https://github.com/micahbrun/continuous-complex-valued-cellular-automata

Continuous Complex-Valued Cellular Automata: Inspired by Lenia and quantum mechanics, this cellular automaton evolves according to the Schrödinger equation, with a convolutional Hamiltonian. This project uses SFML for visualisation.
https://github.com/micahbrun/continuous-complex-valued-cellular-automata

cellular-automata cellularautomata convolution eigen opencv quantum-mechanics sfml

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Continuous Complex-Valued Cellular Automata: Inspired by Lenia and quantum mechanics, this cellular automaton evolves according to the Schrödinger equation, with a convolutional Hamiltonian. This project uses SFML for visualisation.

• Host: GitHub
• URL: https://github.com/micahbrun/continuous-complex-valued-cellular-automata
• Owner: MicahBrun
• Created: 2023-11-28T20:40:29.000Z (over 2 years ago)
• Default Branch: main
• Last Pushed: 2024-01-04T21:55:34.000Z (over 2 years ago)
• Last Synced: 2024-01-04T22:51:29.429Z (over 2 years ago)
• Topics: cellular-automata, cellularautomata, convolution, eigen, opencv, quantum-mechanics, sfml
• Language: C++
• Homepage:
• Size: 35.2 KB
• Stars: 0
• Watchers: 1
• Forks: 0
• Open Issues: 0
• Metadata Files:
  • Readme: README.md

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README

          
# 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. 

https://github.com/MicahBrun/ContinuousComplexValuedCellularAutomata/assets/55516590/c8526b67-51bc-4f8e-9663-20c422b3e83e

## 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.

## Installation

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

- [OpenCV](https://opencv.org/)