The Cost of Curvature
Companion to Dimensional Landauer Bound
Simulation
Manifold Projection Cost
The cost of maintaining a low-dimensional representation. Particles (microstates) are confined to a 1D manifold. Bending the manifold (drag left/right) increases geometric curvature, requiring larger control forces to suppress orthogonal fluctuations. This manifests as increased thermodynamic work, visualized by the red shift.
The Physics
Particles undergo 2D Brownian motion but are constrained to a 1D manifold (the white curve). This is dimensional reduction in action—projecting high-dimensional dynamics onto a lower-dimensional representation.
The control force required to keep particles on the manifold depends on the manifold's curvature. Higher curvature = more force = more heat dissipated.
The Dimensional Landauer Bound
Standard Landauer says erasing a bit costs kT ln 2. But there's a second cost: the geometric contraction costof projecting dynamics onto a lower-dimensional manifold.
W = kT ln 2 × bits + kT × CΦ
Try This
- Straight manifold: Drag to the left edge. Particles flow smoothly with minimal dissipation (green status).
- Curved manifold: Drag to the right edge. Watch particles heat up (turn red) as control forces fight to keep them on track.
- Release: The manifold relaxes back to flat. Dissipation drops. This is why biology favours coherent, low-curvature dynamics.
Key Insight
Curvature = Heat. When you bend a manifold, you're not just changing coordinates—you're increasing the thermodynamic cost of maintaining that representation against thermal noise. This explains why neural systems use oscillatory synchronization: coherent dynamics naturally flatten the effective manifold, reducing dissipation.