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Lect07 - Lecture 7 The Boltzmann Distribution Concept of a...

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Physics 213: Lecture 7, Pg 1 Lecture 7 Lecture 7 The Boltzmann Distribution The Boltzmann Distribution Concept of a thermal reservoir The Boltzmann distribution Paramagnetic Spins – MRI Elasticity of a Polymer Supplement: Proof of equipartition, showing its limits Reference for this Lecture: Elements Ch 8 Reference for Lecture 8: Elements Ch 9

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Physics 213: Lecture 7, Pg 2 Breaking nature down into its elements Breaking nature down into its elements These questions (and many more) involve the exchange of energy or particles between a small system (atom or molecule) and a much larger system (the environment) -- a basic problem in statistical mechanics What is the range of kinetic energies of an O 2 molecule? Under what conditions does O 2 break up into two O atoms? What is the probability that a DNA molecule will unfold and replicate? What is the capacity of a myoglobin molecule to carry oxygen to the muscles? What is the vapor pressure of a solid or liquid? When do the vibrations of a molecule become important? Vitruvian Man, 1490, by Leonardo da Vinci
Physics 213: Lecture 7, Pg 3 Averages from Probabilities of States Averages from Probabilities of States 1. If you could list every quantum state of some system. 1, 2..n… o Realistic for small, independent parts (e.g., a SHO, an atom) 2. And knew the properties of each state (E n , magnetic moment, optical density, etc) 1. And knew the probability of each state (P 1 , P 2 , ...P n …) IF 1. You could calculate the average of E, magnetic moment, optical density, etc. for each part . E.g., <E>= P 1 E 1 +P 2 E 2 + ...P n E n o And their s.d., etc. too, if you want. 2. Now if you have a big system, made up of simple little parts as above, to get <E>, <m>, etc. for the big system, you just add all the parts ! THEN We can figure out how things behave starting from scratch. The key step is 3: the Boltzmann factor gets us the probabilities.

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Physics 213: Lecture 7, Pg 4 Concept of a Thermal Reservoir Concept of a Thermal Reservoir Consider a small system in thermal contact with a large system ( a thermal reservoir ): E n E l + E m + E q = U R = energy of reservoir 0 1 2 3 4 ε ε ε ε Total energy U = E n + U R Set U = 3 ε Key problem : What is the probability, P n , that the small system is in the state n with energy E n ? (Here E n = n ε , i.e., n quanta)? Answer : The probability of finding the small system in state n is proportional to the corresponding R . Small Reservoir system R n P
Physics 213: Lecture 7, Pg 5 1 SHO exchanging with 3-SHO reservoir P n 0.5 E n 0 1 2 3 x ε Probability decreases with E n because # states of the large system increases rapidly with U R = U tot - E n . ( 29 ( 29 ! 1 N ! q ! 1 N q R - - + = * N = 3 E n U R = U - E n R ( U R ) P n = R / Σ R 0 3 ε 10 * 10/20 = 0.5 ε 2 ε 6 6/20 = 0.3 2 ε ε 3 3/20 = 0.15 3 ε 0 1 1/20 = 0.05 20 = total # states for combined system = Σ R (Three oscillators) = q ε ( 29 ( 29 ! 1 N ! q ! 1 N q - - + = N = 4 q = 3 Why is zero energy most probable?

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Physics 213: Lecture 7, Pg 6 The Boltzmann Distribution The Boltzmann Distribution Consider a small system exchanging E with a big reservoir : n = label of state n (quantum #) E n
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