Apr 30, 2010

Living Within Limits - Intro (Tragedy of the Commons)

Author: Garrett Hardin

"I teach only two things: the cause of human sorrow and the way to become free of it" -The Buddha

If you are like me, you do not know who Garrett Hardin is.  However, like me, you may have heard of his most famous work: Tragedy of the Commons (1968)

For those unfamiliar with the reference, this was an ecology article, the heart of which can be summed up by a parable.  In the story, there was a piece of pasture land shared among many herdsman, but owned by none.  Each herdsman is free to use as much of the pasture to raise as many animals as his heart so desired.  Hardin then described the 'tragedy' to follow: 

Living within Limits: Ecology, Economics, and Population TaboosIt is to be expected that each herdsman will try to keep as many cattle as possible on the commons. Such an arrangement may work reasonably satisfactorily for centuries because tribal wars, poaching, and disease keep the numbers of both man and beast well below the carrying capacity of the land. Finally, however, comes the day of reckoning, that is, the day when the long-desired goal of social stability becomes a reality. At this point, the inherent logic of the commons remorselessly generates tragedy. 

As a rational being, each herdsman seeks to maximize his gain. Explicitly or implicitly, more or less consciously, he asks, "What is the utility to me of adding one more animal to my herd?" This utility has one negative and one positive component.
  • Negative: the pasture is slightly degraded by each additional animal. - accrued evenly to all herdsman (importantly, the man in question would only accrue a small fraction of actual costs)
  • Positive: the herder receives all of the proceeds from each additional animal
[T]he rational herdsman concludes that the only sensible course for him to pursue is to add another animal to his herd. And another; and another . . . . But this is the conclusion reached by each and every rational herdsman sharing a commons. Therein is the tragedy. Each man is locked into a system that compels him to increase his herd without limit -- in a world that is limited.
This conclusion has been played out not only in agriculture, but some non-environmental areas of study as well.  Consider the following example.

FDIC Insurance was created as a result of bank-runs during the Depression.  The U.S. government (rightly) surmised that by giving an explicit Federal-Guarantee on the funds of depositors at banks, the probability of bank panic would decrease.

This theory played out for a long time, exactly as the federal government believed.  However, some criticized the creation of the FDIC because it essentially created a 'common pasture' for the banks and depositors.  The depositors no longer had an incentive to analyze the credit-worthiness of their local credit unions, thrifts, and banks - all deposits became identical (backed by the FDIC).  The banks no longer had to worry about depositors withdrawing money if the bank was being 'too risky' - so indeed they began buying up riskier assets and giving out riskier loans.  Quite likely, the FDIC played a large role in creating the 'Too Big to Fail' problem we now face, because there essentially could not be a failure - when banks succeed, more profits; when they fail, FDIC pays... Tragedy
(Tragedy did ensue... some believe the S&L crisis was caused by FDIC insurance, and perhaps even the current recession, which originated from excessive bank lending/risk-taknig)
But I digress.  Living Within Limits carries on the discussion began in Tragedy of the Commons, and Hardin makes one more plea to the rest of us for constraint (for our own sakes).  In the tradition of thermodynamics, here is something to leave you with in this introduction/rant:

Hardin's Second Law of Human Ecology:
There's no away to throw to.

Apr 27, 2010

Evolution is Not Optimal Solution?

While reading about Evolutionarily Stable Strategies (ESS), I came across an example... and then I got to thinking...

Background:  One species with 2 members, with differing traits
(Points corresponding to action performed in parenthesis)
  1. Hawks - behave aggressively
    • If opponent runs, eat food (+50)
    • If opponent fights back, they will fight to near-death
      • 1 always wins (+50), 1 always loses (-100)
  2. Doves - behave submissively
    • If attacked, will run away (+0)
    • If not attacked, will make threatening display
      • 1 always wins (+40), 1 always loses (-10)

Scenario 1 - Doves Only
  • Average of 15 points per confrontation to each member
  • However, susceptible to Hawk mutation
    • One Hawk will receive 50 points per encounter
    • More food will result in increased Hawk population

Scenario 2 - Hawks Only
  • What if Hawks take over?
  • Average of -25 points per confrontation to each member (DISASTER)
  • Population would go extinct (Fighting is costlier than the food)

Scenario 3 - Equilibrium
  • 5:7 Dove to Hawk ratio
  • Stable population balance
  • Average of 6.25 points per confrontation to each member

Why is this so interesting?

