¹ directly?
² amount of activity? mass?
The idea that all neighbors move to the next tick simultaneously is a fundamental assumption in cellular automata in general. If you try changing that, the optimizations that allow us to simulate CAs at any kind of reasonable speed ... all stop working, pretty much. It's kind of painful even to think about.
Which means there are probably very interesting rules out there somewhere, where CAs run faster/slower depending on pattern density -- it's just going to be very tricky to explore that particular search space.
(A Doppler effect does show up in Conway's Life sometimes, but that's about as far as we get with analogies to the physical universe...!)
Quantum discord: https://en.wikipedia.org/wiki/Quantum_discord :
> In quantum information theory, quantum discord is a measure of nonclassical correlations between two subsystems of a quantum system. It includes correlations that are due to quantum physical effects but do not necessarily involve quantum entanglement.
From "Convolution Is Fancy Multiplication" https://news.ycombinator.com/item?id=25194658 :
> FWIW, (bounded) Conway's Game of Life can be efficiently implemented as a convolution of the board state: https://gist.github.com/mikelane/89c580b7764f04cf73b32bf4e94...
Conway's Game is a 2D convolution; without complex phase or constructive superposition.
Convolution theorem: https://en.wikipedia.org/wiki/Convolution_theorem :
> In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain). Other versions of the convolution theorem are applicable to various Fourier-related transforms.
From Quantum Fourier transform: https://en.wikipedia.org/wiki/Quantum_Fourier_transform :
> The quantum Fourier transform can be performed efficiently on a quantum computer with a decomposition into the product of simpler unitary matrices. The discrete Fourier transform on 2^{n} amplitudes can be implemented as a quantum circuit consisting of only O(n^2) Hadamard gates and controlled phase shift gates, where n is the number of qubits.[2] This can be compared with the classical discrete Fourier transform, which takes O(n*(2^n)) gates (where n is the number of bits), which is exponentially more than O(n^2).