StatMech

1AE/ME 6765 Enumeration of Microstates-1School of Aerospace Engineering Copyright © 2009 by Jerry M. Seitzman. All rights reserved.Statistical Mechanics •Goal – calculation/prediction of TD (macroscopic) properties from molecular (microscopic) properties •Overall approach – use QM to describe mol ecular properties and invoke statistical connection between microscopic and macroscopic properties •Options 1) most general - canonical and grand canonical ensembles (ensemble or Gibbs method ) 2) isolated system of independent particles →find connection to entropy, then use TD state equations (Maxwell-Boltzmann method ) AE/ME 6765 Enumeration of Microstates-2School of Aerospace Engineering Copyright © 2009 by Jerry M. Seitzman. All rights reserved.M-B Method Outline • Given isolated macroscopic system with specified energy (Eor U) and volume ( V) made up of Nmicroscopic particles (with Nlarge) •Microscopic model – each particle has well-defined, quantized/ discrete energy levels (ε1, ε2, ε3, …εm) – total energy is sum of energies of particles •Probability and Statistics – number ways to distribute Nparticles over individual energy levels (εi) while maintaining same overall energy E? – which is most “likely”? •Macroscopic TD properties – answer will lead to entropy then other TD properties•implicitly assuming energy levels of one particle are independent of presence of other particles ⇒energy levels not perturbed by molecular interactions •equivalent to perfect gas, ideal solution assumptions 2AE/ME 6765 Enumeration of Microstates-3School of Aerospace Engineering Copyright © 2009 by Jerry M. Seitzman. All rights reserved.Enumeration of Microstates • Macroscopic TD state –E, V, N sufficient to define TD state •M i c r o s c o p i c s t a t e – need more to define microstate – how are particles distributed over the molecular energy levels? • large number of microstates (unique quantum states ) consistent with macrostate, e.g., total Eand N • example: four ε,N=4, E=8 ••••ε4=4ε3=2ε2=2ε1=0 • •••••••••••••••••••••••••• •• ••••• •••••• E≠8 Same as microstate 1 if indistinguishable particles1 23456789 1011 AE/ME 6765 Enumeration of Microstates-4School of Aerospace Engineering Copyright © 2009 by Jerry M. Seitzman. All rights reserved.Enumeration of Microstates • Could also divide into 3 energy levels with one having degeneracy of 2 • So each microstate is a unique quantum state of the system, but has same TD E, N constraints –define Ω= total number of microstates with same E, N • Want to find Ω – will later relate to S – use indistinguishable particlesε4=4ε3=2ε2=2ε1=0 •••• •ε3=4ε2=2ε1=0 ••• ∑=Ω withs microstate1 ∑=∑= E NN N iii ε 3AE/ME 6765 Enumeration of Microstates-5School of Aerospace Engineering Copyright © 2009 by Jerry M. Seitzman. All rights reserved.Particle Statistics • Approach to finding Ω – consider indistinguishable particles ( balls ) – each must exist in some energy level ( big box ) – each energy level can have degeneracy ( little boxes ) • can be “true” degeneracy εa=εb • or can be near degeneracy εa≈εb•ε3=4ε2=2ε1=0 ••• AE/ME 6765 Enumeration of Microstates-6School of Aerospace Engineering Copyright © 2009 by Jerry M. Seitzman. All rights reserved.Matter Models • Examine 5 situations 1) Distinguishable balls in set of boxes with number of balls in each box ( Ni) prespecified →Boltzmann statistics without degeneracy (model for crystals) 2) Same as (1) with degeneracy →Boltzmann statistics with degeneracy 3) Same as (2) but indistinguishable particles and dilute (gi>>Ni)=low probability of >1 particle in small box →Corrected Boltzmann statistics 4) Same as (3) but no restrictions on # particles per small box →Bose-Einstein statistics 5) Same as (3) but only one particle per small box →Fermi-Dirac statistics (follow Pauli Exclusion Principle)if dilute, reduce to (3) 4AE/ME 6765 Enumeration of Microstates-7School of Aerospace Engineering Copyright © 2009 by Jerry M. Seitzman. All rights reserved.Macrostate • Definitions – macrostate ≡given distribution of Nparticles across energy levels (big boxes), i.e., given Ni distribution –W(Ni)≡# of microstates in specific macrostate • So multiple macrostates per TD state • Total number of microstates in TD state related to W ()∑=Ω withs macrostateiNW ∑=∑= E NN N iii ε AE/ME 6765 Enumeration of Microstates-8School of Aerospace Engineering Copyright © 2009 by Jerry M. Seitzman. All rights reserved.Counting Microstates • State with N“balls” (particles) and M“large boxes” (energy levels) • Imagine 1) lining up the Nballs then 