Thermodynamics and Statistical Mechanics

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Thermodynamics and Statistical Mechanics

• Probabilities

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Pair of Dice

• For one die, the probability of any face coming
  up is the same, 1/6. Therefore, it is equally
  probable that any number from one to six will
  come up.
• For two dice, what is the probability that the
  total will come up 2, 3, 4, etc up to 12?

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Probability

• To calculate the probability of a particular
  outcome, count the number of all possible
  results. Then count the number that give the
  desired outcome. The probability of the desired
  outcome is equal to the number that gives the
  desired outcome divided by the total number of
  outcomes. Hence, 1/6 for one die.

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Pair of Dice

• List all possible outcomes (36) for a pair of
  dice.
• Total Combinations How Many
• 2 11 1
• 3 12, 21 2
• 4 13, 31, 22 3
• 5 14, 41, 23, 32 4
• 6 15, 51, 24, 42, 33 5

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Pair of Dice

• Total Combinations How Many
• 7 16, 61, 25, 52, 34, 43 6
• 8 26, 62, 35, 53, 44 5
• 9 36, 63, 45, 54 4
• 10 46, 64, 55 3
• 11 56, 65 2
• 12 66 1
• Sum 36

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Probabilities for Two Dice
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Probabilities for Two Dice
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Microstates and Macrostates

• Each possible outcome is called a microstate.
• The combination of all microstates that give the
  same number of spots is called a macrostate.
• The macrostate that contains the most microstates
  is the most probable to occur.

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Combining Probabilities

• If a given outcome can be reached in two (or
  more) mutually exclusive ways whose probabilities
  are pA and pB, then the probability of that
  outcome is pA pB.
• This is the probability of having either A or B.

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Combining Probabilities

• If a given outcome represents the combination of
  two independent events, whose individual
  probabilities are pA and pB, then the probability
  of that outcome is pA pB.
• This is the probability of having both A and B.

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Example

• Paint two faces of a die red. When the die is
  thrown, what is the probability of a red face
  coming up?

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Another Example

• Throw two normal dice. What is the probability of
  two sixes coming up?

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Complications

• p is the probability of success. (1/6 for one
  die)
• q is the probability of failure. (5/6 for one
  die)
• p q 1, or q 1 p
• When two dice are thrown, what is the probability
  of getting only one six?

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Complications

• Probability of the six on the first die and not
  the second is
• Probability of the six on the second die and not
  the first is the same, so

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Simplification

• Probability of no sixes coming up is
• The sum of all three probabilities is
• p(2) p(1) p(0) 1

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Simplification

• p(2) p(1) p(0) 1
• p² 2pq q² 1
• (p q)² 1
• The exponent is the number of dice (or tries).
• Is this general?

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Three Dice

• (p q)³ 1
• p³ 3p²q 3pq² q³ 1
• p(3) p(2) p(1) p(0) 1
• It works! It must be general!
• (p q)N 1

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Binomial Distribution

• Probability of n successes in N attempts
• (p q)N 1
• where, q 1 p.

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Thermodynamic Probability

• The term with all the factorials in the previous
  equation is the number of microstates that will
  lead to the particular macrostate. It is called
  the thermodynamic probability, wn.

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Microstates

• The total number of microstates is

For a very large number of particles
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Mean of Binomial Distribution
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Mean of Binomial Distribution
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Standard Deviation (s)
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Standard Deviation
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Standard Deviation
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For a Binomial Distribution
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Coins

• Toss 6 coins. Probability of n heads

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For Six Coins
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For 100 Coins
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For 1000 Coins
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Multiple Outcomes
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Stirlings Approximation
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Number Expected

• Toss 6 coins N times. Probability of n heads
• Number of times n heads is expected is
• n N P(n)