Unformatted text preview: PY 410, Statistical and Thermal Physics, Spring 2012 Syllabus Lecture1 Lecture2 Lecture3 Homework 1 (Due Feb. 2nd), problems from Reif: 1.1; 1.4 (analyze the answer at large N), 1.5, 1.9, 1.10, 1.12, 1.14 Additional Problem.* A person throws dice N times. Find the probability distribution of the maximum score for different N. Write down your results explicitly in a table for N=1,2,3 (use decimal numbers not fractions). Discuss the asymptotical form of this distribution when N is large. Solutions Lecture4 Homework 2 (Due Feb 9th), problems from Reif: 1.16, 1.17, 1.18, 1.23, 1.26*, 1.28* M&M problem. This problem consists of three parts 1. Each of you will get a Pack of M&M mini. Each pack contains candies of six colors: Blue (Bl), Brown (Br), Green (G), Orange (O), Red (R) and Yellow (Y). You will need to count number of candies of each color and send the data to a designated person. The sample data you need to send should look like this Bl – 27 Br – 23 G – 18 O – 25 R – 31 Y – 34 Please send the data by Thu, Feb. 2nd. Please do not invent the data; it is important that you do actual counting. 2. Now you can eat your candies, discard them, or share with your friends. 3. After all data is collected you will get back a table containing the results from the whole group. Each of you need to independently perform the statistical analysis of the data and answer the following questions. • Estimate the variance and the mean for the number of candies of each color. Are these results consistent with the Poisson distribution? • Estimate the variance of the sum of B
candies (i.e. blue + brown). Is it approximately equal to the sum of variances? Are the blue and brown colors statistically independent? • Estimate the variance of the total number of candies (i.e. blue + brown + green + orange + red + yellow). Are all colors statistically independent? Explain your result. • Plot the distribution function (in mathematica, excel, or any other software) for all colors (or all colors except red) from all students, i.e. the horizontal axis should be the number of candies and the vertical axis is the normalized frequency of occurrence of this number. The total number of data points should be 6 x number of analyzed packs. Fit this distribution to the Poisson distribution and Gaussian distribution. Which if the distributions work better? Argue why. A
lue
B
Brown
Green
Orange
Red
Yellow
Total B C D E F G H I J K L M N O 43
28
22
33
43
33 26
38
30
19
46
39 27
30
33
36
41
33 24
42
27
38
39
31 25
30
35
32
41
36 27
29
25
30
54
36 32
36
30
18
47
31 35
30
28
30
34
42 31
31
33
28
35
39 27
34
26
40
41
32 38
27
30
32
45
27 27
39
20
37
36
41 25
40
30
27
47
33 27
34
33
37
41
23 31
36
31
26
34
40 202 198 200 201 199 201 194 199 197 200 199 200 202 195 198 note: 12/18 packs have red as max.! Lecture5 Lecture6 Homework 3 (due Feb. 16th) problems from Reif: 2.1, 2.2, 2.4, 2.8,2.10 Lecture7 Lecture8 Homework 4 (due Feb. 23rd) problems from Reif: 3.1, 3.3, 3.4 *Repeat the following problem considered in the notes but for spin 1 particles. Assume that we have a system of large number N of noninteracting particles described by the Hamiltonian: ! = −ℎ ! !! , where !! = −1,0,1 . Find the approximate expressions for the density of states ! ! and then find the temperature as a function of the energy and the magnetic field. Invert this relation and express energy in the system as the function of temperature. Show that your result is consistent with the Boltzmann distribution where the probabilities of spins to be in one of the three possible states are ! ! !!! = ! ! ! , !! = ! , !!! = ! ! !! , ! !! = 1 + 2 cosh !
! Hint: number of configurations such that !! .spins have magnetization 1, !! .
have magnetization 0, and !! .have magnetization
1 is given by the polynomial distribution !!
! !! , !! , !! =
, !! ! !! ! !! !
where !! + !! + !! = !. Because the total energy is fixed ! = −ℎ(!! − !! ) we have an additional constraint. Using the Stirling’s approximation maximize the number of configurations with respect to the remaining variable. This is approximately ! ! you need to find. Lecture9 Lecture10 ...
View
Full
Document
This note was uploaded on 03/17/2012 for the course PHYS 204 taught by Professor Atkin during the Spring '12 term at Amity University.
 Spring '12
 atkin
 Work

Click to edit the document details