Continuous Complex-Valued Cellular Automata

Inspired by Lenia and quantum mechanics, this cellular automaton evolves according to the Schrödinger equation, with a convolutional Hamiltonian.

The kernel and initial state are defined within the relevant funtions inside main.cpp.

Cellular.Automata.Example.mp4

Visualisation

SFML is used for the visualisation of the system at a certain time. The system is made up of a 2-D grid where each point stores a complex value. In the visualisation, the complex number is converted into a colour. The hue of the colour corresponds to the phase of the complex number; the brightness is proportional to the arctangent of the magnitude.

Theory

The system transforms according to: $$i \frac{\partial \psi}{\partial t} = h * \psi$$ Where $h*\psi$ represents the convolution of the functions $h$ and $\psi$. $h$ is a real valued function while $\psi$ may be complex valued.

From this we can derive that: $$\psi(t + \Delta t) = \mathrm{exp}(-i \Delta t ~ h *) \psi(t) $$

In the frequency domain, making use of the convolution theorem, this becomes: $$\tilde{\psi}(t + \Delta t) = \mathrm{exp}(-i \Delta t ~ \tilde{h}) \tilde{\psi}(t) $$ Where $\tilde{f}$ represents the Fourier transform of a function.

If $h$ is a real valued function, than the operator $h*$ is hermition. This fact means that $\mathrm{exp}(-i \Delta t ~ h *)$ is a unitary operator. As a consequence the total square magnitude (probability density in quantum mechanics) of the space is conserved. This is useful as it ensures a stable evolution of the system.

Installation

Ensure you have the following dependency installed before running the project:

• OpenCV