The Ising model is a fundamental mathematical model in statistical mechanics used to describe ferromagnetism—the phenomenon where atomic spins tend to align in the same direction, producing a net magnetic field.
In its simplest form, the Ising model consists of a lattice of discrete “spins”, where each spin can be in one of two states: +1 (up) or –1 (down). Spins interact with their neighbors, and the model calculates the total energy of a configuration based on:
- Spin alignment (neighboring spins prefer to align),
- And possibly an external magnetic field.
Key Features:
- Captures the phase transition between ordered (magnetized) and disordered (non-magnetized) states.
- In 1D, it does not show spontaneous magnetization at non-zero temperature.
- In 2D (solved by Lars Onsager), it exhibits a critical temperature where magnetization appears or disappears.
- Forms the basis for more complex models in magnetism, lattice gases, and even neural networks and social dynamics.
Examples of Applications:
- Modeling the behavior of magnetic materials near the Curie point.
- Simulating binary alloys, adsorption on surfaces, or protein folding.
- Used in computational physics and Monte Carlo simulations.
The Ising model, despite its simplicity, provides profound insights into collective behavior, phase transitions, and critical phenomena in many-body systems.