The real world is probably closer to Scenario 3 than Scenario 1.  However, Scenario 1 is BY FAR, the better world to live in (15 points vs 6.25 points).  Evolution led us to where we are, but quite likely there are better ways to improve life in our communities through concerted efforts to 1) cooperate, and 2) resist the urge to act in our own self-interest at the expense of others - Love Thy Neighbor

Note:  if the doves decide to make no threatening display, the expected value (per member, per encounter) would increase from 15 to 20 - Ninth Commandment (You shall not bear false witness against your neighbor)


Apr 26, 2010

Deep Simplicity - Chapter 7

Life Beyond
  • Jim Lovelock - developed idea that living and nonliving components of terrestrial environemnt interact in network that maintains conditions suitable for life on Earth (Gaia)
    • Suggest that the best way to search for life was 'entropy reduction'
      • Life creates islands of order in the midst of chaos
      • Search for non-equilibrium gases in atmosphere
      • Search for 1/f noise (information giving)
  •  Gaia - idea of Earth as a self-regulating system
    • Gaia: A New Look at Life on Earth
    • NOT synonymous with idea of Gaia as God or Mother Earth
    • Simply states idea of living & nonliving processes working together in self-organizing matter to create and preserve order
    • Evolutionarily speaking, change in biological species not only affect other species, but the entire planetary system
  • Daisyworld (Faint Young Sun Paradox)
    • Paradox - During earlier periods, the Sun existed at only 70% of present heat levels.  Around 4.5B years ago, the Sun settled down to present temperature.  However liquid water (and life) existed prior to the rise in temperature, and persisted throughout the ordeal.  How??
    • Life manipulated the physical environment for its own benefit
    • Simulation - Phase 1
      • Initially - Sun is weak, surface is barren and gray
      • As Sun increases, black daisies grow and thrive
        • Absorb radiant energy
      • As heat increases, white daisies also grow
      • Two population eventually reaches equilibirium and work together to regulate surface temperature (black/white surface to absorbe/reflect light)
    • Simulation - Phase 2
      • If temperature increases, white daisies gain advantage
        • Increase in white daisies will cause decrease in temperature
        • Equilibrium is restored
        • Vice versa
    • Simulation - Phase 3
      • Temperature increases to the point where even white daisies can no longer survive
      • Surface becomes barren
    • Key - Feedback allows life to manipulate the larger environment
  • Real world examples of Daisyworld-type scenarios exist
    • Dimethyl sulfide (DMS)
    • Microscopic life in oceans regulate Earth's climate
    • If algae become active, cloud cover increases
    • Less sunlight available for photosynthesis
    • Biological activity declines

Deep Simplicity - Chapter 6

Facts of Life

Darwinian Evolution
  • Offspring resemble parents - traits passed to next generation
  • Imperfect process - variations exist in each generation
  • More individuals born than survive to reproduce
    • Selects for those who are best 'fit'
  • Competition is between members of same species
  • Evolutionarily stable strategies (ESS) - Hawk vs Dove (p192)
    • Evolution arrives at ESS 
    • ESS is NOT OPTIMAL OUTCOME
    • Applies to stability
    • Red Queen Effect - run as fast as possible to stay in same place (Leigh Van Valen p194)
      • Same number of frogs & flies, but w/ stickier tongues & slipperier bodies
    • Fitness Landscape
      • Hills: favorable landscapes
      • Valleys: unfavorable (evolutionarily behind)ultiple 'local' peaks may exist
      • Co-evolution constantly changes the landscape
    • Most successful species become better at evolving
    • Punctuated Equilibrium - long intervals of no evolutionary change punctuated by short intervals of dramatic change
    • Least fit species most likely to change (die or adapt) - vice versa
    • Bak & Sneppen developed evolution game
      • Fitness b/w 0 and 1
      • Continuously adjusting least fit led to higher overall fitness for landscape
    • Amarel & Meyer revised game to include predator & prey
      • Found mass extinctions occur obeying a power law, without help of any outside intervention (extinctions w/o meteors)
    • Frequent small shocks, occasional large shocks, and no outside shocks all produced the same extinction pattern

    Buy Cheap, Sell Dear

    PE as a Predictor of 20-yr Returns (via Prof Robert Shiller)

