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