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Probability v3

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My public lecture on randomness from 2013.

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Probability v3

  1. 1. Does God play dice? On probability and randomness. Adam Kleczkowski A random walk through Mathematics and Computing Science 21st February 2013 Computing Science and Mathematics
  2. 2. Plan   •  Determinism and randomness •  Mechanics as deterministic science •  Randomness •  Deterministic chaos •  Quantum weirdness •  Summary
  3. 3. What is Mathematics about •  Not really about remembering big numbers or being able to multiply/divide quickly •  Real-world problems •  Describe and synthesise knowledge •  Predict •  Generalisation of the assumptions •  How general can we make the theory? •  Can we question its fundamental assumptions? •  Back to real-world •  Does the new theory make it possible to answer even more complicated questions?
  4. 4. Mathematics How to describe, analyse and predict such different phenomena like: •  pendulum and clock movements •  planet movements •  movement of dust in air •  weather •  light and electron emission and interference
  5. 5. Deterministic and random •  These phenomena can be broadly described as being either •  Deterministic: •  any state is completely determined by prior states •  If we know the history and all parameters, we can exactly determine the state of the system or •  Random: •  states cannot be exactly predicted •  but there are regularities in the occurrences of events even those which outcomes are not certain
  6. 6. Deterministic approach Deterministic approach has a very long history It is very intuitive and usually associated with Mathematics 2+2=4 (usually) In a plane, through a point not on a given straight line, at most one line can be drawn that never meets the given line. (5th Euclidean postulate)
  7. 7. Euclid  of  Alexandria   •  Greek mathematician •  ca. 300BC •  wrote Elements, a treatise in 13 books •  a single, logically coherent framework, including a system of rigorous mathematical proofs •  it remains the basis of mathematics 23 centuries later Built upon work of Thales (ca 600BC) and Pythagoras (ca 500BC)
  8. 8. Galileo Galilei (1564-1642) Galileo was one of the first modern thinkers to clearly state that the laws of nature are mathematical. Philosophy is written in this grand book, the universe ... It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures;.... Put forward the basic principle of relativity: All laws of mechanics are the same in any system moving with constant speed
  9. 9. Example: Pendulum •  Galileo was the first one to describe regular movement of pendulum •  For small swings, period is independent of the different size of swings •  To double the period, the length needs to be quadrupled The movement is highly predictable
  10. 10. Isaac Newton (1642-1727) 1st law: If no forces act on an object, it moves with constant velocity and in the straight line 2nd law:Application of a force results in acceleration 3rd law:Action of a force results in counteraction Newton’s laws of mechanics are a foundation for our view of the natural world – and they are deterministic!
  11. 11. Newton’s laws of gravity In 1687 Newton published the law of gravity: •  the higher the mass, the stronger the force •  the further away the two masses, the weaker the force
  12. 12. Example: Planetary motion Very regular motion, at least on the scale of many thousands years Kepler (1571-1630)
  13. 13. Planetary orbits… z-axis direction) at the initial and final parts of the integrations N±1. The axes units are au. ular momentum. (a) The initial part of N+1 (t = 0 to 0.0547 × 109 yr). (b) The final part t of N−1 (t = 0 to −0.0547 × 109 yr). (d) The final part of N−1 (t = −3.9180 × 109 to 7 Newton’s laws of gravity and mechanics can be used to predict how the Solar system will look like 109 years from now, that is: 1 000 002 013 AD Ito & Tanikawa, 2002
  14. 14. n-body problem Henri Poincare (1854-1912) When more than two bodies interact by gravitational force, the movement becomes very complex… … and unpredictable For planets and moons, this might take a very long time! But not for comets or asteroids…
  15. 15. Crazy planets
  16. 16. Crazy planets
  17. 17. Determinism and randomness Deterministic approach has been very successful, but many real-world phenomena can be very unpredictable Is random outcome completely determined, and random only by virtue of our ignorance of the most minute contributing factors? – randomness due to ignorance Or, is the system deterministic but we cannot control its initial conditions? – randomness due to uncertainty in initial state
  18. 18. Randomness Means ‘lack of predictability or pattern’ In ancient times associated with ‘fate’ Fortuna: a Roman goddess associated with ‘luck’ Ancients also played games where ‘luck’ was important
  19. 19. Randomness Ordinary people face an inherent difficulty in understanding randomness, although the concept is often taken as being obvious and self-evident. Bennett (1999) Scientific – rigorous – study came late even though games of chance played since antiquity
  20. 20. Randomness Democritus, ca. 400 BC •  The world is governed by the unambiguous laws of order; •  Randomness is a subjective concept •  originates from the inability of humans to understand the nature of events. Epicurus, ca. 300 BC •  Everything that occurs is the result of the atoms colliding, rebounding, and becoming entangled with one another, •  There is no purpose or plan behind their motions.
