Manna Sandpile — Stochastic Toppling & SOC

The Manna model is a stochastic sandpile where each toppling site redistributes its 2 grains to randomly chosen neighbors (not fixed like BTW). Despite the stochasticity, the model self-organizes to a critical state with power-law avalanche distributions. Manna universality class is distinct from BTW: exponents τ≈1.27 (avalanche size) vs τ≈1.0 (BTW mean-field).

Toppling rule:
If z(i) ≥ 2: site i fires
z(i) → z(i) − 2
Each grain → random neighbor

BTW vs Manna:
BTW: deterministic (fixed dirs)
Manna: stochastic (random dirs)
Same threshold (z_c = 2)

Critical exponents (2D):
τ ≈ 1.27 (avalanche size)
D ≈ 2.75 (fractal dim)
z ≈ 1.77 (dynamic exp)

Conservation:
Bulk: conserved
Boundary: grains dissipate
→ SOC maintained by balance

Statistics:
Grains added: —
Avalanches: —
Largest: —
Mean size: —