Physics 6311: Statistical Mechanics – Homework 1

Problem 1: Exact differentials (14 points)

a)

Test whether the following differentials are exact.

$$\begin{array}{rcll}d{u}_a& =& (x^2-y^2)\mathit{ 𝑑𝑥}-2\mathit{𝑥 𝑑𝑦}& \\ d{u}_b& =& y^2\mathit{ 𝑑𝑥}+2\mathit{𝑥𝑦 𝑑𝑦}& \end{array}$$

b)

If the differential is exact, calculate the indefinite integral.

c)

Check the dependence of the integral on the path of integration by explicitly integrating both differentials from point $(x_i,y_i)=(0,0)$ to $(x_f,y_f)=(2,2)$ on two different path, $(0,0)\to (2,0)\to (2,2)$ and $(0,0)\to (0,2)\to (2,2)$. Compare the results of the two path and that of a calculation using the indefinite integral (if it exists).

Problem 2: Properties the $\delta$ function (10 points)

Compute the following integrals by manipulating the $\delta$ function

Problem 3: Gaussian integrals (10 points)

Compute the following integral in terms of $A$ and $B$.

Problem 4: Equilibrium states (6 points)

Decide which of the following states is in an equilibrium state, a non-equilibrium steady state, or not a steady state. Explain your reasoning. In some cases, the state is not a true steady or equilibrium state but close to one. Discuss under what conditions it can be treated as a steady or equilibrium state.

a)

a cup of hot tea, sitting on the table while cooling down

b)

the wine in a bottle that is stored in a wine cellar

c)

the sun

d)

the atmosphere of the earth

e)

electrons in the wiring of a flashlight switched off

f)

electrons in the wiring of a flashlight switched on