lec8 - x x 2 2 e ( s <s> ) 2 / 2 2 (...

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MIT OpenCourseWare http://ocw.mit.edu 8.044 Statistical Physics I Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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Facts about sums of RVs Exact expressions for <s> and Var( s )i f S.I. p ( s )= p ( x ) p ( y )i f S.I. p ( s
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• ⊗ usually changes functional form But not always Fourier techniques are very useful 8.044 L8B2
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Very important special case: Central Limit Theorem RVs are S.I. All have identical densities p ( x i ) Var( x )is finite but <x> could be zero n is large p(s) Central Limit Theorem: p(s) is Gaussian n s n 8.044 L8B3
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If x is continuous 1 2 πσ 2 e ( s <s> ) 2 / 2 σ 2 p ( s )= <s> = n<x> σ 2 = 2 x 8.044 L8B4
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± ²³ ´ If x is discrete in equal steps of ∆
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Unformatted text preview: x x 2 2 e ( s &lt;s&gt; ) 2 / 2 2 ( s i x ) p ( s ) = i comb envelope p(s) s x 8.044 L8B5 Example K.E. of an ideal gas 1 2 p ( v x ) = = kT /m 2 2 e v x / 2 2 1 K.E. x = 2 mv 2 x 1 2 1 mean = 2 m v x = kT 2 2 mean sq.= ( 1 4 2 m ) 2 v x 3 4 3 = 4 ( kT ) 2 p (K.E. x ) variance = 1 2 ( kT ) 2 K.E. x 8.044 L8B6 3 directions, N atoms E = total K.E. E = 3 N K.E. x = 3 NkT 2 U Variance( E ) = 3 N Variance( K.E. x ) = 3 N ( kT ) 2 2 1 exp[ ( E (3 / 2) NkT ) 2 p ( E ) = 2 { (3 / 2) N ( kT ) 2 2 { (3 / 2) N ( kT ) 2 ] } } 3 s.d. 2 N kT 1 = = 3 mean 2 N kT 3 2 N 8.044 L8B7...
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This note was uploaded on 11/08/2011 for the course PHYSICS 8.004 taught by Professor Staff during the Spring '08 term at MIT.

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lec8 - x x 2 2 e ( s &amp;amp;lt;s&amp;amp;gt; ) 2 / 2 2 (...

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