Potts Model Simulation

The Potts model is a generalization of the famous Ising model in statistical physics. It describes interacting magnetic spins on a lattice, where each lattice site can be in one of q possible states (instead of just two in the Ising model).

More can be found on Wikipedia:

https://en.wikipedia.org/wiki/Potts_model

Key Concepts

• Lattice Sites: Each cell in the grid represents a magnetic spin
• Q-States: The number of possible spin orientations (colors in the visualization)
• Temperature: Controls thermal fluctuations - higher temperature means more random behavior
  • Low Temperature: Spins align to minimize energy, creating ordered domains
  • High Temperature: Thermal energy dominates, leading to random configurations
  • Phase Transitions: At critical temperatures, the system undergoes sudden changes between ordered and disordered phases
• Energy: Lower energy configurations are more stable
• Magnetization Measures ordering in the system (how aligned the spins are)

Interface Overview

Interaction Modes

1. Pan Mode (Key: 1) - Default mode for viewing and navigation

• Mouse Wheel: Zoom in/out
• Left Click + Drag: Pan around the lattice
• Reset Zoom Button (Q): Return to default zoom level

2. Q-Draw Mode (Key: 2) - Paint lattice sites with specific spin states

Opens color picker window with available q-states. Works while simulation is running for real-time interaction.

• Left Click/Drag: Paint cells with selected q-state
• Brush Size: Adjustable from 1×1 to large brushes
• Color Selection: Click color buttons to choose q-state

3. Temperature Draw Mode (Key: 3) - Paint local temperature variations

Opens temperature chooser window. Creates "hot spots" and "cold spots" in the lattice.

• Temperature Range: 0.1 to 5.0 (adjustable)
• Left Click/Drag: Paint temperature coefficients
• Visual Feedback: Darker = colder, brighter = hotter

4. Field Source Mode (Key: 4) - Place magnetic field sources that bias local spins

• Left Click: Place field source at clicked position
• Field Parameters:
  • Target Q-State: Which spin state the field promotes
  • Strength: How strong the bias is (0.1 to 5.0)
  • Radius: Spatial extent of the field effect
  • Falloff: Whether field strength decreases with distance
• Field List Window: Manage existing field sources
  • Delete Sources: Use trash button in field list

Parameter Settings

Basic Parameters

• Grid Size (5-100): Physical size of the lattice (L×L)
• Q-States (2-12): Number of possible spin orientations
• Temperature (0.1-5.0): Global temperature of the system
• Speed (1-6): Simulation speed level

Advanced Parameters

• Neighbor Range (1-10): How many nearest-neighbor shells participate in interactions
• Distance Falloff: Whether to use 1/r² falloff for distant neighbors
• Interaction Matrix: Custom interaction strengths between q-state pairs

Interaction Matrix Editor (Key: M)

Advanced feature for custom spin interactions.

• Matrix Values: Set how strongly different q-states interact
  • Positive Values: Attractive interactions (like prefers like)
  • Negative Values: Repulsive interactions (like repels like)
  • Zero Values: No interaction
• Load/Save: Import/export interaction matrices
• Reset: Return to identity matrix (standard Potts model)

Keyboard Shortcuts

• 1: Pan Mode
• 2: Q-Draw Mode  
• 3: Temperature Draw Mode
• 4: Field Source Mode
• Space: Play/Pause toggle
• R: Reset simulation
• Z: Single step
• Q: Reset zoom
• M: Open Interaction Matrix Editor
• Escape: Close focused popup window