    This graph presents the relationship between P/E and 20-yr annualized returns of the S&P 500 (stock market).
    • P/E (Price/Earnings) - The price you pay for each dollar earned/profit 
      • Higher PE = more expensive
      • Lower PE = less expensive
      • Prof Shiller uses an adjusted PE which is the a price paid for the average annual dollars earned over the past 10 years
    •  20-yr Annualized Returns - the average annual return over the next 20 years, after the purchase 
    Irrational ExuberanceProfessor Shiller has separated his data into 5 time periods (20-25 yrs per period).  The graph seems to provide evidence that, purchasing stocks at a lower PE (less expensive) typically results in a higher 20-yr return, than otherwise.  Just as importantly, this relationship between cheaper stocks and higher returns appears to be true for each of the 5 time periods, which covers the period from 1890-1985.

    Where are stocks now?  At today's S&P value of approximately 1200, the 10-yr PE Ratio is currently at 22.  Looking at the graph, evidence indicates that we should perhaps expect 20-yr returns of less than 5% per year if we were to purchase stocks at current prices.  Macro-level indicators such as these are important for making investment decisions because they can often help us set the right expectations about future returns.  They also act as a guiding light during oscillating periods of Fear & Greed, affording us the fortitude to buy when prices are low, and the discipline to differ purchases (or sell) when valuations become frothy.

    Deep Simplicity - Chapter 5

    Earthquakes, Extinctions, and Emergence
    • Complex system are made up of simple components
      • The way component interact is very important (creates complexity)
    • Earthquakes - Richter scale is logarithmic; each unit = factor of 30
      • Follows power-law in frequency-size
        • 10:1 ratio of size-1 to size-2 earthquakes (so on)
        • Power-law is present in fractals (coast of Normandy)
      • Scale-free: no difference b/w small & large earthquake except for size
        • Property of fractals
        • Causes are the same, only difference is magnitude & frequency
      • Ubiquity: Why Catastrophes Happen
    • Throwing rock or frozen potatoes at a wall - scalefree
      • Power-law relationship b/w frequency of large and small fragments
    • Landscape of Moon - created by impacts, which imply meteor impacts are scale-free
    • 1/f Law - size of an even is proportional to 1/(power of frequency f)
      • Soundwaves
        • White noise (completely random) - useless
        • 1/f - provides information
        • Mononote (completely uniform) - useless (ie one single musical note in perpetuity)
    • "Hundred Year Droughts" - can be followed by another "drought", or a "freeze" next year
      • Memory-less
    •  Traffic Jams - frequency and size follow power-law
      • There are jams within jams (fractals)
      • Large jams can be caused by large triggers (crashes) or small triggers (braking)
      • As density of cars increase, lowering the speed limit will decrease traffic
        • Function of time it takes to slow down (less) & speed up (more)
    • Commodity & stock prices follow 1/f & power-law (p.160-161)
      • Relatively unpredictable - small changes in interest rates can have no or much effect
    • Extinctions - KT event refers to extinction of dinosaurs (end of Cretaceous)
      • Big Five - global extinction events in past 600M years
      • Are mass extinctions 'special' or just large versions of small extinctions (scalefree)
      • Jack Sepkoski & David Raup - proved that extinctions occur with 1/f noise
        • Life is a complex system, self-organizing, with energy source, at edge of chaos
    •  Per Bak - Sandpile Model
      • On a tabletop, slowly add grains of sand one at a time (source)
      • Pile slowly forms
      • Avalanche(s) eventually appear (edge of chaos)
      • Sand falls off (sink) - 'equilibrium'
      • Amount of sand stay relatively constant - self-organized criticality
      • Each grain can trigger an avalanche or do nothing
    • Earthquakes: crusts slide against each other, strain builds (like sand), rocks deform, energy may be stored or released, marginally or heavily
    • How Nature Works: The Science of Self-Organized Criticality
    • Sandpile model can be simulated on chess board
      • Randomly stack/drop blocks on squares
      • Once a stack reach 4-high, dispense 1 block to each neighbor
        • Some will eventually begin to fall off the board
      • Creates system at precipice of chaos, with energy source and sink (self-organized)
    • Sand/rice obeys power-law (frequency and size) in avalanches
    • It also displays fractal patterns
    • Use red to denote 'energetic' grains (more chaotic)
      • Initial stages show little red
      • As red grows, density also grows
        • Density determines likelihood of large avalanche
      • Even after large avalanche, not all energy is removed
        • System remains at edge of chaos
    • Stuart Kauffman (Santa Fe Institute) - button analogy
      • Initially unconnected (uncomplex)
      • Use string to randomly connect one button to another randomly (button = node)
      • Each cluster of connected buttons is a network
        • Size of largest cluster network determines complexity of the system
        • Growth of largest cluster is slow at first
        • At approximately (# of buttons)/2, cluster grows exponentially (super cluster)
          • Phase transition
    • Grain/sand connection (thread) - gravity, friction, angles
    • Asteroid connection - Sun, planets, moons, other asteroids
    • Kauffman - developed theory on origin of life describing chemicals engaged in autocatalytic reactions
      • Form network
      • With sufficient connections, life emerges
        • Phase transition
        • Either/or (no ambiguity)
    • Kauffman - DNA cell theory
      • 30K to 100K genes in humans - 256 different kinds of specialized cells
        • Despite every cell having the same DNA
      • With 2 connections per node (DNA)
        • Sqrt(nodes) as number of cycles
        • Sqrt(nodes) as number of steps
      • 1 connection freezes
      • >2 connections lead to chaos (butterfly effect)

