lec2 - ( x x < x > = x 0 Var( x ) = 2 Controlled...

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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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Gaussian density (memorize) p(x) ( x x 0 ) 2 1 p ( x )= 2 πσ 2 e 2 σ 2 x x + σ x h he = 0.61 h 1 2 0 0 2 parameters 8.044 L1B7
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Example Atom escaping from a cavity A Hole Atom escapes after n th wall encounter p ( n )=( A H )(1 A H ) n A T A T A Total n =0 , 1 , 2 , ··· 8.044 L2B1
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± p n ( x )= ( A H )(1 A H ) n δ ( x n ) A T A T n =0 Called a geometric or a Bose-Einstein density P n (x) p n (x) 0 2 4 x 0 2 4 x 8.044 L2B2
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Example Mixed, t dependent RV Chemical adsorption Physical adsorption x p(x) x e -t/ τ P(x) 1 Given: atom on bottom at t =0 e -t/ τ p ( x )= e t/τ δ ( x )+(1 e t/τ ) f ( x ) 8.044 L2B3 x
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Averages <f ( x ) > ± f ( x ) p ( x ) dx −∞ p(x) std. dev. <x> x <x> is the mean <x 2 > is the mean square < ( x <x> ) 2 > = < ( x 2 2 x<x> + <x> 2 ) > = <x 2 > 2 <x> 2 + <x> 2 = <x 2 > <x> 2 is the variance ( standard deviation) 2 8.044 L2B4
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Gaussian 1 0 ) 2 / 2 σ 2 p ( x )= 2 πσ 2 e
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Unformatted text preview: ( x x < x > = x 0 Var( x ) = 2 Controlled separately Exponential p(t) e- t / 1 < > t < t > = Var( t ) = 2 Determined by same parameter 8.044 L2B5 Example Mean free path n d L=d/cos / 2 < L > = ( d/ cos ) p ( ) d 0 / 2 = ( d/ cos )2 sin cos d = 2 d / 2 sin d 0 = 2 d [ / 2 ( cos ) = 2 d 0 8.044 L2B6 Poisson density Events occur randomly along a line at a rate r per unit length L x x p (1) r x as x 0 Events are statistically independent 1 p ( n ) = ( rL ) n e rL = 1 < n > n e <n> n ! n ! 8.044 L2B7 Examples of Poisson probability densities 0.6 0.25 0.5 <n> = 0.5 0.2 <n> = 2.3 0.4 0.15 0.3 0.2 0.1 0.1 0.05 1 2 3 4 5 2 4 6 8 10 0.12 0.1 <n> = 10 0.08 0.06 0.04 0.02 5 10 15 20 25 30 10 20 30 40 50 0.01 0.02 0.03 0.04 0.05 0.06 0.07 <n> = 30 8.044 L2B8...
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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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lec2 - ( x x < x > = x 0 Var( x ) = 2 Controlled...

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