Ising Model Simulation

Exploring Ferromagnetism, Phase Transitions, and Critical Phenomena

Real-Time Simulation
Phase Transition Sweep
Magnetization Plot

The Ising Model

The Ising model is a mathematical model of ferromagnetism in statistical mechanics. Each site on a lattice has a magnetic spin that can be either up (+1) or down (-1). Spins interact with their nearest neighbors, preferring to align. At low temperatures, all spins align (ordered/ferromagnetic). At high temperatures, thermal fluctuations randomize spins (disordered/paramagnetic).

Critical Temperature Tc = 2.269: The system undergoes a continuous phase transition from ordered to disordered state. At this precise temperature, the system exhibits scale-invariant fluctuations and critical opalescence - domains of all sizes appear and disappear.

2.27
Temperature (kT/J)
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Magnetization (M)
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Energy (E/J)
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Susceptibility (χ)
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Monte Carlo Steps
Critical
Phase

Phase Transition Analysis

Watch the dramatic phase transition as temperature sweeps from cold to hot. Below Tc ≈ 2.269, the system spontaneously magnetizes (symmetry breaking). Above Tc, thermal fluctuations destroy order. At Tc, the system exhibits critical behavior with diverging correlation length and susceptibility.

  • Low T (T < 2.0): Ordered phase - spins aligned, high magnetization
  • Critical T (T ≈ 2.27): Critical point - domains of all sizes, maximum fluctuations
  • High T (T > 2.5): Disordered phase - random spins, zero net magnetization
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Current Temperature
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Current Magnetization

Time Evolution of Order Parameter

The magnetization M = |Σ spins| / N is the order parameter for the ferromagnetic transition. At low temperatures, M approaches 1 (all spins aligned). At high temperatures, M fluctuates around 0. Near Tc, you'll see critical fluctuations where M oscillates dramatically as the system explores both ordered and disordered configurations.

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Average |M|
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Fluctuations (σ)