    Deep Simplicity - Chapter 4

    From Chaos to Complexity
    • Thermodynamics
      • Closed system - poincare recurrence (time reversibility)
      • Open system - irreversible arrow of time; dissipation of energy
    • Lars Onsager - theory of reciprocal relations
      • Temperature gradient produces concentration gradient (vice versa)
    • Ilya Prigogine - dissipative system in linear regime doesn't end in (settle at) entropy, but when entropy (dissipation) is occurring slowest
      • Linear: adult human beings
      • Nonlinear: growth of fertilized egg
        • Nonlinearity is most important when feedback is present
    • System can only be held in an 'interesting' state away from equilibrium if it is dissipative and open to the outside environment, w/ an outside energy source (sun)
      • Equilibrium is NOT interesting
      • Chaos is NOT interesting
      • Something in between is interesting (Order in the midst of Chaos)
    • Gravity is the key to creating order
    • Gravitational energy of any object that has mass is initially negative (potential energy)
      • Initial states of objects infinitely far apart has 0 gravitational energy (Gmm/r^2)
      • As they move closer, they gain kinetic energy (speed increases)
        • Comes from gravitational field
      • This leads to negative energy for the gravitational field
        • How much for a given point mass?
        • -MC^2 (exact opposite of mass energy)
    • When gravitational energy is ignored (closed box), entropy results (uniformly spread gas)
    • When it is not ignored (large clusters of gas & dust), gravity can create (pull) order and reduce entropy
      • Due to negative energy of gravitational field
    • Gravity ultimately determines direction of time
    •  Alan Turing - tried to invent universal computer (Turing Machine)
      • Read On Growth and Form as a kid (inspiration)
      • Studied how structures emerge from a spherical, featureless egg cell
        • Broken symmetry problem (convection)
        • "Chemical Basis of Morphogenesis" - reversal of thermodynamics
        • Catalysis - presence of chemical is autocataytic, inhibitive, or both
          • B(inhibitor) must diffuse more quickly than A(catalyst), but not universal
          • Runaway production of A = autocatalytic
        • Pattern created is stable (even tho its not at 'equilibrium') - dynamic process
          • As long as continual 'source' of new chemicals (catalysts) is present
          • As long as there is a 'sink' where end-products are removed
    • Belousov - concoted solution which continuously changes colors (confirmed Turing)
      • 2nd law of thermo suggests that solution should eventually settle into even distribution
      • Lotka (1910) - created formula for this type of chemical oscillation
      • Bray (1921) - showed chemical reaction based on Lotka, using iodine + oxygen
      • Volterra (1930's) - described changes in fish population as prey + predator numbers oscillate
      • They all contradicted 2nd law of thermodynamics - results not well-received
    • Zhabotinsky - reproduced Belousov reaction (BZ Reaction)
    • By changing rate of chemical inputs + removal of waste products, one can create changes in the number of 'stable' state (chaos)
    • "How Leopards Get Its Spots" - James Murray
    • Animal skin development follows Turing Model (melanin)
      • Small effects produce big changes
      • Happens at embryo stage
      • Development in embryo stage is tantamont to development
        • Even w/ identical DNA

