Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Lecture 12—Ideas of Statistical Lecture 12—Ideas of Statistical Mechanics Chapter 4, Mechanics Chapter 4, Monday February Monday February 44thth•Finish the model for a rubber band•Demonstration•Spins on a latticeReading: Reading: All of chapter 4 (pages 67 - 88)All of chapter 4 (pages 67 - 88)***Homework 4 due Thu. Feb. 7th*******Homework 4 due Thu. Feb. 7th****Assigned problems, Assigned problems, Ch. 4Ch. 4: 2, 8, 10, 12, : 2, 8, 10, 12, 1414Exam 1: Exam 1: Fri. Feb. 8th (in class), chapters 1-4Fri. Feb. 8th (in class), chapters 1-4Review:Review:Thu. 7th at 5:30pm, tentatively in Thu. 7th at 5:30pm, tentatively in NPB1220NPB1220Rubber band modelRubber band modeln+ = # of forward links; n = # of backward linksN = n+ + n = total # of linksLength l = (n+ n)d = (2n+ N)dd( )( )! !,! ! ! !N NW N nn n n N n++ - + += =-Dimensionless length:21l nxNd N+� �= = -� �� �Rubber band modelRubber band modeld( )( )! !,! ! ! !N NW N nn n n N n++ - + += =-( ) ( )ln ln ln lnW N N n n N n N n+ + + += - - - -Sterling’s approximation: ln(Sterling’s approximation: ln(NN!) = !) = NNlnlnNN NNln 1 ln 1n n n nNN N N N+ + + +� �� � � � � �=- + - -� �� � � � � �� � � � � ��Rubber band modelRubber band modeld( )( )! !,! ! ! !N NW N nn n n N n++ - + += =-( ) ( )ln ln ln lnW N N n n N n N n+ + + += - - - -Sterling’s approximation: ln(Sterling’s approximation: ln(NN!) = !) = NNlnlnNN NN1 1 1 1ln ln2 2 2 2x x x xN+ + - -� �� � � � � � � �=- +� �� � � � � � � �� � � � � � � ��Rubber band modelRubber band model2nd law:2nd law:dUdU = = TdSTdS + + FdlFdl1ln ln ,2 2 1B B BF k Nd l k k lT d Nd l d d Ndd ddd+ +� � � �= = � =� � � �- -� � � �2Blk TFNdA simple model of spins on a latticeA simple model of spins on a latticen1 = # of ‘up’ spins; n = # of ‘down’ spinsN = n1 + n = total # of spinsEnergy U = (n1 n) = (N 2n1)12BBe me m=-=+Quantum spinsin a magneticfieldB22+-hhMagneticmoment 64Statistical Mechanics – ideas and Statistical Mechanics – ideas and definitionsdefinitionsAn example:An example:Coin toss again!!widthA simple model of spins on a latticeA simple model of spins on a latticen1 = # of ‘up’ spins; n = # of ‘down’ spinsN = n1 + n = total # of spinsEnergy U = (n1 n) = (N 2n1)12BBe me m=-=+Quantum spinsin a magneticfieldB22+-hhMagneticmoment 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= = -A simple model of spins on a latticeA simple model of spins on a lattice1 1ln2 1Bk xT xe-� �=� �+� �-1 0 1-8-6-4-202468 2/kBTx121U nxN Ne= = -0 1 2 30.00.20.40.60.81.0 MV/NB/kBTA simple model of spins on a latticeA simple model of spins on a latticetanhBN BMV k Tm m� �=� ��
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