2D Ising Model: Simulation & Critical Phenomena Analysis

Project Overview

This project implements a high-performance Markov Chain Monte Carlo (MCMC) simulation of the 2D Ising Model to investigate statistical mechanics and critical phenomena. It features a physics-grade simulation engine, comprehensive thermodynamic analysis, and interactive visualization tools.

Key Physics Explored:

• Phase Transitions: Second-order ferromagnetic-paramagnetic transition.
• Critical Phenomena: Divergence of correlation length and susceptibility near $T_c$.
• Finite-Size Scaling: Extraction of critical exponents ($\nu, \gamma, \beta$).
• Hysteresis: Dynamic magnetic memory and coercivity.
• Universality: Validation of the 2D Ising universality class.

Key Results

1. Phase Transition

We observe the classic symmetry breaking at the Onsager critical temperature $T_c \approx 2.269$. The specific heat and susceptibility show sharp peaks that scale with lattice size.

2. Critical Scaling & Universality

Using Finite-Size Scaling (FSS), we collapsed data from lattice sizes $L=16$ to $L=64$ onto a single universal curve, confirming the scale-invariance of the system near criticality.

Metric	Measured	Theory
$T_c$	$2.2677 \pm 0.002$	$2.2692$
$\gamma/\nu$	$1.672$	$1.75$

3. Magnetic Hysteresis

Below $T_c$, the system exhibits magnetic memory. We quantified the "loop area" as a dynamic order parameter, vanishing exactly at the phase transition.

4. Spatial Correlations

We measured the spin-spin correlation function $G(r)$, observing exponential decay in the disordered phase and power-law decay near $T_c$.

Phase 2: Hopfield Networks & Associative Memory

Building on the statistical mechanics simulation of the 2D Ising Model, Phase 2 implements a Hopfield Network to investigate the emergence of associative memory. By extending the Ising formalism to include non-local, programmable couplings ($J_{ij}$), we bridge the gap between Spin Glasses and neural computation.

Theoretical Foundation: The Hopfield network is mathematically isomorphic to an Ising model with long-range interactions. The "Energy" of the spin system is equivalent to a Lyapunov function for the network dynamics, ensuring that the system always evolves towards energy minima. These minima represent stored "memories."

Key Implemented Features

• Hebbian Learning Rule: Weights are constructed via the outer product of target patterns: $W_{ij} = \frac{1}{N} \sum_{\mu} \xi_i^\mu \xi_j^\mu$.
• Asynchronous Dynamics: Neurons update stochastically or sequentially, minimizing the global energy $E = -\frac{1}{2} \sum_{i,j} W_{ij} s_i s_j$.
• Pattern Corruption & Restoration: Capability to recover perfect patterns from inputs degraded by noise (e.g., 30-50% flipped bits).
• Capacity Analysis: Empirical verification of the storage capacity limit ($C \approx 0.14N$).

Experimental Results:

We quantified the network's performance using orthogonal bit patterns on a $10 \times 10$ lattice ($N=100$). The results validate standard Hopfield theory.

Experiment	Condition	Measured Success	Theoretical Expectation
Robustness	Noise $\le$ 25%	$100%$	Perfect Retrieval
Robustness	Noise = 50%	$\approx 0%$	Unstable (Random)
Capacity	Low Load ($P=3$)	$100%$	Global Minima Stable
Capacity	High Load ($P \approx 0.14N$)	$< 20%$	Spin Glass Phase (Overload)

1. Pattern Restoration (Associative Recall)

We demonstrated the network's error-correction capability by initializing it with a corrupted version of a stored pattern (e.g., the letter 'A' with varying noise levels). The network dynamics successfully evolved the state down the energy gradient to the original clean pattern.

2. Network Capacity & Interference

We analyzed the network's performance as the number of stored patterns ($P$) increased.

• Success Regime: For $P < 0.14N$, the network reliably retrieves memories.
• Failure Regime: As $P$ exceeds the capacity limit, "crosstalk" between patterns creates spurious local minima (spin glass phase), causing the network to converge to "hallucinated" mixed states rather than pure memories.

3. Pattern Generation & Visualization

We utilized a custom utility to generate orthogonal bit patterns (e.g., 'Y', 'H', 'E', 'I', 'A') to rigorously test the network's ability to discriminate between distinct memories.

Running the Neural Experiment

To reproduce the Hopfield network experiments:

# 1. Run Single Pattern Validity Test
python experiments/experiment_1_single_pattern.py

# 2. Run Capacity Analysis
python experiments/experiment_2_capacity.py

# 3. View Pattern Utilities
python experiments/test_utils_visual.py

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Interactive Dashboard

Explore the physics in real-time with the included Streamlit dashboard:

pip install -r spin-equilibrium/requirements.txt
streamlit run spin-equilibrium/viz/dashboard.py

Features:

• Live Controls: Adjust Temperature ($T$), Field ($B$), and Coupling ($J$).
• Real-time Plotting: Watch Magnetization and Energy evolve.
• Phase Diagram Tracker: See your current state vs. the Onsager solution.

Usage

1. Run Full Simulation Support

Reproduce all experiments (Thermodynamics, Hysteresis, Scaling):

python experiments/run_simulation.py
python experiments/hysteresis_loop.py
python experiments/fss_run.py

2. Generate Plots

Create publication-quality figures from collected data:

python experiments/generate_plots.py
python experiments/fss_analyze.py

3. New Package Structure (Refactored)

A clean, installable version of the core logic is provided in ising_simulation/.

cd ising_simulation
pip install -e .

Repository Structure

• spin-equilibrium/: Original source code and modules.
• experiments/: Scripts for running physics experiments.
• results/: Data, Figures, and Animations.
• ising_simulation/: Refactored professional Python package.

License

MIT License.