Critical Mass: How One Thing Leads to Another by 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
Slide 2
Simulation Models and Social Systems Crowds Traffic Growth of
Cities Formation of Firms Political and Business Alliances Voting
Crime Rate Transmission of Culture Social networks Wars The Economy
The Internet Collective Behavior Phase Transitions Emergence
Slide 3
Balls Theory Phase Transitions and Emergent Patterns are
generic properties of collective systems: Type of particle doesnt
matter Free will doesnt matter (at least not much)
Slide 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 Balls 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 dont reflect all the variables Susceptibility to random
fluctuations makes systems inherently unpredictable
Slide 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
Slide 6
Relationship Between Characteristics of Gases Boyles Law: For a
fixed amount of gas kept at a fixed temperature, P and V are
inversely proportional (while one increases, the other decreases)
Robert Boyle mid-1600s
Slide 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.
Slide 8
Kinetic Theory of Gases Maxwells Probability Distribution Began
working with Bernouillis theory Intuited it wasnt necessary to know
the details only the probability distribution Made physics
statistical in concept. Boltzmann did the mathematics More than a
century later: 1859 James Maxwell Ludwig Boltzmann
Slide 9
Phase Transitions Change over Time Under normal atmospheric
conditions, as temperature increases over time, the state changes
from ice, to liquid, to vapor Phase Diagram for Water Johannes
Diderik van der Waals, 1873
Slide 10
Phase Transitions 1 st Order 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? Constant Pressure
Slide 11
Continuous Phase Transitions 2 nd 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
Slide 12
Continuous Phase Transitions Critical 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 - = Critical Exponent
Slide 13
Phase Diagram of Water CompressibilityHeat Capacity
Slide 14
Phase Transitions Universality 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
Slide 15
Continuous Phase Transitions Magnets 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
Slide 16
Continuous Phase Transitions Supercritical 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.
Slide 17
Continuous Phase Transitions Supercritical Fluids
Slide 18
Critical Transitions Cooling 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
Slide 19
Critical Transitions Magnets Magnet Water LiquidGas With
magnets, random fluctuations determine the direction of spin when
magnetism is restored
Slide 20
Review Equilibrium States Phase Transitions: 1 st OrderAbrupt
change from one stable state to another as threshold point is
reached Continuous Phase Transitions: 2 nd OrderGradual change from
one of two stable density states to indeterminate density as
critical point is reached. Critical Exponent reflects the rate of
change for certain properties approaching the critical point.
Different substances having the same Critical Exponent are said to
be of the same Universality Class. Critical Transitions: 2 nd
OrderChange from indeterminate state to one of two possible density
states. Choice of state is determined by random fluctuations.
Slide 21
Non-Equilibrium Growth Processes 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 Ilya Prigogene, 1970s
Slide 22
Non-Equilibrium Growth Processes Snowflake 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
Slide 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
Slide 24
Fractals in Living Systems 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
Bacillus subtilis bacteria Computer- generated from model of DLA
process Electrodeposition (Diffusion-Limited Aggregation
Process)
Slide 25
Phase Transitions in Living Systems Morphology Diagram 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 Fractal Branching Dense
tumor-like pattern Thin radiating branches Broadly spread
Concentric Circles
Slide 26
Emergence in Living Systems Slime Mold Fish Emergent behavior
occurs whether or not there is volition on the part of the
particles
Slide 27
1 st Order Phase Transitions Traffic Patterns Predicted by
Computer Model Variables: Inflow on Main Road Inflow from
On-Ramp
Slide 28
1st Order Phase Transitions Crime Rate Variables: Criminal
Percentage of Population Level of Social/Economic Deprivation
Severity of Criminal Justice System
Slide 29
1 st Order Phase Transition Marriage Rate Variables: Proportion
of Population Married Economic Incentive to Stay Married
Slide 30
1 st Order Phase Transition Marriage Rate Variables: Proportion
of Population Married Economic Incentive to Stay Married Strength
of Social Attitudes
Slide 31
Continuous Phase Transition Formation of Alliances
