p211p1
1
Problem set I, Physics 211, University of California, Berkeley Due Friday, Feb 5, 5 pm in 251 Le Conte (see box) 1. St. Petersburg lottery: Suppose that you pay a fixed fee of x dollars to enter the following game. A fair coin will be tossed repeatedly until it comes up tails. If it first comes up tails on the nth flip, you get 2 n dollars. (a) If the game can go on forever, how much should you be willing to pay to enter (i.e., what is the expectation value of your return from the game)? (b) If the bank running the lottery can pay you at most 1 trillion dollars, estimate the expec- tation value of your return from the game. 2. Log-normal distribution: Suppose that you start investing with x 0 dollars. At the end of each year, your investment either increases by 10 percent or goes down by 10 percent (i.e., x n+1 = ( 1.1x n with p =1/2 0.9x n with p =1/2. (1) Write a probability distribution for the number of dollars after N years (you may assume that N is large enough to go to the continuum approximation, and you may write an approximate form valid near the distribution’s peak). What is the expected number of final dollars? Write an integral for the probability that your final wealth is greater than your initial wealth. Using a computer or a table of standard functions, estimate this probability for N = 50. Sketch the distribution of final wealth for N = 50. Hint: if you get stuck, there is an article in Wikipedia with the same title as this problem that may be helpful. 3. Reif 2.7 (you can check your answer to (b) in the back of the book). “Consider a particle confined within a box...” 4. Reif 3.6. This should not require much calculation. “A glass bulb...” 5. 2D harmonic oscillator: Compute the density of states for a two-dimensional harmonic oscillator with frequency ω, i.e., E nx,ny = ~ω(1 + n x + n y ) (2) for nonnegative integers n x and n y . What is the entropy S (E) in the continuum limit? 6. Characterization of entropy: Show that the entropy S = - ∑ N i=1 p i log p i is maximized (with fixed N ) for the case of equal probabilities. Then show that combining two independent random processes leads to addition of the entropy. It turns out that these two properties, plus some basic assumptions about continuity and symmetry, are enough to determine the entropy formula uniquely. 1
-
Upload
patrick-sibanda -
Category
Documents
-
view
213 -
download
0
description
Assignment on statistical mechanics
Transcript of p211p1
Problem set I, Physics 211, University of California, BerkeleyDue Friday, Feb 5, 5 pm in 251 Le Conte (see box)
1. St. Petersburg lottery: Suppose that you pay a fixed fee of x dollars to enter the followinggame. A fair coin will be tossed repeatedly until it comes up tails. If it first comes up tails on thenth flip, you get 2n dollars.
(a) If the game can go on forever, how much should you be willing to pay to enter (i.e., whatis the expectation value of your return from the game)?
(b) If the bank running the lottery can pay you at most 1 trillion dollars, estimate the expec-tation value of your return from the game.
2. Log-normal distribution: Suppose that you start investing with x0 dollars. At the end ofeach year, your investment either increases by 10 percent or goes down by 10 percent (i.e.,
xn+1 =
{1.1xn with p = 1/20.9xn with p = 1/2.
(1)
Write a probability distribution for the number of dollars after N years (you may assume thatN is large enough to go to the continuum approximation, and you may write an approximate formvalid near the distribution’s peak). What is the expected number of final dollars?
Write an integral for the probability that your final wealth is greater than your initial wealth.Using a computer or a table of standard functions, estimate this probability for N = 50. Sketchthe distribution of final wealth for N = 50.
Hint: if you get stuck, there is an article in Wikipedia with the same title as this problem thatmay be helpful.
3. Reif 2.7 (you can check your answer to (b) in the back of the book). “Consider a particleconfined within a box...”
4. Reif 3.6. This should not require much calculation. “A glass bulb...”
5. 2D harmonic oscillator: Compute the density of states for a two-dimensional harmonicoscillator with frequency ω, i.e.,
Enx,ny = ~ω(1 + nx + ny) (2)
for nonnegative integers nx and ny. What is the entropy S(E) in the continuum limit?
6. Characterization of entropy: Show that the entropy S = −∑N
i=1 pi log pi is maximized (withfixed N) for the case of equal probabilities. Then show that combining two independent randomprocesses leads to addition of the entropy. It turns out that these two properties, plus some basicassumptions about continuity and symmetry, are enough to determine the entropy formula uniquely.
1
