Critical Mass: How One Thing Leads to Another by Philip Ball (2004)

1. Critical Mass: How One Thing Leads to Anotherby Philip Ball (2004) Thesis: • It is possible to develop a “science of society” by applying theories from statistical physics to explain (and sometimes predict) collective human behavior • This is not a unified theory of human behavior, but rather the application of specific tools for specific purposes • Models are interesting, but cannot explain real world behavior without an underlying theory. A “convenient allegory” is not sufficient

2. Simulation Models and Social Systems • Crime Rate • Transmission of Culture • Social networks • Wars • The Economy • The Internet • Crowds • Traffic • Growth of Cities • Formation of Firms • Political and Business Alliances • Voting Collective Behavior Phase Transitions Emergence

3. Ball’s Theory • Phase Transitions and Emergent Patterns are generic properties of collective systems: • Type of ‘particle’ doesn’t matter • Free will doesn’t matter (at least not much)

4. Application • Simulation models allow us to specify the “rules” governing the system • Models produce patterns that are similar, and sometimes mathematically equivalent to those seen in Natural and Living Systems • Ball’s theory provides a basis from which to argue that similarities result from the same rules, and can explain some aspects of collective human behavior • Prediction is problematic: • Models don’t reflect all the variables • Susceptibility to random fluctuations makes systems inherently unpredictable

5. Outline • How did Statistical Physics evolve? • Shift from Newtonian Determinism to Statistical Science: “The Law of Large Numbers” • Mechanics of Phase Transitions • Leap from Equilibrium systems to Non-Equilibrium growth processes (non- living) • Leap from Non-Living to Living non-equilibrium growth processes (single-cell organisms) • Leap from Single-cell organisms to Humans

6. Relationship Between Characteristics of Gases Robert Boyle mid-1600’s • Boyle’s Law: For a fixed amount of gas kept at a fixed temperature, P and V are inversely proportional (while one increases, the other decreases)

7. Kinetic Theory of Gases • Daniel Bernoulli, 1738: • Pressure: is a result of collisions between molecules moving at different velocities • Temperature: altering the temperature changes speed of molecules Derived Boyle's law using Newton's laws of motion. His work was ignored. Most scientists believed that the molecules in a gas stayed more or less in place, repelling each other from a distance, held somehow in the ether.

8. Kinetic Theory of GasesMaxwell’s Probability Distribution More than a century later: 1859 James Maxwell Ludwig Boltzmann • Began working with Bernouilli’s theory • Intuited it wasn’t necessary to know the details – only the probability distribution • Made physics “statistical” in concept. Boltzmann did the mathematics

9. Phase TransitionsChange over Time Johannes Diderik van der Waals, 1873 Phase Diagram for Water Under normal atmospheric conditions, as temperature increases over time, the state changes from ice, to liquid, to vapor

10. Phase Transitions1st Order Constant Pressure • Density does not change gradually as temperature increases. At a transition point, it changes abruptly. • Same particles but different arrangement. This phenomenon is not a “tendency” of the individual particles. It is a property of the whole, caused by attractive/repulsive forces between particles. • Similar to “Tipping Point” or catastrophe?

11. Continuous Phase Transitions2nd Order • At a certain temperature and pressure (“Critical Point”), it becomes possible to gradually transform a gas to a liquid without going through an abrupt phase transition • High temperature disrupts the forces of attraction and repulsion between molecules Phase Diagram for Water

12. Continuous Phase TransitionsCritical Exponent • At the Critical Point, certain properties “diverge” off to zero or infinity • Approaching the Critical Point, the rate of change of these properties increases exponentially (Power Law) • Critical Pressure: Compressibility (resistance to reducing volume) • Critical Temperature: Heat Capacity (energy needed to raise temperature by one degree) • Difference in Density between Liquid and Gas • Rate of Divergence is called the “Critical Exponent” Power Law Expressed Mathematically: g(x) = x-τ τ = Critical Exponent

13. Phase Diagram of Water Compressibility Heat Capacity

14. Phase TransitionsUniversality • Liquids have different Critical Point values • But all have the same Critical Exponent (rate of change approaching the Critical Point) • Same Critical Exponent, same Universality Class

15. Continuous Phase TransitionsMagnets • Magnets lose their magnetism when heated, and regain it when cooled. • Rate of change increases exponentially approaching “Critical Point” and when that point is reached, drops to zero. • A certain class of magnets also has the same Critical Exponent as Liquids: Same Universality Class

16. Continuous Phase TransitionsSupercritical Fluids • As the Critical Point is approached, the distinction between liquid and gas dwindles steadily to nothing. • Beyond the Critical Temperature and Pressure, substance becomes a “Supercritical Fluid”: neither Liquid nor Gas. • Density is not uniform throughout: random motions of atoms cause chance fluctuations Computer Simulation: Black regions represent Liquid, white regions represent Gas.

17. Continuous Phase TransitionsSupercritical Fluids

18. Critical TransitionsCooling Down a Supercritical Fluid • As the Critical Point is approached from the other direction: • Extreme sensitivity to random fluctuations • Long-range correlations – all particles act together • Density of Supercritical Fluid is not uniform. Random Fluctuations determine whether Supercritical Fluid transitions to a Liquid or Gas when passing through the Critical Point

19. Critical TransitionsMagnets Magnet Water Liquid Gas With magnets, random fluctuations determine the direction of ‘spin’ when magnetism is restored

20. ReviewEquilibrium States

21. Non-Equilibrium Growth Processes IlyaPrigogene, 1970’s • Similarities between Bifurcations and Critical Transitions: • A sudden global change to a new steady state • Random fluctuations determine path at each bifurcation point • Different outcomes despite same initial conditions

22. Non-Equilibrium Growth ProcessesSnowflake Formation • Crystals are formed during transition from one equilibrium state (Vapor) to another (Ice) • During the transition, system is far from equilibrium • Uniqueness reflects the different paths between the Vapor and Ice state

23. Phase Transitions in Snowflake Formation Under unusual atmospheric conditions (e.g. extremely low temperatures, humidity levels), strikingly different snowflake patterns begin to form at certain thresholds

24. Fractals in Living Systems Bacillus subtilis bacteria Computer-generated from model of DLA process Electrodeposition (Diffusion-Limited Aggregation Process) • Certain bacteria produce a fractal pattern of growth • Same fractal dimension as patterns produced by non-biological Diffusion Limited Aggregation (DLA) growth processes • Suggests these formation processes share same essential features

25. Phase Transitions in Living SystemsMorphology Diagram Concentric Circles Dense tumor-like pattern Broadly spread Fractal Branching Thin radiating branches Changing nutrient levels and mobility produce abrupt changes in the growth pattern Dotted Line = transition from immobile to mobile particles Grey Lines = phase transition boundaries

26. Emergence in Living Systems Slime Mold Fish Emergent behavior occurs whether or not there is volition on the part of the “particles”

27. 1st Order Phase Transitions Traffic Patterns Variables: Inflow on Main Road Inflow from On-Ramp Predicted by Computer Model

28. 1st Order Phase TransitionsCrime Rate Variables: Criminal Percentage of Population Level of Social/Economic Deprivation Severity of Criminal Justice System

29. 1st Order Phase TransitionMarriage Rate Variables: Proportion of Population Married Economic Incentive to Stay Married

30. 1st Order Phase TransitionMarriage Rate Variables: Proportion of Population Married Economic Incentive to Stay Married Strength of Social Attitudes

31. Continuous Phase TransitionFormation of Alliances