Monte Carlo Simulation of the Ising Model
Long before Jupyter Notebooks or Google Colab existed, Professor Young was a pioneer in using visualization techniques to aid scientific understanding. He developed a Java applet that allowed researchers and students to explore and visualize the Monte Carlo simulation of the Ising model—one of the earliest interactive tools for studying statistical mechanics in a hands-on way. This simulation visually represents the behavior of spins in a lattice and how they evolve based on temperature fluctuations.
Modern Accessibility
As modern-day browsers no longer support Java applets, this short video serves as supplemental material to showcase what the original visualization looked like. The original source code is preserved on the Internet Archive, ensuring that future researchers and students can access and study this pioneering work.
Understanding the Ising Model
Each small box in the spin field represents one of two possible states:
- Si = -1 (white)
- Si = 1 (blue)
The energy of the system is given by the formula:
E = – \sum<i,j> Si Sj
where <i,j> denotes all pairs of nearest neighbors on the lattice. The energy per spin (E/N, where N = L² is the number of spins) varies with temperature:
- At infinite temperature, E/N = 0 (random spin configuration)
- At zero temperature, E/N = -2 (all spins aligned)
Adjusting Temperature
Users can control the temperature in two ways:
- Typing a positive real number in the temperature field.
- Adjusting the thermometer using the mouse.
The critical temperature of the two-dimensional Ising model is:
T_Crit = 2 / ln(1 + \sqrt{2}) \approx 2.269
Initially, the temperature is set to this value.
Observations at Different Temperatures
Above the Critical Temperature (T > T_Crit)
- Spin arrangement becomes nearly random, regardless of the initial state (“Init cold”, “Init warm”, or “Init hot”).
- The system fluctuates quickly, and magnetization is approximately zero.
- At infinite temperature, the spin arrangement is completely random.
Below the Critical Temperature (T < T_Crit)
- Starting with “Init cold” (all Si = -1):
- A few small clusters of blue (Si = 1) appear.
- The system has a nonzero negative magnetization.
- If started with all blue (Si = 1), there would be a net positive magnetization.
- Two thermodynamic states emerge: up spin (positive magnetization) and down spin (negative magnetization).
- The system stays in one state depending on initialization.
- Starting with “Init hot” or “Init warm”:
- The system initially fluctuates between up and down spin states.
- Large clusters of each spin type form.
- Over time, one state dominates due to random thermal fluctuations.
- The final state (up spin or down spin) is determined with equal probability.
Near the Critical Temperature (T \approx T_Crit)
- Large clusters of aligned spins form.
- Fluctuations occur slowly due to an increasing “correlation length.”
- At the critical point, an infinitely large system’s correlation length diverges.
Interactive Exploration
When running the Java applet, users can input different temperatures and observe:
- How spin configurations evolve over time.
- The impact of temperature on spin clustering and magnetization.
- The transition from an ordered to a disordered state as temperature increases.
This simulation provides an intuitive and visual way to understand phase transitions and statistical mechanics principles within the Ising model.