    Deep Simplicity - Chapter 3

    Chaos Out of Order

    • Iterative Process - dependent on prior state
    • Logistic Equation - can be used to illustrate period-doubling
    • Example: how the population of a species changes
      • x (next) = B * x (current) * (1 - a)
        • B = Birth rate
        • x = Population
        • a = Premature death rate (prior to mating)
      • If B<1:  population dies
      • If B>1 & B<3:  x eventually settles at roughly 0.66 (2/3 of max population)
      • If B>3:  single attractor state becomes two (period doubles to 2)
        • Practically speaking, this results in 1 year of overpopulation, followed by 1 yr of underpopulation, etc - this cycle continues back and forth (2 stable states)
    • Robert May - first to conduct tests of behavior of logistic equations
      • B=3.4495 leads to 4 states
      • B=3.56 leads to 8 states
      • B=3.569 leads to 16 states
      • B=3.5699 (or higher) leads to infinite states
        • Mostly chaotic (infinite) beyond 3.5699, although there are small pockets of order, which eventually lead to chaos, on a smaller scale (self-similar)
        • Chaos to order, chaos to order, etc.
    • James Yorke & Tien Yien Li - "Period Three Implies Chaos"
    • Mitchell Feigenbaum - Period-doubling is not unique to logistic equations; it is product of iterative feedback process (self-referential)
      • Feigenbaum's Number - 4.669:1 is the universal ratio of period-doubling in all self-referential systems
    Fractals
    • Giuseppe Peano - constructed a curve (Peano Curve) that completely fills a plane (w/o ever crossing)
      • Pattern is self-similar and infinitely long, but contained in finite area
      • Curves (nonintersecting) are 1 dimensional and planes are 2 dimensional
      • Mandelbrot -  realized that the Peano Curve is somewhere between 1 & 2 dimensions
    •  George Cantor - Cantor Set
      • Produced by iteration; self-similar
      • Divide a line by thirds
      • Divide the two outer thirds by thirds, repeat, etc.
    •  Warclaw Sierpinski - Sierpinski Gasket
      • Self-similar
    • Helge Von Koch - Koch Curve
      • Infinitely long (although it has end points)
      • Koch snowflake
      • Koch island
        • Richardson realized measurements of coastlines erred by up to 20%
        • Due to progressively smaller scales of measurement
      • Each segment can be scaled by a factor of 3 to replicate original (self-similar)
     Richardson's coastline paradox

    Apr 25, 2010

    Deep Simplicity - Chapter 2

    The Return of Chaos

    Phase Space
    • Approximate solutions to N-body problem can be arrived at through performing successive iterations of calculus
      • However, this does not always work
        • Some infinite series converge, others do not
    •  Williams Hamilton - developed idea of phase space
      • Used position & momentum in description of particles (instead of force)
      • Six dimensional space which describes position (3 dimensions) & momentum (3 dimentions)
      • 2 particles need 12-dimensional phase space
      • Single point in phase space represents entire state of a system
        • Position & momentum of every point in the box
        • Poincare Recurrence Time - when a particle passes over the same point in phase space
    •  Poincare - analyzed only cross section (small part) of phase space (Poincare Section)
      • Found that orbits of 3 bodies typically diverge (instability is normal)
      • If a particle trajectory cut Poincare Section even a tiny distance away from the original point, the system can follow completely different patterns
        • Many systems are very sensitive to initial conditions, and move away non-linearly
        • Can't predict future (Laplace wrong)
    Weather
    • Lewis Fry Richardson - predictions of weather (forecasting)
      • Based on work of Vilhelm Bjerknes
        • Measured important properties such as temperature & pressure
        • Apply physics to determine how conditions will affect each other
      • Weather Prediction by Numerical Process
      • Contrasted to 'synoptic' forecast (chart reading)
        • Ever heard of people trying to read stock charts (technicals) to predict the future?  That's basically the same concept as 'synoptic' forecasting.  Thanks to Richardson, weather forecasters realized long ago that the weather from prior periods does not have much bearing on what the weather for future periods will be.  Instead, meteorologists now use various readings such as temperature and pressure.  This would be akin to market forecasters who rely on indicators such as consumer confidence, inflation, factory output, etc. (although these indicators are often forecasts themselves, hence perhaps explaining the unreliability of most 'market forecasts' - forecasts built on forecasts)
    •  Edward Lorenz - Butterfly Effect (real atmosphere is extremely sensitive to the starting position)
      • Does the Flap of a Butterfly's Wings in Brazil Setoff Tornadoes in Texas
      • Sometimes weather is more/less chaotic (predictable)
      • The New Scientists Guide to Chaos - Franco Vivaldi
      • Daniel Kirkwood - influence of Jupiter & Saturn sometimes causes asteroids to swing out of normal orbit (chaos)
        • Theoretically works in reverse, but in practice asteroids collide w/ inner planets
        • Calculations can be used to approximate historical & future orbits (but not exactly due to rounding)
      •  Irrational numbers are the reason for chaos
        • Small deviations subvert Laplace's attempts to "predict" universe
        • Only universe (God) itself can predict the universe (freewill?)
          • Also shows that universe cannot be reversed
            • Must show particles (all) move back to the present state, which cannot be stated (irrational numbers have inifinite decimals)

