Lect06 - Lecture 6 Energy Exchange Statistics of energy...

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Physics 213: Lecture 6, Pg 1 Lecture 6 Lecture 6 Energy Exchange Energy Exchange Statistics of energy exchange General definition of temperature Why heat flows from hot to cold Reference for this Lecture: Elements Ch 7 Reference for Lecture 7: Elements Ch 8
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Physics 213: Lecture 6, Pg 2 Basic reminders and scope of thermodynamics Basic reminders and scope of thermodynamics When an isolated system can explore some number of microstates , they become equally likely. The probability that you find some macro state A is just (A)/ Ω, i.e., the fraction of all the microstates that look like A. o in terms of entropy: P(A) is proportional to e σ (A) , since σ (A)=ln( (A)) o The entropy of a system is ln( ),where counts all possible states, but for big systems ln( A ) for the most likely macrostate is not too much less than ln( ) » E.g., for 1000 spins: σ = 1000 ln(2)=693.147 (why?) σ ( 500 ups) = ln(1000! / 500! 500!)= 689.467 Thermodynamics applies to systems big enough so that the entropy of the most likely macrostate (e.g., 500 up spins) isn’t too much different from the total entropy (e.g., any # up)
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Physics 213: Lecture 6, Pg 3 If the internal parameter is initially placed near its most likely position, x e , then it will likely remain near x e . σ is a maximum: If the internal parameter x is initially placed far from the most likely position, x e , then x will evolve in time towards x e (where most states are located.) 0 dt d σ 0 dt d σ The Entropy of any isolated system can only increase (or remain constant) as a function of time: The Second Law of Thermodynamics The Second Law of Thermodynamics All available micro states become equally likely, but for big systems that lets us predict with near certainty what the macro state will be, because the vast majority of microstates look pretty similar (e.g., about 50% ‘ups’ for two-state). Macroscopic systems have very sharply peaked distributions (much sharper than shown here): x e (or σ 29 Internal parameter, x (e.g.,position of partition) 0 dt d = σ
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Physics 213: Lecture 6, Pg 4 Lessons from Volume exchange Lessons from Volume exchange The number of states of the whole was the product of the number of states of the parts. The log of the total number is the sum of the logs of the numbers of the parts. o We call the ln(number of microstates) the entropy σ For big systems, in equilibrium we almost certainly see the macrostate that maximizes total number o To find it we maximize entropy by maximizing the sum of the entropies of the parts. If parts can exchange volume, in equilibrium each must have the same derivative of its entropy with respect to its volume. o Otherwise volume could shift and increase net entropy » argument doesn’t rely on the parts being the same at all We now use the same principles for systems that trade ENERGY TOT 1 2 = Ω Ω TOT 1 2 ln( ) Ω ≡ σ σ = σ + σ 1 2 1 2 @fixed N, etc.
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