# lec22 - 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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Canonical Ensemble p(E) p ( E ) e E/kT NOT! E p ( { p, q } ) e −H ( { p,q } ) /kT 8.044 L22B1
8.044 L22B2 ADVANTAGES OF CANONICAL OVER MICROCANONICAL ENSEMBLE MICROCANONICAL CANONICAL 1) ONE INTEGRATES OVER ALL PHASE SPACE SURFACE OF CONSTANT E Φ Ζ

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2) SEPARATION e −H /kT = e −H a /kT e −H b /kT let H = H a + H b , then p ( { p, q } )= p ( { p, q } a ) p ( { p, q } b )( a&ba re S I) Z = Z a Z b F = F a + F b S = S a + S b etc. 8.044 L22B3
For N similar, non-interacting systems Z =( Z 1 ) N ,F = NF 1 ,S = NS 1 N indistinguishable particles ( Z 1 ) N Z = , correct Boltzmann counting N ! 8.044 L22B4

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Example Non-interacting classical monatomic gas N i · i = i =1 H i N Z = ( Z 1 ) N N ! H = i =1 2 m 2 p 2 + p y + p 2 z H 1 ( ± r )= x p, ± 2 m p, ± 2 + p y + p z ) / 2 mkT /Z 1 x p 1 ( ± r e ( p 2 2 Gaussian p x p · ± 2 <± p> = <p 2 + p y + p 2 > =3 mkT x z < H 1 > / 2 kT 8.044 L22B5
±² Z 1 = ³ 2 2 2 e ( p x + p y + p z ) / 2 mkT dp x dp y dp z dxdydz h 3 ´ 3 / 2 =(2 πmkT ) 3 / 2 L x L y L z /h 3 = V µ 2 h 2 N ´ 3 N/ 2 ´ 3 / 2 1 V µ 2 Z ( T, V, N )= 1 V N µ 2 = N ! h 2 N ! h 2 units of cm 3 · ´ 3 / 2 N 1 V µ 2

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lec22 - MIT OpenCourseWare http/ocw.mit.edu 8.044...

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