      Apr 24, 2010

      Statistical Mechanics - Entropy is Not Certainty

      Statistical Mechanics

      Model - Closed box full of gas particles
      Hypothesis - Gases will eventually distribute themselves in more or less even concentrations across the volume of the box
      Puzzle - Newtonian laws of mechanics do not forbid the gases from moving into one half of the box, or alternative otherwise uneven concentrations

      Then Why?- to answer this, its helpful to look at the scenarios below

      2 Particle Scenario
      • Imagine a box with 2 particles inside
      • Particles are moving, constantly bouncing off walls, or each other (Newton's 2nd Law)
      • Possible Arrangements (at any given time)
        • 0 particles on left side, 2 particles on right side (25%)
          • xx-AB
        • 1 particle on left side, 1 particle on right side (50%)
          • Ax-Bx
          • Bx-Ax
        • 2 particles on left side, 0 particles on right side (25%)
          • AB-xx
      4 Particle Scenario
      • Imagine a box with 4 particles inside
      • Particles are moving, constantly bouncing off walls, or each other (Newton's 2nd Law)
      • Possible Arrangements (at any given time)
        • 0 particles on left, 4 on right (6.3%)
          • xxxx-ABCD
        • 1 particle on left, 3 on right (25%)
          • Axxx-BCDx
          • Bxxx-ACDx
          • Cxxx-ABDx
          • Dxxx-ABCx
        • 2 particles on left, 2 on right (37.5%)
          • ABxx-CDxx
          • ACxx-BDxx
          • ADxx-BCxx
          • BCxx-ADxx
          • BDxx-ACxx
          • CDxx-ABxx
        • 3 particles on left, 1 on right (25%)
          • ABCx-Dxxx
          • ABDx-Cxxx
          • ACDx-Bxxx
          • BCDx-Axxx
        • 4 particles on left, 0 on right (6.3%)
          • ABCD-xxxx
      As you can see from the above scenarios, at N=2 and N=4, the most likely outcomes are the ones where the particles are arranged evenly into the two sides of the box.  The shape that is revealed is what in statistics is referred to as the bell-shaped curve (x representing the distinct outcome and y representing the corresponding probability).



      The most common outcomes are the ones that lie near the peak of the curve.  So, in statistical mechanics, we find that outcomes were the particles are more evenly distributed are probabilistically speaking more likely to occur.

      This characteristic would hold true for progressively larger numbers of particles (as N approaches infinite).  Therefore, in the words of Gribbin, Boltzmann's gases spread out to fill the box because that is the more likely outcome, while gases huddling close together in one corner, while far from impossible, is very unlikely.

