Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Lecture 16 — The Canonical Ensemble Lecture 16 — The Canonical Ensemble Chapter 5, Chapter 5, Friday February 15Friday February 15thth•Review of previous class•Boltzmann distribution•Partition function•Entropy in the canonical ensemble•The bridge to thermodynamics through Z•Two-level systems•Single-particle in a box (quantum mechanics)Reading: Reading: All of chapter 5 (pages 91 - 123)All of chapter 5 (pages 91 - 123)Homework 5 due next Friday (22nd)Homework 5 due next Friday (22nd)Homework assignments available on Homework assignments available on web pageweb pageAssigned problems, Assigned problems, Ch. 5Ch. 5: 8, 14, 16, 18, : 8, 14, 16, 18, 2222Review of main results from lecture 15Review of main results from lecture 15Canonical ensemble leads to Boltzmann distribution function:( )( )( )exp / exp /exp /i B i Bij BjE k T E k TpZE k T- -= =-�Partition function:( )exp /j j BjZ g E k T= -�Degeneracy: gjEntropy in the Canonical EnsembleEntropy in the Canonical EnsembleM systemsni in state i1 2!! !.. !..MiMWn n n=ln lni iM B B i ii in nS k M k M p pM M� � � �=- =-� � � �� � � �� �lnB i iiS k p p=-�Entropy per system:The bridge to thermodynamics The bridge to thermodynamics through through ZZ( )exp / ;j BjZ E k T= -�js represent different configurationslnBF k T Z=-( ) ( )ln lnlnB BVV VT Z ZFS k k Z TT T T� �� �� � � ��� �� �=- =- = +� �� �� � � �� � �� �� �� � � ��( ) ( )2 2ln lnln lnB B BV VZ ZU TS F k T Z T k T Z k TT T� �� �� � � �� �= + = + - =� �� � � �� �� �� � � ��22VV VVU S FC T TT T T� �� � �� � � �= = =-� � � �� �� � �� � � �� �The bridge to thermodynamics The bridge to thermodynamics through through ZZ( )exp / ;j BjZ E k T= -�js represent different configurations1/Bk Tb =A simple model of spins on a latticeA simple model of spins on a lattice-1 0 10.00.10.20.30.40.50.60.7 S/NkBx1 1 1 1ln ln2 2 2 2Bx x x xS Nk+ + - -� �� � � � � � � �=- +� �� � � � � � � �� � � � � � � ��121U nxN Ne= = -n1 > n2T > 0n1 < n2T < 0n1 = n2T = ∞Be m=( )tanh /Bx k Te=-A simple model of spins on a latticeA simple model of spins on a lattice( )2 /2 /2 /ln 11BBk TBBk Tk TS Nk eeeee-� �= + +� �+� �Be m=-10 -8 -6 -4 -2 0 2 4 6 8 100.00.10.20.30.40.50.60.7 ln(1 + e x ) + x/(1 + e+x )x = 2/kBTA simple model of spins on a latticeA simple model of spins on a lattice( )222sech /V BBNC k Tk Tee� �=� �� �Be m=0 1 2 3 4 5 6 70.00.20.4 CV/NkB /kBTSchottkyanomalyA single particle in a boxA single particle in a boxV(x)V = ∞V = 0V = ∞xx = Lsinnn xLpy� �=� ��
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