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lec8 - x µ ∆ x √ 2 πσ 2 e − s −<s> 2 2 σ 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 ) if S.I. p ( s ) = p ( x ) p ( y ) if S.I. p ( s ) slightly more complicated if not S.I. 8.044 L8B1
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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 = n σ 2 x 8.044 L8B4
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Unformatted text preview: x µ ∆ x √ 2 πσ 2 e − ( s − <s> ) 2 / 2 σ 2 δ ( s − i ∆ x ) p ( s ) = i ± ²³ ´ comb envelope p(s) s ∆ x 8.044 L8B5 ± Example K.E. of an ideal gas 1 2 p ( v x ) = σ = kT /m √ 2 πσ 2 e − v x / 2 σ 2 1 K.E. x = 2 mv 2 x 1 2 1 mean = 2 m ± ² v ³´ x ² µ = kT 2 σ 2 mean sq.= ( 1 4 2 m ) 2 ± ² v ³´ x ² µ 3 σ 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 ² K.E. x ± = 3 NkT 2 U ≡ ² Variance( E ) = 3 N Variance( K.E. x ) = 3 N ( kT ) 2 2 1 exp[ − ( E − (3 / 2) NkT ) 2 p ( E ) = ² 2 π { (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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