      Apr 23, 2010

      Deep Simplicity - Chapter 1

      Order Out of Chaos

      Mechanical Physics
      • Copernicus - 1st to publish Sun-centered 'universe' theory (completed 1530); previous paradigms were Earth-centered
      • Kepler - Showed that the orbit of planets were elliptical; previous paradigms were circular
      • Galileo - found evidence to support Copernicus's theory
        • Laid down principles of science (theory to experimentation)
        • Developed theories of motion, ie gravity/acceleration, friction, etc
        • Developed models of theoretical/perfect/ideal conditions in order to simplify real world
          • Realized that 'ideal' models did not express reality perfectly
      • Newton - "Mathematical Principles of Natural Philosophy" (1687)
        • Used Galileo's concept of approximations to reality (point masses)
        • Developed calculus (Leibniz as well)
        • Universal Gravitation: F = G*m1*m2/(r^2)
          • Force b/w two objects directly proportional to their mass, and inversely proportional to the distance between the two objects
        • 1st Law: In absence of force, objects stay at rest or remain at constant velocity
        • 2nd Law: F = MA (force is time derivative of momentum)
        • 3rd Law: Every object acting with force F on another object, experiences force -F (equal and opposite)
      • N-Body Problem
        • For systems (planets, atoms, etc) where the number of masses exceeds 2 (N>2), the problem cannot be integrated
          • Universal Gravitation (see above) only applies to two bodies... what happens if there is a m3?
          • Alternative model - if a pool ball strikes two other pool balls, exact outcome is impossible to determine with complete accuracy
        • This is not a shortcoming in people, but in math
        • Lack of analytical solution means that nature itself does not 'know'
        • If that is true, then it would seem to disprove Laplace
          •  Laplace's Demon

            We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.
            —Pierre Simon Laplace, A Philosophical Essay on Probabilities
      • Newton's universe seems to have no distinction b/w past and future
        • Collisions can happen in reverse - no memory of time in planetary bodies, atoms, etc.

      Electro-Magnetics
      • Faraday - developed ideas of electric & magnetic fields
      • Maxwell - later developed the mathematical equations
        • Maxwell's equations suggest the speed of light is constant (root of relativity)
        • Like Newton, Maxwell's equations suggest no build-in arrow of time
          • Light can, in theory, flow away from and into a source
       
      Thermodynamics
      • Fourier (1811) - flow of heat is proportional to the temperature difference with heat flowing from hot to cold
      • Impossible to predict overall properties of an object which is at a human scale by analysis of objects at atomic and particle level - chaos
      • With billions and billions of molecules interacting, orders associated with simple laws appear
      • 1st Law: Conservation of energy (energy cannot be created nor destroyed, but only transformed into heat)
      • 2nd Law: Entropy (disorder always increases in a closed system); perpetual motion machines are impossible
      • 3rd Law: Absolute zero - impossible in real world
      • Entropy measures amount of order in a system
        • Increasing disorder corresponds to increasing entropy
        • Inevitable increase in entropy defines direction of time (macro-level)
        • At micro-level, Newton's laws suggest atoms can arrange themselves into increasing order
        • Attractor states = Equilibrium (ie absolute zero)
      • Boltzman - developed statistical mechanics (p. 32) - this topic is important enough that I shall cover it in a future post
      • Loschmidt - pointed out that molecules become correlated after colliding
        • Boltzman assumed molecules were uncorrelated (no direction in time)
      • Poincare - molecules, given sufficient time, will return to original state (speed & direction); known as Poincare recurrence time or Cycle time
      • According to Cycle time, entropy will probably rise, although it could fall

      Deep Simplicity - Intro

      Author: John Gribbin

      This book was recommended by Charlie Munger, of Berkshire Hathaway fame.  It's a very scientific book written by an astrophysicist (nuff said).  The topic is seductive: Chaos Theory.  Who doesn't want to learn some rudimentary chaos theory to regurgitate at a dinner conversation in order to impress members of the opposite gender?

      Deep Simplicity: Bringing Order to Chaos and ComplexityNaturally, it turned out to be one of the most difficult books I have read in awhile, due to the sheer number of ideas covered, my inadequate recollection of college level physics, and the complexity of the topics covered (chaos theory, butterfly effect, fractals, etc).  Despite my deliberate pace (over the course of probably 20+ hours), I probably only grasped about 50% of the material in here - too meaty for my digestion.  The material covered here is initially very scientific... definitely not for everyone.  It would be apt to think of the first part as a brief refresher on Physics 101, along with some concepts from Physics 901.  Nevertheless, the latter part of the book is not as much about chaos theory, and thermodynamics, so much as it is about Gribbin's insights into how chaos plays a role in our lives, and our world. 

      In that view, it's a very enjoyable, yet challenging read.  It covers topics such as how we forecast the weather, the 'true' length of the coast of Normandy, things that are 1.5 dimensional, evolutionary theory, the possibility of a major global extinction event...

      Pretty fascinating stuff!