Phys 160 Thermodynamics and Statistical

Physics

Lecture 8

Randomness and Probability

• 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 un-

certainties? If not what best to do?

• How to deal with uncertains and unknowns In some effective way- this is the realm of probability.

• Probability gives a meaningful description and a numerical measure of these uncertainties.

• These afford us to act in a reason-able and effective way

• 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

• 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.

• 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.

• 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 = 311 875 200!

• Is the counting correct? We did distinguish the order of the cards treating 1 2 3 4 5 as different from 2 3 4 5 1 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.

• The number of 5 cards hand is • 52x51x50x49x48 / 5x4x3x2x1 • = 311 875 200/120 = 2 598 960 • 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.

• 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.

• 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 .

• (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 2100. How many macrostate?

• Only 101! 0 head, H, HH, HHH, upto 100 heads.

• 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

• 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.