Time Arrow, Statistics and the Universe II

1. Time Arrow,Statisticsand the Universe II Physics Summer School 17 July 2002 K. Y. Michael Wong Outline: * Counting and statistical physics * Explaining physical properties of gases * Boltzmann’s entropy equation * Second law of thermodynamics * Counting energy

2. Basic assumption of statistical physics All microstates are equally probable. Hence the macrostate with the largest number of microstates is the most probable one. Hence in the example of 4 molecules in a box, the molecules tend to distribute evenly in the box, since the corresponding macrostate has the largest number of microstates.

3. The haunted castle The situation is like a drunken ghost wondering inside a haunted castle with many rooms of different sizes. (It can penetrate all walls.) Even if he starts from a small room, he will most likely be located in the room with the largest area after a long time. The largest room corresponds to the macrostate having the largest number of microstates.

4. momentum The phase space position Physicists often describe the motion of particles in a phase space, whose coordinates are the positions and momenta of the particles. Phase space can be rather complicated. For 3-dimensional motion of a particle, the phase space is 6-dimensional. For 3-dimensional motion of N particles, the phase space is 6N-dimensional. Every point in the phase space is a microstate!

5. momentum position Drifting microstates However, physicists proved an elegant theorem about this complicated space: Consider a cluster of microstates in the phase space. It will move around the phase space as time goes on. However, the volume of this cluster will remain the same, although its shape may be distorted.

6. Like an ink drop Hence the motion of this cluster of microstates is drifting like an ink drop diffusing in water. Its regular shape will gradually diffuse into thin threads filling every region, subregion, subsubregion … in the phase space. For any point in the phase space, the distorted ink drop can get arbitrarily close to that point.

7. Making ramen and candies Another analogy for this phase space motion: Chefs making Peking ramen repeatedly stretch and fold the noodles until the texture becomes smooth enough. Candy factories have machines which repeatedly stretch and fold the caramel until the texture becomes smooth enough.

8. Many ghosts in the castle If a small room in the haunted castle is filled with a cluster of ghosts, they will eventually get arbitrarily close to any location in the castle when they wonder around. This is the basis of the probability of all microstates, and hence the largest probability of the macrostate with the largest volume in phase space.

9. Discrete and continuous All microstates are equally probable. • Hence either: • When the microstates are discrete (e.g. gas chamber with 2 sides, or quantum systems with discrete energy levels): • The macrostate with the largest number of microstates (i.e. the highest degeneracy) is the most probable one. • When the microstates are continuous (e.g. molecules in a gas chamber): • The macrostate with the largest volume in phase space is the most probable one.

10. Free Expansion The macrostate with molecules filling only the left box has a low degeneracy. The macrostate with molecules filling both the left and right boxes has a high degeneracy. Hence given enough time, the gas will reach the macrostate which fills both left and right boxes. The macrostate which only fills the left box is not completely impossible, but it is highly improbable.

11. Modern Examples The counting of microstates have become important in many scientific problems nowadays. For example, consider the shape of a polymer or a DNA chain. Comparing the 2 chains, the shorter one containing several folded units has a larger number of microstates than the straight one. Hence a folded chain is more likely to be found in nature than a straight one (neglecting energy for the time being).

12. All of the following has an increase in the degeneracy (or the volume in phase space) of the macrostate except: 1) A straight DNA chain molecule coils up like a spaghetti. 2) Ice melts into water. 3) Water vaporizes to steam. 4) Oxygen gas mixes up with nitrogen gas. 5) A steel bar expands on heating. 6) None of them. Question 4

13. Explanation 1) A straight DNA chain molecule coils up like a spaghetti. Degeneracy increases. 2) Ice melts into water. Water molecules have more freedom to move. Hence the volume in the phase space of the macrostate increases. 3) Water vaporizes to steam. Molecules in steam have more freedom to move. Hence the volume in the phase space of the macrostate increases. 4) Oxygen gas mixes up with nitrogen gas. This is similar to the problem of mixing gases separated in two chambers. Degeneracy increases. 5) A steel bar expands on heating. Iron atoms have more freedom to move after expansion. The volume in the phase space of the macrostate increases. Answer: 6) None of them.

14. The large N limit No. of atoms: deg (half left, half right) deg (all left) N=4 6 1 N=10 252 1 N=100 1029 1 When N is large, nearly all microstates correspond to roughly equal division of molecules between left and right.

15. Logarithm For large N, it is often more convenient to simply specify the number of zeroes in the degeneracy rather than the number itself. This is called the logarithm of the degeneracy. e.g. deg = 1029Þ log deg = 29 When “log deg” increases from 29 to 30, “deg” increases tenfold! Related example: the Richter scale for earthquakes is logarithmic. When the Richter scale increases by 1, the energy released increases tenfold!

16. Mathematicians find a consistent way to define logarithms even when the number is not exactly powers of 10. log (1  1029) = 29, log (3  1029) = 29.5, log (10  1029) = log 1030 = 30. e.g. The logarithm can be easily computed by pressing the “log” button on your calculator.

17. Boltzmann’s entropy equation Entropy S: S = constant  log W W is the degeneracy of a macrostate (for discrete systems), or the volume of a macrostate in the phase space (for continuous systems). Macrostates with large degeneracy or phase-space volume also have large entropy.

18. Second law of thermodynamics (1st alternative form) The entropy of a closed system never decreases. Since disordered macrostates have higher entropy, this means that order tends to disorder.

19. Suppose a pendulum is placed inside a box containing many air molecules. If a molecule has 10 quanta of energy in the beginning, chances are that after a waiting period, 1) the molecule continues to keep the energy. 2) all molecules share the energy. 3) the molecules and the pendulum share the energy. 4) the pendulum possesses the energy. 5) the molecules and the pendulum exchange energy periodically. Question 5

20. Counting energy In quantum mechanics, energy is quantized. 6 quanta of energy shared between gases A and B, each of 3 molecules: If gas A has 3 energy quanta, we count its degeneracy by: there are 3 microstates with molecular energies 3+0+0, there are 6 microstates with molecular energies 2+1+0, there is 1 microstate with molecular energies 1+1+1. total degeneracy = WA = 3 + 6 + 1 = 10.

21. Temperature The temperature of an ideal gas is proportional to the average energy per molecule. If we choose units with c=1,

22. Total degeneracy of the 2 gases: W = WA× WB Results 1. In the most probable macrostate, TA = TB. 2. If the system starts with TA > TB, the probability will increase in the direction of decreasing EA and increasing EB, i.e. heat is transferred from the hot gas to the cold one.

23. Example 2 4 quanta of energy shared between gases A and B, each of 2 molecules: If gas A has 4 energy quanta, we count its degeneracy by: degeneracy = WA = 5

24. Considering other possible macrostates, we can obtain this table of degneracy and entropy: Results 1. In the most probable macrostate R, EA = EBÞ TA = TB. 2. If the system starts with TA > TB, the probability will increase in the direction of decreasing EA and increasing EB, i.e. heat is transferred from the hot gas to the cold one.