Phys 160 Thermodynamics and Statistical Physics. Lecture 8 Randomness and Probability

Size: px

Start display at page:

Download "Phys 160 Thermodynamics and Statistical Physics. Lecture 8 Randomness and Probability"

Rodger Moore
5 years ago
Views:

1 Phys 160 Thermodynamics and Statistical Physics Lecture 8 Randomness and Probability

2 All life and even science provide examples of situations where we are confronted with possibilities whose outcomes we do not know. Examples: Lottery ticket, hit by lightning, hurricane will hit New York, etc.,all involve uncertains and unknowns. Can we completely get rid of uncertainties? If not what best to do?

3 How to deal with uncertains and unknowns In some effective waythis is the realm of probability. Probability gives a meaningful description and a numerical measure of these uncertainties. These afford us to act in a reasonable and effective way

4 The predictions come one way or other and still is regarded as correct. Our understanding of the world comes down to understanding processes and outcomes that are probabilistic in nature due to randomness. Can we make predictions about these events? Quantum Mechanics, Biology

5 Probabilistic descriptions are taking a central role in science. Random happenings are things where the individual outcomes of one trial is unknown; but repetitions or aggregates have some regularity. Role of probability is to describe the operation of random occurrences in the aggregate.

7 In a dice, suppose we ask what is the probability of rolling even number? This is an event. It has 3 outcomes. Th total number of outcomes is 6. So the probability of rolling an even number 3/6 = 0.5 If all outcomes are equally likely, the probability is just a question of counting.

8 What is the probability of getting a poker hand-all 4 aces out of 5 cards? What are the number of cards that have 4 aces in 5 cards? After the 4 aces, the last card can be any one of the remaining 48. i.e. 48 outcomes The total outcomes is 52x51x50x49x48 = !

9 Is the counting correct? We did distinguish the order of the cards treating as different from and so on-multiple times. They are in fact the same hand! We have to correct for the overcounting. The number of ways of ordering the 5 cards is 5x4x3x2x1 = 120.

10 The number of 5 cards hand is 52x51x50x49x48 / 5x4x3x2x1 = /120 = This is called combinatorics. What has this to do with Thermodynamics? Why so many thermodynamic processes go in one direction but never the reverse. This is the Big Question. The quick answer is: Irreversible processes are not inevitable but overwhelmingly probable.

11 Two State systems: Suppose we flip three coins, a penny, a dime and a quarter. How many possible outcomes are there? Let us count them by brute force. HHH, HHT,HTH,THH, HTT,THT,TTH, TTT Each outcome is now called a microstate. To specify a microstate, the state of each individual particle has to be stated.

12 If we specify the state more generally, say we want two heads, this is like an event. We call it a macrostate. How many microstates are in this macrostate? THH, HTH, HHT. If we know the microstate, we also know the macrostate. But not the reverse. The number of microstates in a macrostate is called the multiplicity.

13 (HHH) = 1; (HH) = 3; (H) = 3 (0) = 1. (All) = 1+3=3+1 = 8. Thus the probability of any parti cular macrostate is (n) / (all) Suppose there are 100 coins. The total number of microstates is How many macrostate? Only 101! 0 head, H, HH, HHH, upto 100 heads.

14 What is the value of (1) or (H) Start with all coins T up. To have one H, any one of these coins can be turned up. There are 100 ways. So (1) = 100. For (2), the first coin has 100 choices and the second one 99. Hence the number of distinct pair is (2) = (100 x 99)/2

16 The above formula gives the number of ways of choosing n objects out of N. Problem 2.1 Suppose we flip four coins. a) List all possible outcomes b) List all macrostates and their probabilities c) Check the multiplicity of each macrostate. There are 16 outcomes.

19 Probability gives a meaningful description and a numerical measure of these uncertainties. Random happenings are things where the individual outcomes of one trial is unknown; but repetitions or aggregates have some regularity. Role of probability is to describe the operation of random occurrences in the aggregate. A collection of outcomes is an event Each outcome is now called a microstate. To specify a microstate, the state of each individual particle has to be stated. If we specify the state more generally, it is a macrostate.

29 Very Large Numbers There are three kinds of numbers that commonly occur in statistical mechanics: small numbers, large numbers, and very large numbers. Small numbers are small numbers, like 6,23, and 42. You already know how to manipulate small numbers. Large numbers are much larger than small numbers, and are frequently made by exponentiating small numbers.

30 The most important large number in statistical mechanics is Avogadro's number, which is of order The most important property of large numbers is that you can add a small number to a large number without changing it. For example, = (The only exception to this rule is when you plan to eventually subtract off the same large number: = 42.)

33 One common trick for manipulating very large numbers is to take the logarithm. This operation turns a very large number into an ordinary large number, which is much more familiar and can be manipu lated more straightforwardly. Then at the end you can exponentiate to back the very large number. I'll use this trick later this section.

40 Approximation for the factorial given by Ramanujan as log(n!) nlogn n+log(n(1+4n(1+2n)))6+log(π)2 In Hardy's words: I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways." The two different ways are these: 1729 = =

P [(E and F )] P [F ]

P [(E and F )] P [F ] CONDITIONAL PROBABILITY AND INDEPENDENCE WORKSHEET MTH 1210 This worksheet supplements our textbook material on the concepts of conditional probability and independence. The exercises at the end of each