Particle Dynamics Simulation and Animation

This project implements the modelization made in A mathematical perspective on Transformers (2023) paper. In this paper, authors propose a new mathematical model for transformers, based on the dynamics of particles. This repository contains the code to generate the following animation to illustrate the findings.

Project Structure

• simulation.py: Contains the code to simulate the particle dynamics and compute the trajectory and energy of the system.
• animation.py: Contains the code to animate the particle movement on a sphere and the evolution of interaction energy using Manim.
• trajectory_full.npy: Numpy file storing the computed trajectory of particles.
• energy_full.npy: Numpy file storing the computed interaction energy over time.
• simulation_params.txt: Text file storing the simulation parameters.
• energy_evolution.png: Plot of the energy evolution over time.

Requirements

• Python 3.x
• Numpy
• Matplotlib
• Manim

Running the Simulation

To run the simulation and generate the trajectory and energy data, execute:

python simulation.py

This will create trajectory_full.npy, energy_full.npy, and energy_evolution.png files in the project directory.

Running the Animation

To visualize the particle movement and energy evolution, execute:

manim -pql animation.py FullModelScene
manim -pql animation.py EnergyGraphScene

This will generate and display the animations for the particle dynamics and energy evolution.

Simulation Parameters

The simulation parameters can be adjusted in simulation.py:

• n: Number of particles

• d: Dimensions (for $\mathbb{S}^2$, use d=3)

• beta: Inverse temperature

• dt: Time step

• steps: Number of integration steps