  21. 21. Gerolamo Cardano (1501-1576) Liber de ludo aleae – Book on Games of Chance is the first known book on probability also contains a section on cheating…
  22. 22. Andrey Kolmogorov (1903-1987) In 1933 Andrey Kolmogorov defined probability in an axiomatic way, finally producing a systematic approach to the probability theory; as Euclid did 2300 years earlier for geometry In fact, there are still many unsolved problems in probability and its relationship to ‘real-world’: a subject of another talk…
  23. 23. Deterministic child with a pie A walker moving left or right according to rules Case 1: going straight Case 2: moving periodically
  24. 24. Deterministic child with a pie •  Keep moving in one direction, or •  Moves some steps to the right and then some steps to the left •  but we always know where he will be at a given time… 0 200 400 600 800 1000 0 200 400 600 800 1000 Time Position 0 200 400 600 800 1000 0 200 400 600 800 1000 Time Position
  25. 25. A random child Case 3: a child in a crowd
  26. 26. A random child Case 3: a child in a crowd We do not know exactly where he will be at a given time. 0 200 400 600 800 1000 -100-50050100 Time Position
  27. 27. A random child But we can roughly say: •  study a lot of children and see how far they spread from the point of the origin… 0 200 400 600 800 1000 -100-50050100 0 2 4 6 8 10-100-50050100 Time Position +1
  28. 28. A random child But we can roughly say: •  study a lot of children and see how far they spread from the point of the origin… 0 200 400 600 800 1000 -100-50050100 0 200 400 600 800 1000 -100-50050100 0 2 4 6 8 10-100-50050100 +1
  29. 29. A random child But we can roughly say: •  study a lot of children and see how far they spread from the point of the origin… 0 200 400 600 800 1000 -100-50050100 0 200 400 600 800 1000 -100-50050100 0 200 400 600 800 1000 -100-50050100 0 2 4 6 8 10-100-50050100 +1
  30. 30. 0 2 4 6 8 10-100-50050100 A random child But we can roughly say: •  study a lot of children and see how far they spread from the point of the origin… 0 200 400 600 800 1000 -100-50050100 0 200 400 600 800 1000 -100-50050100 0 200 400 600 800 1000 -100-50050100 0 200 400 600 800 1000 -100-50050100 +1
  31. 31. A random child But we can roughly say: •  study a lot of children and see how far they spread from the point of the origin… 0 50 100 150 200 -100-50050100 0 200 400 600 800 1000 -100-50050100
  32. 32. A random child This requires a different way to describe the position of a child: what is a chance of meeting him at a given position? -100 -50 0 50 100 050100150200 Position Chance/probability Probability!
  33. 33. Brownian motion Motion of pollen grains suspended in water (1827) Robert Brown (1773-1858)
  34. 34. Brownian motion The same mechanism is responsible for dispersion of colour in water
  35. 35. Brownian motion Albert Einstein (1879-1955) Marian Smoluchowski (1872-1917) Theory of Brownian motion (1905, 1906) Fundamental for random processes A large particle is being acted upon by many small, randomly moving particles: This results in a random motion of the large particle
  36. 36. Quantum cloud The steel sections were arranged using a computer model with a random walk algorithm starting from points on the surface of an enlarged figure based on Gormley's body that forms a residual outline at the centre of the sculpture. Antony Gormley (1950-)
  37. 37. Turbulence •  Water and air motion can in principle be described in terms of deterministic equations •  These equations are deterministic, so that the future behaviour can be fully predicted by laws of motion •  But the behaviour can be very unpredictable due to sensitivity to initial conditions and perturbations CL Navier (1785-1836) G Stokes (1819-1903)
  38. 38. Weather •  Weather prediction is probably one of the hardest problems in physics •  Forecasts are usually phrased in terms of probability Latest 15 days ensemble forecast wind for London, meteogroup.co.uk
  39. 39. Weather EN Lorenz (1917-2008) •  Lorenz was studying weather models. •  In 1950s he discovered that small rounding-off errors in some equations can produce completely different results in the future. •  This has been called a ‘butterfly effect’
  40. 40. Chaos theory Poincare and later Lorenz discovery opened a new area of research, called Deterministic ChaosTheory In this theory, the rules are deterministic and each trajectory is well described and predictable, but uncertainty in initial conditions means a big and increasing uncertainty – randomness – later on. M Feigenbaum (1944-)
  41. 41. Double pendulum Double pendulum motion is highly unpredictable, even though it can be described by the same equations of motion as the single one!
