cdet=1/10, dis=1.3
The general pattern of lightning is called a Lichtenberg figure. The pattern in such figures is a fractal pattern and this same pattern can be found on tree-branches, tree-roots, high-voltage sparks, lung bronchi, blood vessels and arteries.
Here's some Maple 9 code which simulates a Lichtenberg figure, based on the code on this web page.
> p:={};cdet:=1/10;
> L:=proc(z1,z2,dis)
> local l,nz,midx,midy;
> global p,cdet;
> if dis < cdet then
> p:=p union {plot([[Re(z1),Im(z1)], [Re(z2),Im(z2)]],x=-1..1,scaling=constrained,color=magenta)};
> else
> midx:=evalf((Re(z1)+Re(z2))/2);
> midy:=evalf((Im(z1)+Im(z2))/2);
> midx:=midx + (RandomTools[Generate](float(range=0..1,digits=2))-1/2)*dis;
> midy:=midy + (RandomTools[Generate](float(range=0..1,digits=2))-1/2)*dis;
> nz:=evalf(midx+midy*I);
> L(z1,nz,dis/2);
> L(nz,z2,dis/2);
> fi;
> p;
> end:
> display(L(-1,1,1.3));
Model some injective lightning path using a Lichtenberg pattern f(z). What is the field strength throughout f's domain and range? Is the field strength complex analytic?
The answer to the above question comes as a corollary from the author's father's Ph.D. thesis, which examines the problem of the "crack" in the generally anisotropic disk.
This pattern is also the pattern exhibited when isotropic materials "crack" under high tensile pressure, for example, like when glass or crystal matter breaks under tension.
Accordingly, the dielectric material can be seen as an insulating isotropic (or anisotropic) disk on the complex plane, normal to the two electrodes E1=(-1,0) and E2=(1,0).
A "spark" is easily seen to consist of a sequence of linear elements Δs which form a chain from E1 to E2. Hence considering a single such linear element Δs, it can be seen as a "crack" on the (insulating) disk, having orientation n (see page 10 on the Ph.D.).
The field strength on this linear element will then correspond to the (tensile) tensors σx, σy and τxy (on the same page).
Our case corresponds to uniformly distributed normal charge, which are cases B.2 and C.2, of normal charge on the boundary. The field strength at the edges of Δs will then correspond to points σ1 and σ2 on the unit circle, after the corresponding Muskhelishvili transformation, which correspondingly will have arguments θ1 and θ2.
The final answer then comes from the diagrams on page 47. Specifically, at the edges of each linear segment Δs the field strength becomes unbounded on both the isotropic and the anisotropic cases, hence it cannot possibly be analytic there on either case.
Here's an animation done with the above Maple 9 code, with the electrodes E1 and E2, as above:
Here is a different approach using a Gaussian process that approximates a "Brownian bridge" by Robert Israel, using the following Maple 13 code:
> N := 100; c := 0.2; nframes:= 20;
> C:= Matrix(N,N,(i,j) -> evalf(i*(N+1-j)/(N+1)^2),shape=symmetric);
> A:= LinearAlgebra[LUDecomposition](C, method=Cholesky);
> for k from 1 to nframes do
> Vx:= Statistics[Sample](Normal(0,1),N);
> Vy:= Statistics[Sample](Normal(0,1),N);
> Wx:= A . Vx^%T; Wy := A . Vy^%T;
> pts:= [[0,0], seq([i/(N+1)+c*Wx[i],c*Wy[i]],i=1..N),[1,0]];
> frame[k]:= plot(pts,colour=black);
> end do:
> BB:= plots[display]([seq(frame[k],k=1..nframes)], insequence=true):
>
>
> plots[display]([BB,plottools[disk]([0,0],0.05,colour=black),
> plottools[disk]([1,0],0.05,colour=black)],
> scaling=constrained, axes=none);
Brownian bridge simulation of Lichtenberg patterns