Add Fokker-Planck Decapode

examples/chemistry/fokker_planck.jl

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+# We use here the formulation studied by Jordan, Kinderlehrer, and Otto in "The
+# Variational Formulation of the Fokker-Planck Equation" (1996).
+
+# The formualation they studied is that where the drift coefficient is the
+# gradient of (some potential) Ψ.
+
+# Load libraries.
+using Catlab, CombinatorialSpaces, Decapodes, DiagrammaticEquations
+using CairoMakie, ComponentArrays, LinearAlgebra, MLStyle, MultiScaleArrays, OrdinaryDiffEq
+using GeometryBasics: Point3
+Point3D = Point3{Float64}
+using Arpack
+
+# Specify physics.
+Fokker_Planck = @decapode begin
+  (ρ,Ψ)::Form0
+  β⁻¹::Constant
+  ∂ₜ(ρ) == ∘(⋆,d,⋆)(d(Ψ)∧ρ) + β⁻¹*Δ(ρ)
+end
+
+# Specify domain.
+s = loadmesh(Icosphere(6));
+sd = EmbeddedDeltaDualComplex2D{Bool, Float64, Point3D}(s);
+subdivide_duals!(sd, Barycenter());
+
+# Compile.
+sim = eval(gensim(Fokker_Planck))
+fₘ = sim(sd, nothing)
+
+# Specify initial conditions.
+# Ψ must be a smooth function. Choose an interesting eigenfunction.
+Δ0 = Δ(0,sd)
+Ψ = real.(eigs(Δ0, nev=32, which=:LR)[2][:,32])
+# We require that ρ integrated over the surface is 1, since it is a PDF.
+# On a sphere where ρ(x,y,z) is proportional to the x-coordinate, that means divide by 2π.
+ρ = map(point(sd)) do (x,y,z)
+  abs(x)
+end / 2π
+
+constants_and_parameters = (β⁻¹ = 1e-2,)
+u₀ = ComponentArray(Ψ=Ψ, ρ=ρ)
+
+# Run.
+tₑ = 20.0
+prob = ODEProblem(fₘ, u₀, (0, tₑ), constants_and_parameters)
+soln = solve(prob, Tsit5(), progress=true, progress_steps=1)
+
+# Verify that the PDF is still a PDF.
+s0 = dec_hodge_star(0,sd);
+@info sum(s0 * soln(0).ρ)
+@info sum(s0 * soln(tₑ).ρ) # ρ integrates to 1
+@info any(soln(tₑ).ρ .≤ 0) # ρ is nonzero
+
+# Save solution data.
+@save "fokker_planck.jld2" soln
+
+# Create GIF
+function save_gif(file_name, soln)
+  time = Observable(0.0)
+  fig = Figure()
+  Label(fig[1, 1, Top()], @lift("ρ at $($time)"), padding = (0, 0, 5, 0))
+  ax = LScene(fig[1,1], scenekw=(lights=[],))
+  msh = CairoMakie.mesh!(ax, s,
+    color=@lift(soln($time).ρ),
+    colorrange=(0,1),
+    colormap=:jet)
+
+  Colorbar(fig[1,2], msh)
+  frames = range(0.0, tₑ; length=21)
+  record(fig, file_name, frames; framerate = 10) do t
+    time[] = t
+  end
+end
+save_gif("fokker_planck.gif", soln)
+