2) sorting in order into large boxes •1ststep – how many ways to line up Nballs? •1stball: Nchoices •2ndball: N−1 choices • etc. ⇒N! •2ndstep – sorting into boxes4 2 1 5 3 N1=2 N2=3 123 132 213 231 312 3216=3! ; 5AE/ME 6765 Enumeration of Microstates-9School of Aerospace Engineering Copyright © 2009 by Jerry M. Seitzman. All rights reserved.Sorting into Energy Levels • Case 1) Boltzmann Statistics w/o degeneracy –e x a m p l e N=7, M=4 – one of N! lineups – no different (can’t “order” molec. in same state) –N i! “lineups” are identical for each i –h e r e4 7 1 5 3 62N1=1 N2=3 N3=1 N4=2 6 7 1 5 3 42 () ∏ ==M iii NNNW 1!! total # lineups total # equiv. lineups420!2!1!3!1!7==W lots of microstates even for only a few particles and boxes AE/ME 6765 Enumeration of Microstates-10School of Aerospace Engineering Copyright © 2009 by Jerry M. Seitzman. All rights reserved.Another Example •N=5, M=3 • Every four configurations are the “same” –N1! ⋅N2! ⋅N3! = 1⋅2⋅2=4 ;3 45 21 3 54 21 4 35 21 4 53 21 2 35 41 2 53 412 45 31 2 54 31 5 34 21 5 43 21 2 34 51 2 43 51N1=1N2=2 N3=2 6AE/ME 6765 Enumeration of Microstates-11School of Aerospace Engineering Copyright © 2009 by Jerry M. Seitzman. All rights reserved.Sorting into Energy Levels • Case 2) Boltzmann Statistics w/ degeneracy –Nparticles, Mlarge boxes, gismall boxes in large box – ignoring gi, already know – but now within each large box (energy level) giplaces to put a particle – how many particles allowed in each small box? • with no exclusion rule, as many as NiN1 ;g1=7() ∏ ==M iii NNNW 1!! N2 ;g2=3 () ⎟⎠⎞⎜⎝⎛×= ∏ =boxes small into ballsarrange to ways# !! 1M iii NNNW AE/ME 6765 Enumeration of Microstates-12School of Aerospace Engineering Copyright © 2009 by Jerry M. Seitzman. All rights reserved.Sorting into Energy Levels • Case 2) Boltzmann Statistics w/ degeneracy – how many was to arrange Niballs into gismall boxes? – e.g., Ni=2, gi=3 gives 9 (=32) ⇒giNi – since each big box (energy level) independent ()∏ ∏= ==M iN i M iiiig NNNW 1 1!!12 12 121 21 21 22 112 12 7AE/ME 6765 Enumeration of Microstates-13School of Aerospace Engineering Copyright © 2009 by Jerry M. Seitzman. All rights reserved.Sorting into Energy Levels • Case 3) (Corrected) Boltzmann Statistics – now make balls/particles indistinguishable • Does not matter how we initially lineup balls • BUT note from previous example we have now overcounted • Not a problem if chance of overcounting negligible ⇒corrected Boltzmann statistics only valid for gi>>Ni()∏ ∏ == iN i M iiiig NNNW 1!! 471 53 62 671 53 42 () ∏∏ = ==M iiM iN i i CB N g N Wi 1 1! 1 22 1 Ni=2,gi=3now the same microstate or number of quantum states available >> number of particles AE/ME 6765 Enumeration of Microstates-14School of Aerospace Engineering Copyright © 2009 by Jerry M. Seitzman. All rights reserved.Sorting into Energy Levels • Case 4) Bose-Einstein Statistics – indistinguishable particles – no limit on number of particles per quantum state (small box) Bosons – to avoid dilute requirement consider one large box (energy level) with Ni=7, gi=4; but use gi−1=3 partitions to mark them – gives us Ni+gi−1 (10) things to arrange • if distinguishable ( Ni+gi−1)!ways to line them up – but both balls and partitions are indistinguishable •Ni!(gi−1!) overcounts (balls distingui shable from partitions) – since each big box independent()() ()∏ =−−+=M i i ii i i BEgNg NN W 1 !1 !!1 8AE/ME 6765 Enumeration of Microstates-15School of Aerospace Engineering Copyright © 2009 by Jerry M. Seitzman. All rights reserved.Sorting into Energy Levels • Case 5) Fermi-Dirac Statistics – now only one particles pe r quantum stat e (small box) Fermions (e.g., e-spin) – place Ni(e.g, 3) particles in gi(e.g., 7) •gi(=7) places to put 1stball •gi−1 (=6) places to put 2ndball • continue until no balls left gi-Ni+1 (=5) – but particles indisti nguishable, overcounted by N! – since each big box independent ()()∏ =−=M i i i ii i FDNgNgN W 1 ! !!()()() !!1 ...1 i ii i i i iNggg N gg−=−+− Note: requires gi≥Nior would have more than on particle per quantum state AE/ME 6765 Enumeration of Microstates-16School of Aerospace Engineering Copyright © 2009 by Jerry M. Seitzman. All rights reserved.Boltzmann Limit • Look at B-E and F-D cases for gi>>Ni Boltzmann Limit • So in Boltzmann limit, no practical difference between Boson and Fermion statistics()()()∏ =⋅⋅⋅−+−+=M i ii i i i i Ng N g N g 1 !2 1()∏ =−=M i i i ii FDNgNgW 1 ! !! CBW=()()∏ =+−⋅⋅⋅−=M i ii i i i NN g gg 1 !1 1()∏ =≤≈M i iN i Ngi 1! () ()∏ =−−+=M i i ii i BEgNg NW 1 !1 !!1()∏ =≥≈M i iN i Ngi 1!