lec4 - MIT OpenCourseWare http:/ocw.mit.edu 8.044...

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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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Poisson Random Variable r(x) S.I. events p (1) in ∆ x r x r is constant x p ( n =0; L )= e rL 1 st element of p ( n ; L ) 8.044 L4B1
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LL + L x 0 p ( n ; L +∆ L ) p ( n ; L ) p (0; ∆ L )+ p ( n 1; L ) p (1; ∆ L ) ± ²³ ´ ± ²³ ´ (1 r L ) ( r L ) dp ( n ; L ) p ( n ; L L ) p ( n ; L ) = rp ( n 1; L ) ( n ; L ) L dL 8.044 L4B2
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± ²³ ´ ± ²³ ´ dp ( n ; γ ) Let γ rL + p ( n ; γ ) = p ( n 1; γ ) 1 st order, linear DE already known DE: γ is the variable and n is an index Probability: n is the variable and γ is a parameter 1 p ( n ; γ )= γ n e γ n ! 8.044 L4B3
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Example Jointly Gaussian random variables t Random processes: Noise voltage or current v ( t ) ,i ( t ) Thermodynamic variable P ( t ) ,T ( t ) ( t ) Thermal radiation E ( t ) ,B ( t ) Seismic background signal s ( t ) 8.044 L4B4
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t 1 t 2 =t 1 + τ v 1 v 2 t v(t) 2 ± 1 1 ρ 2 exp v 1 2 ρv 1 v 2 + v 2 p ( v 1 ,v 2 )= 2 2 πσ 2 2 σ 2 (1 ρ 2 ) σ is constant ρ = ρ ( τ ) 1as τ 0as τ →∞ 8.044 L4B5 0
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v 1 v 1 v 1 v 2 v 2 v 2 ρ 1 ρ = 0 ρ -1 v 2 and v 2 are S.I. 8.044 L4B6
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lec4 - MIT OpenCourseWare http:/ocw.mit.edu 8.044...

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