1. Complexity: Ch. 3 Information Complexity in Systems 1

2. Energy and Entropy Two laws of thermodynamics 1.Energy is conserved a)in a static universe b)total energy increases in an expanding universe 2.Entropy always increases a)because the universe used to be much smaller b)unless the universe shrinks in the future? Complexity in Systems 2

3. A Book about Entropy, the Universe and Everything Complexity in Systems 3 From Eternity to Here: The Quest for the Ultimate Theory of Time Sean Carroll

4. Is the World Reversible or Not? Physics is microscopically reversible. We can play a micro-movie either way and can’t say that either direction is impossible The Universe is macroscopically irreversible. Landslides never run “backwards” What gives? Complexity in Systems 4

5. A box half-full of gas gas in a box 5 2000 moleculesno molecules

6. Or maybe it’s half-empty stipulate no energy lost to collisions, etc, so the system is reversible gas in a box 6

7. 2000 things $20 in pennies roughly the population of Sharon, VT molecules in 65 cubic inches of interstellar space Complexity in Systems 7

8. 2000 moleculesno molecules open a hole A box half-full of gas with a hole gas in a box 8

9. counting states Assume the hole has been open for a long time, meaning that the system has settled into dynamic equilibrium. Give every particle a distinct name, such as 1, 2, … gas in a box 9

10. counting states Definition: a microstate consists of k particular molecules on the right and all the rest on the left. Definition: a macrostate consists of any k molecules on the right and all the rest on the left. gas in a box 10

11. Counting micro states How many ways can there be 2000 on the left and none on the right? Answer: 1 How many ways can there be 1999 on the left and 1 on the right? Answer: 2000 gas in a box 11

12. dealing with large systems As the number of particles increases, it becomes difficult, then impractical, and finally impossible to solve for the motion of every particle. gas in a box 12

13. dealing with large systems Statistical mechanics was developed to understand the properties of the aggregate without knowing the behavior of each individual particle. gas in a box 13

14. counting states (cont.) gas in a box 14 For the hard core: Number of ways to have k on the right and (N – k) on the left

15. Every State is Equally Likely gas in a box 15 Number on LeftNumber on RightMicrostates  Entropy 2000010 199913.3 199826.3 19901026.44 1900100171.04 1500500486.75 1100900595.79 1000 600.31 Quantity has a quality all it’s own.  micro

16. Complexity in Systems 16

17. Could we ever track this system numerically Imagine that every electron, proton. or neutron in the visible universe is a computer, which runs at 1 trillion operations per second How large a system of particles could we track? Complexity in Systems 17

18. Number of neutrons, protons, and electrons in the observable universe Complexity in Systems 18

19. Upper bound on operations per second Complexity in Systems 19

20. Upper bound on operations per second (cont) Complexity in Systems 20

21. Upper bound on operations per second (cont) Complexity in Systems 21

22. probability of unbalanced Populations gas in a box 22 10 -17 pressure fluctuations

23. What’s going on here? The system is still reversible: it could conceivably someday pass through its original state of all molecules on the left. The probability of that happening is so low it could not possibly ever happen. As the system size increases things that were possible start to become effectively impossible. gas in a box 23

24. what about entropy? gas in a box 24 10 -17

25. Takeaway Systems with a lot of particles (2000) behave differently in kind from systems with a few particles. We have to use different techniques with large systems (statistical mechanics). gas in a box 25

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27. entropy Is very unlikely to decrease significantly, especially as the system size (number of particles) becomes macroscopically large. gas in a box 27

28. James Clerk Maxwell Complexity in Systems 28 Katherine Toby

29. Maxwell’s Demon Complexity in Systems 29 Appears to defy the Second Law of thermodynamics by decreasing entropy

30. Entropy of… Complexity in Systems 30