Lecture 3: Boltzmann distribution

Transcription

1 Lecture 3: Boltzmann dstrbuton Statstcal mechancs: concepts Ams: Dervaton of Boltzmann dstrbuton: Basc postulate of statstcal mechancs. All states equally lkely. Equal a-pror probablty: Statstcal vew of thermal equlbrum Quantum system sharn enery wth a ervor The canoncal ensemble. Partton functon p ( ) Z( β ) e Z( β ) β e February 05 Lecture 3 1

2 Equal a-pror a probablty Basc Prncple of Statstcal Mechancs All mcroscopc quantum states are equally lkely. Probablty of a macroscopc state s proportonal to the number of mcro-states that can ve rse to t (that s, to the quantummechancal deeneracy). Aruments n favour: Aruments n favour: Gven enouh tme, the system wll explore all the accessble states. Erodc hypothess. Quantum mechancal prncple of detaled balance. Quantum transtons equalse occupaton (See problem 5, sheet 1) Predctve: t ves ood ults. We cannot thnk of anythn better (and t s beautfully smple). February 05 Lecture 3 2

3 Dffcultes Hstorcal note: Contrary to the preva, postvst orthodoxy around 1900, e.. Mach ( ) paraphrased.. Statements n physcs must relate to observables. Current vew: e.. Wenber (1993) paraphrased.. The postvsts mstake was not reconsn success when t happens. Dffcultes wth a-pror a probablty: Tme to explore all deenerate states: Example: N as atoms n a contaner. How lon wll t take to explore all confuratons when atoms are n ether left or rht half? N molecules 2 N states (left or rht) Transton tme ~10-4 s/per molecule ~10-4 /N s for the system. Tme of explore all the states, T 2 N x10-4 /N s. Take T T unverse s N 70. Eenstates are statonary (by defnton) so there can be no transtons. February 05 Lecture 3 3

4 How to overcome the dffcultes Tme to explore all states: We do not need to explore all the states. Most mcrostates roup around the most probable state. Also, the number (densty) of states ncreases enormously as the sze of the system ncreases. e oscllators; No. of No. of quanta arranements ~ ~ Statonary states Swtchn on an nteracton to cause transtons does not help t smply enerates a new set of non-deenerate quantum states. Aan, we are saved by the hue densty of states. Gven a small spread of enery u~h/t trans there are lots of accessble states. February 05 Lecture 3 4

5 Ludw Boltzmann ( ) 1906) Much crtcsed n hs lfetme: I am conscous of ben only an ndvdual stru weakly aanst the stream of tme (1898) Eht years later depsed and suffern headaches he commtted sucde. Reconton followed soon afterwards. February 05 Lecture 3 5

6 Equlbrum and Temp. Two systems n contact (.e. sharn enery) Total enery fxed: u u 1 + u 2. 1 (u 1 ) - deeneracy of 1st system 2 (u 2 ) - deeneracy of 2nd system For the whole system (u) 1 (u 1 ) 2 (u 2 ) 1 (u 1 ) 2 (u-u 1 ) Equlbrum state maxmses (u), so evaluate d((u))/du 1. d ( 1( u1)) d ( 2( u u1)) + 0 d u1 d u1 d ( 1( u1)) d ( 2( u2)) d u d u 1 Condton for thermal equlbrum It follows that the Temperature of a system s related to β so β s some functon of thermodynamc temp. 2 d ( u) d u February 05 Lecture 3 6

7 Lnk between β and temperature Is β a reasonable measure of T? Take N oscllators wth m quanta. ( m) d d m ( N ( N + m 1) ( N 1 )! m! + m 1) ( N 1)( N ( N N + m 1 m d d m ( N + m 1) 1) m m + m 1) + 1 m N 1 > m d d m 0 1/β ncreases steadly wth Temperature February 05 Lecture 3 7

8 Boltzmann dstrbuton System n contact wth a ervor. Total enery u. Enery n system,. p 1 Probablty of system ben n state of enery : (u-) 1 ( ) p ( u ) Ae Ae ( u ) Deeneracy of of system d ( u) d u ( u) e ( u) e e β ( ) ; Z( β ) e e Z( β ) β Ae Boltzmann dstrbuton Partton functon February 05 Lecture 3 8

9 Partton functon, Z Canoncal ensemble Systems n equlbrum wth a ervor are sad to be n ther canoncal state (standard state). The dstrbuton for a number of such systems s the canoncal ensemble. Boltzmann dstrbuton Our proof shows how the Boltzmann dstrbuton arses. The probablty of the systems havn a ven enery,, decreases exponentally wth because the deeneracy of the ervor ncreases exponentally wth. Partton functon Many thermodynamc propertes follow from Z. Example: Averae enery n our system e 1 d Z e Z d β d Z d β February 05 Lecture 3 9

10 Boltzmann Gbbs Entropy, I Boltzmann entropy (S( k ) Our dscusson has been for an solated system,.e. at constant enery, U, say. At fxed enery all the mcrostates ultn n a total enery, U, were equally lkely. What s the formula for the entropy of a system n contact wth a ervor,.e. at constant temperature, T? The canoncal ensemble Consder an ensemble of a suffcently lare number, M, of dentcal systems so that, for each system, the others act as a heat ervor. The ensemble s solated (wth fxed enery) but each member of the ensemble s at the same temperature. E * * labels labels the the states states of of each each system system n systems n n state state.. M Σ n February 05 Lecture 3 10

11 Boltzmann Gbbs entropy, II Boltzmann Gbbs entropy Apply the Boltzmann entropy to the ensemble as a whole. The number of ways of arrann n 1 systems (n state 1), plus n 2 systems n (state 2) etc.. s Entropy of of all allm systems Usn Usn Str s approxmaton Entropy of of one one system, SS SS M /M M /M S S M M M! n! n! k 1 2 k ( ) ( M M n n ) km n M N.B. N.B. M Σ n n M p n /M /M S k p p Entropy ven n terms of the probablty of occupaton of each quantum state,. Often ths form s more useful than the ornal Boltzmann form, especally for small systems. February 05 Lecture 3 11

12 Brde to classcal thermodynamcs Partton functon and the free enery The Boltzmann Gbbs equaton allows us to determne the Free enery, usn the Boltzmann dstrbuton to ve us the probabltes, p. Recall, F U - TS. Boltzmann probabltes are p exp( / kt ) Z ( p ) kt ( Z ) S k p + ( ) Z kt 1 S p + ( ) Z p T U p 1 p S U U TS / T + k kt F ( Z ) ( Z ) kt ( Z ) Free Enery n n terms of of the Partton Functon February 05 Lecture 3 12