  42. 42. Strange number π Are the digits ‘predictable’? Is every digit 0-9 equally represented? There is also randomness even where we normally do not expect it…
  43. 43. A child with a pi Move up or down by: di − mean di( ) e.g.: 3 – move left by 2 1 – move left by 4 4 – move left by 1 1 – move left by 4 5 – move right by 1 9 – move right by 4 3.14159265358979
  44. 44. A child with a pi Move up or down by: di − mean di( ) e.g.: 3 – move left by 2 1 – move left by 4 4 – move left by 1 1 – move left by 4 5 – move right by 1 9 – move right by 4 0 200 400 600 800 1000 -40 -20 0 20 40
  45. 45. A boy with a pi – or rather 1/2013 0 200 400 600 800 1000 -40 -20 0 20 40 •  1/2013 is a rational number •  There is a pattern to the digits – they repeat about every 59 steps •  1/2013 is not a ‘normal’ number
  46. 46. Normal numbers Emile Borel (1871-1956) A number is ‘normal’ if in its sequence of digits, all digits and all subsequences of digits are equally represented •  Many non-rational numbers appear ‘normal’: •  It has not been proven that π is indeed ‘normal’ but it looks very likely 2 He also stated the ‘infinite monkey theorem’: a monkey typing long enough would eventually produce the complete works of Shakespeare!
  47. 47. Quantum theory So far we looked at systems where randomness comes because we cannot describe all that is going on •  Like throwing a coin so in principle, if we could describe the movement of hand, fingers and coin , we could exactly predict heads or tails But in quantum world, unpredictability is fundamental!
  48. 48. Quantum view of light Light is a wave, caused by electric and magnetic forces pushing back and forth against each other, like water waves that are caused by gravity and water pressure pushing against each other. All waves carry energy.
  49. 49. Light also comes in packets, photons! •  When you shine light onto a metal it can cause an electrical current to flow which is how solar cells work. •  No matter how faint the light is, electrons will flow as long as it is the right color. •  This means that atoms receive light energy in packages, in chunks: You either get one, or you do not. This idea that light is a wave of energy that can only be received in certain specific quantities laid the foundation for the quantum theory.
  50. 50. Randomness of photons and electrons •  On the atomic level, things behave in a fundamentally random way. The location of a photon or an electron cannot be predicted precisely •  Randomness is woven into the fabric of our world! N Bohr (1885-1962) E Schrödinger (1887-1961)
  51. 51. “God does not play dice.” •  Einstein hated the idea that our world behaves in ways that are fundamentally random. •  He thought that the quantum theory was just an approximation to a deeper theory that would allow us to predict exactly where electrons would be. •  No one has found the deeper theory that Einstein was looking for… •  Experimental evidence suggests that such a theory does not exist
  52. 52. Summary •  Our understanding of the world is largely coloured by determinism: cause and effect are very closely linked. •  But the real world is often not deterministic, but random. •  Even deterministic equations can often produce a complicated, apparently random pattern. •  Quantum theory says that the world is fundamentally random at small scales. •  We need to learn to deal with such random and apparently complex systems.
  53. 53. Thank  you  for  your  a5en6on!     A  Random  Walk  through   Mathema3cs  and  Compu3ng  Science   Spring  2013   h5p://www.maths.s6r.ac.uk/lectures/     The  next  talk  is  by  Ken  Turner   28th  February  2013   7pm  

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