From one narrow cloud to several modes
Where does the probability go?
The same small, shifted Gaussian evolves towards a mixture. Move the component means closer together or farther apart, then follow the density on a shared physical clock.
Changing k or the positions updates the target contours immediately and restarts at t = 0. The initial center stays to the left of every component mean.
KL Total variation
Solving the Fokker–Planck equation…
What the controls and numerical approximation mean
The k equally weighted components have covariance 0.65² I and means R(cos(π + 2πj/k), sin(π + 2πj/k)), j = 0,…,k−1. The position slider moves these means radially; for k = 1, it moves the single Gaussian too. At R = 0, every choice of k is exactly the same target. The new contours appear immediately during dragging; only the time evolution is recomputed in the background. Changing either target control resets time to zero.
The initial distribution is always N((−3.6,0), 0.3² I), independently of k and R. Its center is strictly to the left of every component mean throughout the slider range. The isotropic Brownian noise and physical time are unchanged. Initial KL need not be the same for different targets. More components do not automatically mean slower convergence: distances and the arrangement of the modes matter.
A deterministic finite-volume method solves ∂tρ = Δρ + div(ρ∇V), V = −log π, in the physical (x,y) plane. Scharfetter–Gummel fluxes preserve the sampled target; alternating implicit steps preserve positivity and probability. A 151 × 151 grid covers [−6,6]², with reflecting boundaries and time step 0.0125. Saved densities and diagnostics are interpolated in time. KL and total variation use the full computational grid; the drawing crops remote tails. While computing, a 101 × 101 density preview shows the fixed initial cloud and the updated target; time remains disabled until the evolution is ready.
Equal spatial units and the color scale are fixed. Color uses log(1 + density/0.005), with maximum 1/(2π·0.3²), to keep small densities visible without rescaling each frame. Solid contours use fixed absolute density levels; dashed contours show 20%, 50%, and 80% of the target’s peak. These numerical trajectories do not measure the optimal log-Sobolev constant.