A SMALL UNIVERSE, ONE UPDATE AT A TIME
Watch order emerge.
Explore a fluctuating field. Tune its interactions.
Find the patterns hidden in the noise.
S = ∑x [ ½∑μ(φx+μ − φx)² + ½m²φx² + ¼λφx⁴ ]Periodic boundaries · lattice spacing a = 1 · temperature absorbed into S
Live field φ(x, y)
PAUSED−20+2
02 / OBSERVABLES
Mean field M—Spatial average of φ
Mean |M|—Sampled order parameter
χ|M|—V(〈M²〉 − 〈|M|〉²)
Binder cumulant U₄—1 − 〈M⁴〉 / (3〈M²〉²)
Metropolis acceptance—
0 samples collected
03 / MEASUREMENTS
The field, in numbers.
Magnetization distribution P(M)
Spatial correlation G(r) = 〈φ(x)φ(x+r)〉
Average over both axes and sampled fields. Unconnected correlation.
Phase scan 〈|M|〉 vs m²
Independent short runs at 11 masses, using the current size, λ and update method. 300 warm-up + 600 measurement sweeps per mass.
How this experiment works
The field is real-valued, with Boltzmann weight exp(−S). Metropolis proposes φ′ = φ + uniform(−δ, δ), accepting with min(1, exp(−ΔS)). For the displayed action, the local quadratic kinetic coefficient is 2, giving ΔS = (2 + m²/2)(φ′² − φ²) − (φ′ − φ)∑neighbors φ + λ(φ′⁴ − φ⁴)/4.
Hybrid mode follows one Metropolis sweep with three embedded-sign Wolff clusters. Each bond uses the current edge magnitudes: p = 1 − exp(−2|φᵢφⱼ|), between equal-sign sites. Clusters flip signs while Metropolis also changes amplitudes.
This browser version corrects the original local kinetic coefficient (4 → 2) and the original cluster probability, which used the seed value for every edge. Consequently results need not match the original script. −1.1 is an exploration preset, not an established critical mass. A finite lattice has no true singular transition.
Displayed susceptibility uses |M| fluctuations as in your script; the signed definition is V(〈M²〉−〈M〉²). Samples remain correlated, and no error bars or autocorrelation correction are estimated. Short scans are exploratory. For precision, increase thermalization and statistics in a scientific workflow. All simulation runs locally in your browser.