using Plots
using QuadGK
using LaTeXStrings

V(x) = 1 - cos(x)
η = .1

# plot(title="Invariant distribution of perturbed OL") 
plot(title="") 
for η ∈ [0, .5, 1, 2]
    ρ(q) = exp(-V(q)) * quadgk(y -> exp(V(q + y) - η*y), 0, 2π)[1]
    Z = quadgk(ρ, - π, π)[1]
    plot!(q -> ρ(q) / Z, -π, π, label=L"\eta = " * string(η), lw=2)
end
plot!(xlims = (-π, π), grid=false, ylims=(0, .4), framestyle=:box)
savefig("figures/invariant_perturbed_ol.pdf")