Chapter 12 (II)

Chapter 12 (II) - 12.3 Most Probable Distribution Assume...

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12.3 Most Probable Distribution )! ( ! ! 1 1 N N N N Assume that we have N distinguishable particles to occupy two energy levels ε 1 and ε 2 , the probability to find N 1 particles occupying at energy level ε 1 is given by the number of combinations of N particles taken N 1 at a time (N 1 N) ω = ω (N 1 ,N-N 1 )= N C N1 =( 1 2 . 6 ) Here, the ordering of occupation does not matter; otherwise we have to use permutations rather than combinations. Of particular interest to us is to find the maximum value of ω , i.e., ω max . This can be done by letting . 0 1
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During deriving ω max , we need to use the so-called Stirling approximation: lnN! N(lnN-1) for N>>1 . Since ln ω is a monotonically increasing function of ω , to find ω max is equivalent to find (ln ω ) max The solution: when N 1 =N 2 =N/2, ln ω or ω has its maximum. This result is not surprizing at all. We got an exact 50-50 split between the two states. In other words, the most probable configuration (macrostate) is the most random state. In
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This note was uploaded on 02/28/2011 for the course PHYS 359 taught by Professor Dr.asdas during the Winter '10 term at Waterloo.

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Chapter 12 (II) - 12.3 Most Probable Distribution Assume...

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