Statistical Distributions - Thermodynamics and Statistical Mechanics - Lecture Slides, Slides of Thermodynamics

Download Statistical Distributions - Thermodynamics and Statistical Mechanics - Lecture Slides and more Slides Thermodynamics in PDF only on Docsity! Thermodynamics and Statistical Mechanics Statistical Distributions Docsity.com Multiple Outcomes NN N N NNN Nw i i i = ∏ = ⋅⋅⋅⋅ = ∑ ! ! !!! ! 321 Distinguishable particles Docsity.com Most Probable Distribution ∑∑∑ ∑∑ ∏ === == = +−+= −+= = == n j j n j jj n j jjB n j j n j jjB B n j j N j BB NNNgNNw NgNNw w N g Nww j 111 11 1 lnln!lnln !ln ln!lnln 0ln :Instead ! ! where0 δ δ Docsity.com Most Probable Distribution ∑ ∑ ∑ = = =         =                 −+−= +−+= n j j j jB n j j j jjjjB n j jjjB N g Nw N N NNgNw NgNNw 1 1 1 ln)(ln )1ln)(ln(ln )1ln(ln!lnln δδ δ δδ Docsity.com Constraints (Lagrange Multipliers) 0)()(ln)( 0ln 0ln 111 11 11 =−−        =      −− == = ∑∑∑ ∑∑ ∑∑ === == == n j jj n j j n j j j j n j jj n j jB n j jj n j j B NN N g N NNw UNNN w δεβδαδ εβαδ ε δ Docsity.com Quantum Statistics • Indistinguishable particles. 1. Bose-Einstein – Any number of particles per state. Particles with integer spin:0,1,2, etc 2. Fermi-Dirac – Only one particle per state: Particles with integer plus ½ spin: 1/2, 3/2, etc Docsity.com Bose-Einstein •At energy εi there are Ni particles divided among gi states. How many ways can they be distributed? Consider Ni particles and gi – 1 barriers between states, a total of Ni + gi – 1 objects to be arranged. How many arrangements? Docsity.com Bose-Einstein ∑∑∑ ∏ === = −−−−+= − −+ =⋅⋅⋅ − −+ = n j j n j j n j jjBE n j jj jj nBE jj jj j gNgNw gN gN NNNw gN gN w 111 1 21 )!1ln(!ln)!1ln(ln )!1(! )!1( ),,( )!1(! )!1( Docsity.com Constraints (Lagrange Multipliers) 01ln 0ln 0ln 1 11 =−−        + =         −−        + =      −− ∑ ∑∑ = == j j j n j j j jj j n j jj n j jBE N g N gN N NNw βεα βεαδ εβαδ Docsity.com Bose-Einstein j j j j j j j j j j f eg N e N g e N g N g j jj = − = −==+ =−−        + + ++ 1 1 11 01ln βεα βεαβεα βεα Docsity.com Boltzmann Distribution j j e fe g N g N j j j j j j βεα βεα βεα + −− === −−=        1 ln Docsity.com Fermi-Dirac ∑ ∑ ∑∑∑ = = === −−−−= −+−−− +−−= −−−= n j jjjjjjjjFD jjjjjj n j jjjjjjFD n j jj n j j n j jFD NgNgNNggw NgNgNg NNNgggw NgNgw 1 1 111 )]ln()(lnln[ln )]()ln()( lnln[ln )!ln(!ln!lnln Docsity.com Fermi-Dirac ∑ ∑ ∑ = = =                 − =         − − +−+−−= −−−−= n j j jj jFD n j jj jj jj j j jjFD n j jjjjjjjjFD N Ng Nw Ng Ng Ng N N NNw NgNgNNggw 1 1 1 lnln )( )( )ln(lnln )]ln()(lnln[ln δδ δδ Docsity.com Constraints (Lagrange Multipliers) 01ln 0ln 0ln 1 11 =−−        − =         −−        − =      −− ∑ ∑∑ − == j j j n j j j jj j n j jj n j jFD N g N Ng N NNw βεα βεαδ εβαδ Docsity.com Boltzmann Distribution j j egeN fe g N g N jj j j j j j j βεα βεα βεα −− −− = == −−=        stateper particles ofNumber ln Docsity.com Boltzmann Distribution ∑ ∑ ∑∑ = − − = − − = −− = −− = = == = n j j j j n j j n j j n j j jj j j j j j eg eg NN eg Ne egeNN egeN 1 1 11 βε βε βε α βεα βεα Docsity.com Partition Function Z eg NN Zeg eg eg NN j j j j j j n j j n j j j j βε βε βε βε − = − = − − = = = ∑ ∑ FunctionPartition 1 1 Docsity.com Ideal Gas ∫ ∫ ∫∫ ∞ − ∞ − ∞ −∞ −       =       =       ==       = 0 21 2/3 2/3 22 0 21 2/3 2/3 22 0 21 2/3 220 2/3 22 12 4 )()(12 4 2 4 )( 2 4 )( dxexmVZ demVZ demVdgeZ dmVdg x βπ βεβε βπ εε π εε εε π εε βε βεβε     Docsity.com Gamma Function ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 2 3 0 21 2 1 2 1 2 1 2 3 0 1 2 3 0 21 π π =Γ= =ΓΓ=Γ +Γ=Γ =ΓΓ= ∫ ∫∫ ∞ − ∞ −−∞ − dxex nnn dxexndxex x xnx Docsity.com Partition Function for Ideal Gas N UZ CZ CmVZ dxexmVZ x == ∂ ∂ − −= =      =       = ∫ ∞ − ββ β ββ π π βπ 2 3ln ln 2 3lnln 1 2 2 4 12 4 2/32/3 2/3 22 0 21 2/3 2/3 22   Docsity.com Quantum Statistics •When taken to classical limit quantum results must agree with classical. B-E and F-D must approach Boltzmann in classical limit. What is that limit? •Low particle density! Then distinguishability is not a factor. Docsity.com Classical limit kT β e f ef e f g N j j j j j j j j 1 Boltzmann as Same 1 1,1For 1 1 = ≅ >><< ± == + + + βεα βεα βεα Docsity.com Quantum Results Dirac-Fermi Einstein-Bose 1 1 + − ± == kT j j j j ee f g N ε α Docsity.com