Lecture 12—Ideas of Statistical Mechanics Chapter 4, Monday February 4 th
11
Lecture 12—Ideas of Statistical Lecture 12—Ideas of Statistical Mechanics Chapter 4, Mechanics Chapter 4, Monday February Monday February 4 4 th th •Finish the model for a rubber band •Demonstration •Spins on a lattice Reading: 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. 4 Ch. 4 : 2, 8, 10, : 2, 8, 10, 12, 14 12, 14 Exam 1: Exam 1: Fri. Feb. 8th (in class), chapters 1- Fri. Feb. 8th (in class), chapters 1- 4 4 Review: Review: Thu. 7th at 5:30pm, tentatively in Thu. 7th at 5:30pm, tentatively in
-
Upload
dustin-boyle -
Category
Documents
-
view
24 -
download
0
description
Lecture 12—Ideas of Statistical Mechanics Chapter 4, Monday February 4 th. Finish the model for a rubber band Demonstration Spins on a lattice. Reading: All of chapter 4 (pages 67 - 88) ***Homework 4 due Thu. Feb. 7th**** Assigned problems, Ch. 4 : 2, 8, 10, 12, 14 - PowerPoint PPT Presentation
Transcript of Lecture 12—Ideas of Statistical Mechanics Chapter 4, Monday February 4 th
Lecture 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 lattice
Reading: 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 NPB1220NPB1220
Rubber band modelRubber band model
n+ = # of forward links; n = # of backward linksN = n+ + n = total # of linksLength l = (n+ n)d = (2n+ N)d
d
! !
,! ! ! !
N NW N n
n n n N n
Dimensionless length:2
1l n
xNd N
Rubber band modelRubber band model
d
! !
,! ! ! !
N NW N n
n 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 NN
ln 1 ln 1n n n n
NN N N N
Rubber band modelRubber band model
d
! !
,! ! ! !
N NW N n
n 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 NN
1 1 1 1ln ln
2 2 2 2
x x x xN
Rubber band modelRubber band model
2nd law:2nd law: dUdU = = TdSTdS + + FdlFdl
1ln ln ,
2 2 1B B BF k Nd l k k l
T d Nd l d d Nd
2Blk TFNd
A simple model of spins on a latticeA simple model of spins on a lattice
n1 = # of ‘up’ spins; n = # of ‘down’ spinsN = n1 + n = total # of spinsEnergy U = (n1 n) = (N 2n1)
1
2
B
B
Quantum spinsin a magneticfield
B
2
2
Magneticmoment
64Statistical Mechanics – ideas and Statistical Mechanics – ideas and
definitionsdefinitions
An example:An example: Coin toss again!!
width
A simple model of spins on a latticeA simple model of spins on a lattice
n1 = # of ‘up’ spins; n = # of ‘down’ spinsN = n1 + n = total # of spinsEnergy U = (n1 n) = (N 2n1)
1
2
B
B
Quantum spinsin a magneticfield
B
2
2
Magneticmoment
A simple model of spins on a latticeA simple model of spins on a lattice
-1 0 10.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
S/Nk B
x
1 1 1 1ln ln
2 2 2 2B
x x x xS Nk
121
U nx
N N
A simple model of spins on a latticeA simple model of spins on a lattice
1 1ln
2 1Bk x
T x
-1 0 1-8
-6
-4
-2
0
2
4
6
8
2/kBT
x
121
U nx
N N
0 1 2 30.0
0.2
0.4
0.6
0.8
1.0
MV
/N
B/kBT
A simple model of spins on a latticeA simple model of spins on a lattice
tanhB
N BM
V k T
