Criticality Part 2: Bifurcations, Universality, and Self-Organization Introduction Recap of Part 1: tipping point vs. continuous criticality What’s missing: the geometric picture, why different systems share exponents, systems that find criticality on their own Goal: connect phenomenological classification to deeper mathematical frameworks Bifurcations: The Deterministic Skeleton Phase Portraits and Fixed Points State space, trajectories, attractors Fixed point stability: linearization and eigenvalues Basins of attraction: which initial conditions lead where Catalog of Local Bifurcations Saddle-Node Bifurcation Normal form: $\dot{x} = r + x^2$ Two fixed points collide and annihilate The canonical “tipping point”—no smooth return Examples: extinction thresholds, financial crises, regime shifts in ecosystems Transcritical Bifurcation Normal form: $\dot{x} = rx - x^2$ Two fixed points exchange stability Connection to exponential growth model from Part 1 Pitchfork Bifurcation Supercritical: $\dot{x} = rx - x^3$ (smooth symmetry breaking) Subcritical: $\dot{x} = rx + x^3$ (discontinuous jump) Connection to continuous vs. tipping point criticality Hopf Bifurcation Fixed point gives way to limit cycle Supercritical vs. subcritical variants Examples: oscillations in predator-prey systems, electrical circuits Global Bifurcations Homoclinic and heteroclinic bifurcations Why local analysis isn’t always enough Catastrophe theory: Thom’s classification Interactive: Bifurcation Explorer Phase portrait that deforms as control parameter changes Visualize fixed point creation/destruction/stability exchange Show corresponding bifurcation diagram From Bifurcations to Statistical Mechanics The Role of Many Degrees of Freedom Single ODE → lattice of coupled units Fluctuations and noise enter naturally Mean-field theory: averaging over neighbors recovers bifurcation-like equations Landau Theory of Phase Transitions Free energy as a function of order parameter Equilibrium: minimize free energy Taylor expansion near transition: $F(m) = a(T)m^2 + bm^4 + \ldots$ Connection to pitchfork bifurcation Order Parameter Behavior Near Criticality Power-law scaling: $m \sim |T - T_c|^\beta$ Critical exponent $\beta$ characterizes the transition Mean-field prediction vs. reality: fluctuations matter in low dimensions Correlation Length and Susceptibility Correlation length $\xi$: how far do fluctuations propagate? $\xi \sim |T - T_c|^{-\nu}$ diverges at criticality Susceptibility $\chi \sim |T - T_c|^{-\gamma}$: response to perturbations diverges Critical slowing down: relaxation time also diverges Universality: Why Details Don’t Matter The Puzzle Liquid-gas transition in water and magnetization in iron follow same power laws Microscopic physics completely different How can this be? What Determines Universality Class Dimensionality of space Symmetry of order parameter (scalar, vector, etc.) Range of interactions (short vs. long range) What doesn’t matter: lattice structure, precise interaction strengths, chemical details The Renormalization Group Idea Coarse-Graining Block spins: average over small regions Ask: what effective Hamiltonian describes the coarse-grained system? Iterate: zoom out repeatedly RG Flow and Fixed Points Parameters flow under coarse-graining Fixed points of the flow = scale-invariant systems = critical points Behavior near fixed point determines critical exponents Intuitive Picture At criticality, zooming out doesn’t change statistical properties The system “looks the same” at all scales This self-similarity is the origin of power laws Scaling Relations Critical exponents aren’t independent Relations like $\alpha + 2\beta + \gamma = 2$ (Rushbrooke) Follow from self-consistency of RG picture Reduces number of independent exponents Table of Universality Classes Class Examples $\beta$ $\nu$ $\gamma$ 2D Ising Uniaxial magnets, lattice gas 1/8 1 7/4 3D Ising Liquid-gas, binary mixtures ~0.326 ~0.630 ~1.237 Mean-field High dimensions, long-range 1/2 1/2 1 2D Percolation Porous media, networks 5/36 4/3 43/18 3D Percolation Composite materials ~0.41 ~0.88 ~1.80 Fractals at Criticality Why Fractals Emerge No characteristic length scale at $p_c$ or $T_c$ Structures must be self-similar Fractal geometry is the natural description Fractal Dimension Definition: $M(L) \sim L^{d_f}$ where $d_f$ is fractal dimension For percolation cluster at $p_c$: $d_f < d$ (spatial dimension) 2D percolation: $d_f = 91/48 \approx 1.896$ The cluster fills space “incompletely” Fractal Dimension as Critical Exponent Related to other exponents through scaling relations $d_f = d - \beta/\nu$ connects geometry to thermodynamics Measuring $d_f$ gives information about universality class Examples of Critical Fractals Percolation clusters at threshold Ising domain boundaries at $T_c$ Coastlines and river networks (approximately) Diffusion-limited aggregation Interactive: Box-Counting on Critical Cluster Generate percolation cluster at $p = p_c$ Overlay boxes of decreasing size Plot $\log N$ vs. $\log (1/\epsilon)$ Slope gives fractal dimension Self-Organized Criticality The Tuning Problem In standard criticality: must tune control parameter precisely to $p_c$ But power laws appear everywhere in nature Who is doing the tuning? SOC: Criticality Without Tuning Some systems naturally evolve toward critical state Separation of timescales is key: slow drive, fast relaxation System “self-organizes” to the boundary between stability and instability The BTW Sandpile Model Rules 2D lattice, each site has “sand grains” $z_{ij}$ Slowly add grains at random sites If $z_{ij} \geq 4$, site topples: loses 4 grains, neighbors each gain 1 Toppling can trigger neighbors → avalanche Grains fall off at boundaries Emergent Behavior System reaches statistical steady state Avalanche size distribution follows power law: $P(s) \sim s^{-\tau}$ No characteristic avalanche size Always poised at criticality Other Examples of SOC Earthquakes: Gutenberg-Richter law $P(E) \sim E^{-b}$ Forest fires: burned area distribution Mass extinctions: species loss events Neural avalanches in brain activity Solar flares: energy release distribution Controversies and Limitations Is true SOC common, or are power laws often misidentified? Many claimed power laws don’t survive rigorous statistical testing Alternative mechanisms: finite-size effects, superposition, tuning to different points SOC requires specific conditions: slow drive, fast relaxation, no characteristic scale in driving Interactive: Sandpile Simulation Add grains one by one Visualize avalanches as they propagate Plot avalanche size histogram (log-log) Show approach to power-law distribution Synthesis: A Unified View of Criticality The Common Thread Criticality = diverging correlation length = loss of characteristic scale Power laws, fractals, and scale invariance are manifestations of same phenomenon Universality: only symmetry and dimension matter near the fixed point Hierarchy of Descriptions Bifurcation Theory (deterministic, low-dimensional) ↓ add noise, many degrees of freedom Mean-Field / Landau Theory (averaged, approximate) ↓ include fluctuations properly Renormalization Group (full critical behavior) ↓ special dynamics Self-Organized Criticality (criticality as attractor) Mapping the Landscape Framework Tipping Point Continuous Transition Bifurcations Saddle-node, subcritical Supercritical pitchfork, transcritical Phase transitions First-order Second-order (continuous) Order parameter Discontinuous jump Power-law approach to zero Correlation length Finite Diverges Hysteresis Often present Absent Critical exponents Not applicable Universal values Early Warning Signals Approaching tipping points: critical slowing down Variance and autocorrelation increase before transition Flickering between states Potential for prediction and intervention Open Questions Criticality in non-equilibrium systems: active matter, driven systems Quantum criticality: phase transitions at zero temperature Criticality and computation: is the brain critical? Why might this be advantageous? Machine learning on critical systems: can neural networks learn critical exponents? Conclusion Criticality connects dynamical systems, statistical mechanics, and geometry The same mathematical structures appear across physics, biology, and social systems Understanding criticality gives tools for prediction, control, and recognizing universal patterns Much remains unknown, especially in non-equilibrium and living systems Sources Nonlinear Dynamics and Chaos by Steven H. Strogatz Statistical Mechanics by James Sethna (especially Chapter 12 on RG) Scale Invariance and Beyond edited by Dubrulle, Graner, Sornette How Nature Works by Per Bak (original SOC book) Critical Phenomena in Natural Sciences by Didier Sornette Clauset, Shalizi, Newman: “Power-Law Distributions in Empirical Data” (arXiv:0706.1062)