2D Gaussian Scalar Field — Exact Spectral OU WebGL2

Initializing WebGL2 spectral sampler…

running

Controls

Resolution255²

These sizes use N+1 = 2ˢ, so the Dirichlet DST-I is implemented by a power-of-two odd-extension FFT. The displayed canvas may still be full-screen; this is the lattice resolution.

Mass m0.400 Langevin timestep dt0.0200 dt numeric0 to 1e9 accepted Mode dynamicsphysical OU

Use CPU noise texture

One exact Markov update is performed per frame. The inverse DST itself is a multipass GPU transform, not repeated local thermalization.

Display contrast |φ| scale1.50

The diagnostic reads back the GPU mode texture and rendered field. It checks κ|φ̂pq|² shell means, CPU-vs-GPU IDST consistency, and one-step autocorrelation against the selected ρpq.

No diagnostic run yet.

Model implemented

The field lives on interior sites i,j = 1,…,N with zero Dirichlet boundary values outside the square. All parameters are in lattice units, a = 1.

S[φ] = 1/2 ∑ φ (m² − Δ_D) φ

κpq = m² + 4sin²(πp/(2(N+1))) + 4sin²(πq/(2(N+1)))

The GPU stores orthonormal sine coefficients φ̂pq, not real-space φ. For the physical overdamped Langevin equation dφ = −Kφ dt + √2 dW, every mode is updated exactly:

φ̂'pq = exp(−κpq dt) φ̂pq
      + sqrt((1 − exp(−2κpq dt))/κpq) ηpq,
ηpq ~ N(0,1)

The real-space field is then rendered using the orthonormal DST-I:

φij = ∑p∑q φ̂pq up(i) uq(j),
up(i) = sqrt(2/(N+1)) sin(π p i/(N+1)).

Interior painted pins from the local-update demo are intentionally absent: arbitrary held regions destroy this diagonal mode representation. Rectangular Dirichlet boundaries are exactly supported.