Physics 6311: Statistical Mechanics - Homework 3

Problem 1: Box distributions (16 points)

The random variables $X$ and $Y$ are independent and have identical box distributions

a)

Find the averages $⟨x⟩$ and $⟨y⟩$.

b)

Compute the variances ${\sigma }_x^2$ and ${\sigma }_y^2$.

c)

A new random variable $Z$ is defined as $Z=X+Y$. Find its average $⟨z⟩$ and variance ${\sigma }_z^2$

d)

Derive the probability density $P_Z(z)$ of the random variable $Z$. (Hint: Use the method of characteristic functions)

Problem 2: Random window panes (12 points)

A machine in a factory making glass window panes is malfunctioning. As a result, it is producing rectangular windows of random size. Specifically, the horizontal and vertical sizes of the window are independent random quantities. They can take values between 0 and 2 m with a constant probability density.

a)

Calculate the average area $⟨A⟩$ of the produced windows and its standard deviation.

b)

Derive the probability density of $A$. (Hint: Be careful with the integration bounds when transforming and integrating over the $\delta$-function)

c)

What is the most likely area?

Problem 3: Probability of a density fluctuation (12 points)

Consider two identical boxes, A and B.

a)

10 particles are distributed over the two boxes at random. Calculate the probabilities $P(4)$ and $P(5)$ for finding exactly $N_A=4$ and $N_A=5$ particles in the box A, respectively. Calculate $P(4)∕P(5)$.

b)

Repeat the calculations for $1000$ particles. Compare $N_A=400$ and $N_A=500$. (Hint: It may be convenient to first compute $\mathit{ln}[P(400)∕P(500)]$ and then re-exponentiate the result. (For large $n$ the factorial can be approximated by Stirling’s formula $\mathit{ln}(n!)\approx nln(n)-n$)