Phase Transitions · Complex Physics

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Complex Physics

Statistical mechanics, percolation, and fractals

$ Statistical MechanicsBoltzmann distribution, partition functions, and ensemble averages $ Metropolis AlgorithmMonte Carlo sampling using the Metropolis-Hastings method $ Phase TransitionsCritical behavior and order parameters in phase transitions $ Mean-Field ResultsMean field approximations for interacting many-body systems $ 1D Ising Model and Transfer Matrix MethodExact solutions using transfer matrix techniques $ Critical PhenomenaUniversality, scaling laws, and renormalization $ PercolationPercolation thresholds, critical exponents, and Bethe lattice theory $ FractalsFractal dimension, the Mandelbrot set, and box-counting methods $ Self-Organized CriticalitySandpile models and power-law distributions $ NetworksGraph theory, small-world networks, and scale-free topology $ Agent-Based Models and Stochastic SimulationSimulating emergent behavior from individual agent rules $ EconophysicsStatistical physics methods applied to financial markets

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