2. 1 Time Arrow, Statistics and the Universe I Physics Summer School 16 July 2002 K. Y. Michael Wong Outline: * The problem of time arrow * What solution have physicists proposed * ???

3. 2 One way traffic in Nature? 1. The disintegration of the egg will never happen in the reverse direction (re-integration). 2. Air molecules diffusing out of the bottle will never progress in the reverse direction (infusion). Time Arrow

4. 3 Contradiction? Molecular motions are described by the kinetic theory of gases. That is, gas molecules are particles obeying Newton’s Second Law. However, Newton’s Second Law is reversible in time. Question: Why are there so many irreversible processes in a world described by reversible mechanical laws? Before? After? Before?

5. 4 (a) Discuss with your group members to suggest an explanation why physical processes are irreversible. (b) Choose the best explanation using PRS. Question 1

6. 5 1) 2) 3) Scrap Board

7. 6 I know that many of you are giving me “standard answers”, but:  What does your “standard answer” mean?  Is it just another way of describing the same thing?  Or is it really a fundamental explanation?

8. 7 Time Arrow, Statistics and the Universe I Outline: * The irreversibility of time arrow * The work of Boltzmann * How to count arrangement of molecules? * Microstates and macrostates Cover slide revisited: Physics Summer School 16 July 2002 K. Y. Michael Wong

9. 8 Coming Attractions Day 2 Counting and statistical physics Explaining physical properties of gases Boltzmann’s entropy equation Second law of thermodynamics Counting energy Day 3 Alternative forms of the second law Maxwell’s demon Available energy in refrigerators and heat engines Time arrow and cosmology

10. 9 Ludwig Boltzmann (1844-1906) foundational work on the entropy (second law of thermodynamics)

11. 10 Engraved on his tomb in a Vienna cemetery is the statistical expression for entropy. Boltzmann's analysis of entropy was ridiculed by some very powerful figures in the German scientific establishment. Boltzmann was depressed by these attacks and by his own poor health, and took his own life in 1906.

12. 11 Boltzmann formulated a molecular explanation for thermodynamics. 1. The industrial revolution stimulated the study of heat engines. The situation faced by Boltzmann 2. Heat was accepted to be a form of energy. 3. Conservation of energy was accepted (1st law of thermodynamics). 4. Though energy is conserved, the amount of available energy is decreasing. Why? It is now the foundation of statistical physics.

13. 12 How many ways are there to put 2 distinguishable balls into 3 boxes? (Choose your answer from 0 to 9.) A review of counting: Question 2

14. 13 If 2 distinguishable balls are put into 3 boxes, 1st ball: there are 3 ways. 2nd ball: there are 3 ways. Total: there are 3  3 = 9 ways. ans. Explanation

15. 14 How many ways are there to put 2 indistinguishable balls into 3 boxes? (Choose your answer from 0 to 9.) A review of counting: Question 3

16. 15 If 2 indistinguishable balls are put into 3 boxes, Case 1: both balls are in the same box: there are 3 ways. Case 2: the balls are in different boxes: there are 3 ways. Total: there are 3 + 3 = 6 ways. ans. Explanation

17. 16 How to count combinations? How many ways are there to choose 3 balls out of 5? 15234 1 2 1 3 1 3 1 4 5 2 2 3 3 3 2 4 1 2 54 2 3 4 5 4 5 145 5 Answer: 10

18. 17 Do it step by step: 1st step: there are 5 ways to choose a ball. 2nd step: there are 4 ways to choose a ball. 3rd step: there are 3 ways to choose a ball. Total: there are 5 x 4 x 3 = 60 ways to choose the 3 balls. There is an over-count because the order of appearance of the balls does not matter! How to count combinations? How many ways are there to choose 3 balls out of 5? 1 23 1 2 3 123 1 2 3 1 23 1 23

19. 18 If 3 balls are picked, 1st step: there are 3 ways. 2nd step: there are 2 ways. 3rd step: there is 1 way. Total: there are 3 x 2 x 1 = 6 ways of ordering the 3 balls. Since only 1 out of 6 ways is counted as a distinct combination, the number of combinations = 60/6 = 10. 1 23 1 2 3 123 1 2 3 1 23 1 23

20. 19 A useful way to write the result: This can be rewritten as Introduce the factorial symbol: Then we have the result:

21. 20 4 molecules in a box How many ways are there: to place 0 and 4 molecules in L and R respectively? to place 1 and 3 molecules in L and R respectively? to place 2 and 2 molecules in L and R respectively? to place 3 and 1 molecules in L and R respectively? to place 4 and 0 molecules in L and R respectively?

22. 21 Microstates and Macrostates There are 16 microstates. There are 5 macrostates. The degeneracy (multiplicity) of each of the 5 macrostates is 1, 4, 6, 4, 1 respectively. Note that macroststates with a fair distribution between L and R have the largest degeneracy.