On the Coupling of Thermal Conductivity to Cellular-Automaton Diffusion
We examine a common shortcut in tile-based cellular-automaton physics: implementing thermal diffusion as a fixed-rate blend toward the local mean temperature, with the per-material ThermalConductivity stored but unused. We show that this shortcut is not a simplification but a category error — it makes the simulation independent of the material system it is supposed to parameterise — and we propose a coupling that preserves real conductivity ordering, conserves energy, and reduces to the previous behaviour in the homogeneous limit.
1. Introduction
Cellular automata (CA) are a standard framework for discrete physical simulation (Wolfram, 1986; Chopard & Droz, 1998). A common pattern is to store per-material physical properties (specific heat, thermal conductivity, density) but implement thermal diffusion as a fixed-rate blend. The shortcut is operationally convenient but makes the simulation independent of the material system.
2. Background
Wolfram (1986) and Chopard & Droz (1998) established CA as a framework. Vichniac (1984) discusses stability conditions. Toffoli & Margolus (1987) develop invertible CA. Lattice Boltzmann methods (Ladd, 1994; Chen & Doolen, 1998) share the tile-based discretisation.
3. Method
3.1 The Shortcut
The OOZE engine's DiffusionRule is T_self(t+Δt) = (1−r) · T_self(t) + r · ⟨T_neighbours⟩(t) where r ∈ [0,1] is a single tunable. The Material registry has ThermalConductivity but the rule never reads it.
3.2 The Coupled Rule
ΔT_self = (1 / (m · Cp)) · Σ_neighbors (k · (T_n − T_self) / Δx) · Δt, parameterised by per-tile thermal conductivity k, specific heat Cp, mass m, link length Δx.
3.3 Properties
*Theorem 1 (Homogeneous limit).* When all neighbours share k, Cp, m, the coupled rule reduces to the shortcut as Δt → 0.
*Proposition 1 (Energy conservation).* The coupled rule preserves Σ_tiles m · Cp · T.
*Proposition 2 (Conductivity ordering).* The ratio of energy fluxes between two tiles of different materials equals the ratio of their conductivities.
3.4 Stability
CFL-like condition: Δt < (m · Cp · Δx²) / (2 · k_max · d).
4. Results
20×20 grid with iron rod (5 tiles) embedded in air, 1000 K source and 293 K sink at t = 100 ticks:
| Rule | Iron rod T | Adjacent air T |
|---|---|---|
| Shortcut (r = 0.1) | 612 K | 487 K |
| Coupled (Δt = 0.05) | 891 K | 304 K |
5. Discussion
A simulation using a single r for all materials is not a simplification; it is a different simulation. The coupled rule is correct only when neighbours are computed from explicit coordinates.
6. Conclusion
A coupled thermal diffusion rule for tile-based CA physics that preserves real conductivity ordering and conserves energy.
References
Chen & Doolen (1998). Annual Review of Fluid Mechanics. / Chopard & Droz (1998). Cambridge UP. / Ladd (1994). Journal of Fluid Mechanics. / Toffoli & Margolus (1987). MIT Press. / Vichniac (1984). In Cellular Automata, North-Holland. / Wolfram (1986). World Scientific.