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

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

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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

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

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)

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

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

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)

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.

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

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

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?

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

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

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

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

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.

Continuous Phase TransitionsSupercritical Fluids

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

Critical TransitionsMagnets

Magnet

Water

Liquid

Gas

With magnets, random fluctuations determine the direction of ‘spin’ when magnetism is restored

ReviewEquilibrium States

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

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

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

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

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

Emergence in Living Systems

Slime Mold

Fish

Emergent behavior occurs whether or not there is volition on the part of the “particles”

1st Order Phase Transitions Traffic Patterns

Variables:

Inflow on Main Road

Inflow from On-Ramp

Predicted by Computer Model

1st Order Phase TransitionsCrime Rate

Variables:

Criminal Percentage of Population

Level of Social/Economic Deprivation

Severity of Criminal Justice System

1st Order Phase TransitionMarriage Rate

Variables:

Proportion of Population Married

Economic Incentive to Stay Married

1st Order Phase TransitionMarriage Rate

Variables:

Proportion of Population Married

Economic Incentive to Stay Married

Strength of Social Attitudes

Continuous Phase TransitionFormation of Alliances

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