Poincare Diffusion

Welcome to the Poincaré Diffusion Lab, where we explore the fascinating dynamics of particles within the hyperbolic geometry of the Poincaré disk model. Our lab uses computer simulations to visualize and analyze diffusion processes, random walks, and the influence of Lie algebra symmetries on particle behavior. Through these interactive tools students can gain deeper insights into hyperbolic spaces and their unique properties.

Simulation Programs

Our suite of simulation programs is designed to model various aspects of random walk particle dynamics on the Poincaré disk. All simulations are conveniently accessible through our intuitive Poincaré Diffusion Launcher, ensuring a seamless and user-friendly experience.

1. RandomWalk04.exe

Purpose:
Simulates the random walk diffusion of particles within the Poincaré disk. This program models how a single particle moves over time, adhering to the principles of hyperbolic geometry.

Key Features:

• Particle Initialization: Begins with particle centrally located at the origin.

• Random Step Generation: Particle move in randomized directions, constrained by hyperbolic space parameters.

• Real-Time Visualization: Observes the path traced out by the particle.

2. Diffusion04.exe

Purpose:
This program simulates random walk diffusion for a large number of particles. It also tracks the radial distribution function of particles.

Key Features:

• Diffusion Mechanics: Introduces a large number of particles.

• Statistical Analysis: Computes the Radial Distribution Function (RDF) to assess particle distribution.

• Comprehensive Visualization: Offers serval user controlled parameters to adjust the behavior of the system.

3. HyperbolicRepulsion03.exe

Purpose:
Compares and contrasts random walk diffusion on the Poincaré disk with that on the Euclidean disk. The simulation serves to visual an apparent "hyperbolic repulsion'' which is an illusion induced by the  Poincaré metric

Key Features:

• Euclidean Disk Comparison: Implements a radial repulsion to diffusion on the Euclidean disk to visually compare with the behavior on the Poincaré disk.

• Dynamic Vector Fields: Calculates and applies repulsion vectors based on real-time particle positions.

• Interactive Visualization: Displays how repulsive forces influence overall particle distribution and movement.

4. Diffusionsu11generators02.exe

Purpose:
Integrates su(1,1) Lie algebra generators into the diffusion process, introducing structured symmetries and transformations that govern particle dynamics.

Key Features:

• Lie Algebra Integration: Utilizes generators K0K_0K0​, K+K_+K+​, and K−K_-K−​ to define vector fields affecting particle movement.

• Ladder Operators: Implements K+K_+K+​ and K−K_-K−​ with precise phase shifts for complex trajectory modeling.

• Enhanced Analytical Tools: Provides insights into how Lie algebra symmetries influence diffusion patterns and particle behavior.

Resources

Access the essential tools and documentation for the Poincaré Diffusion Lab below. All simulation programs are launched through the Poincaré Diffusion Launcher for ease of use.

• Poincaré Diffusion Launcher:
  The central hub for running all simulation programs. Follow the installation wizzard as with any Windows program

• Lab Manual:
  Comprehensive guide detailing the theoretical foundations, usage instructions, and analytical methods for each simulation program.

• GitHub Repository:
  Explore the source code, contribute to the project, and access additional resources on our GitHub page.

For further inquiries or support, please visit our Poincaré Diffusion Lab webpage or explore our GitHub Repository.