}, 0, 1, 0.01}, {{theta, 0, "Theta"}, 0, 180, 0.1}, {{phi, 0, "Phi"}, 0, 360, 0.1} ] So what would you suggest to change the definitions of a, b, FieldE and FieldB (the first 4 lines of the code above), so to have a stochastic electromagnetic field ?
EDIT 1 : Here's a precision about the plane wave superposition. Consider a single plane wave :
X = {x, y, z} FieldE[t_, x_, y_, z_] := A a Sin[k.X - omega t + phase] FieldB[t_, x_, y_, z_] := A Cross[k, a] Sin[k.x - omega t + phase] where a is a random polarisation vector, that should stay orthogonal to k (the wave vector). A is a random amplitude. We could write k = omega u where u is a random unit vector. And phaseis a random phase constant (between 0 and $2 \pi$).
This plane wave is a solution to Maxwell equations. Now just superpose a large number of these plane waves, each one with a random amplitude A, random orientation of a(that stays orthogonal to k), random orientation of the vector k(or u), and random phase. Now, my only constraint is a distribution of frequencies ($\omega$) that should be Lorentz invariant (this constraint comes from the basic postulate of Stochastic Electrodynamics and has nothing to do with Maxwell equations).
I would like to define at least a superposition of a few (3 or 4 ?) random plane waves, independantly of any frequency distribution. That would already be great.
