1.4 Problems for Chapter 1

Tutors' page

EXERCISES:

1.1 Entropy and irreversibility: (a) Two heat reservoirs are at temperatures $T_{1}$ and $T_{2}>T_{1}$. They are connected by a metal rod that conducts the heat $Q$ per unit time from 2 to 1. Do an entropy balance to show that this process is irreversible.

(b) Using the specific values $T_{1}=300K$, $T_{2}=320K$ and $Q=10 J$, calculate the entropy increase per second.

1.2 Reversible, isothermal expansion: Compute the entropy balance for the experiment described (or shown) in the lecture, using estimated values of the necessary experimental parameters.
Discuss the role of the ideal gas assumption: is it necessary / unnecessary / convenient ?

1.3 Maxwell square:
Draw up Maxwell's square and use it to complete the following equations:
(a) $G=A \pm \dots$; (b) $A=E \pm \dots$;
(c) $P= \dots = \dots$ (in terms of derivatives of $A$ and $E$);
(d) $dG=V dP - \dots$.

1.4 Random directions in 2D: Using your favourite random number generator, sample a number $K \approx 50-100$ of angles $\phi_{i}$ $(i=1, \dots, K)$ equidistributed in $[0, 2 \pi]$. Interpret $\cos \phi_{i}$ and $\sin \phi_{i}$ as the $v_{x,y}-$components of a randomly oriented velocity vector $\vec{v}_{i}$ with $\vert\vec{v}_{i}\vert=1$.
(a) Draw a histogram of the empirical probability (i.e. event frequency) of $p(v_{x})$. Compare the shape of your $p(v_{x})$ to the bottom right histogram in Applet Stadium (see 1.2).
(b) Normalize the histogram such that a sum over all bins equals one. What is the value of $p(0.5)$?

1.5 Binomial distribution: Suggest an experiment with given $p$ and $n$. Perform the experiment $N$ times and draw an empirical frequency diagram; compare with 1.39.

1.6 Density fluctuations in air:
(a) Calculate the mean number of molecules (not discerning between $N_{2}$ and $O_{2}$) that are to be found at a pressure of $10 Pa$ in a cube with a side length of the wave length of light ( $\approx 6 \cdot 10^{-7} m$). What is the standard deviation of the particle number, both absolutely and relative to the mean particle number? (Air is to be treated as an ideal gas at normal temperature $T_{0}=273.15 K$.)
(b) Compute the value of the probability density of the event $N(\Delta V) = k \approx <N>$, i. e. the probability of finding an integer number $k$ next to the mean number of particles in the sample volume? (Hint: Don't attempt to evaluate factorials of large numbers, such as appear in the binomial distribution $p_{k}^{n}$; rather, use that distribution which resembles $p_{k}^{n}$ when $n$ becomes large.)
What is the probability of finding only $95$ percent of the mean particle number in the sample volume?

1.7 Multinomial distribution: A volume $V$ is divided into $m=5$ equal-sized cells. The $N=1000$ particles of an ideal gas may be allotted randomly to the cells.
a) What is the probability of finding in a snapshot of the system the partitioning $\{ N_{1},N_{2}, \dots , N_{5}\}$? Explain the formula.
b) Demonstrate numerically that the partitioning with the greatest probability is given by $N_{i}= N/m = 200$. For example, compare the situations $(201,199,\dots)$, $(202,199,199,\dots)$, $(202,198,\dots)$, $(205,195,\dots)$, and $(204,199,199,199,199)$ to the most probable one.
c) (1 bonus point) Prove analytically that $N_{i}=200$ is the most probable distribution. Hint: minimize the function $f(\vec{k}) \equiv \log p_{n}(\vec{k})$ under the condition $\sum_{i} k_{i}=n$.

1.8 Transformation of a distribution density: Repeat the calculation of Example 3 for the two-dimensional case, i.e. $\vec{v} \equiv \{ v_{x}, v_{y}\}$ and $\vec{w} \equiv \{v, \phi \}$. Draw the distribution density $p_{2D}(v)$.

TEST YOUR UNDERSTANDING OF CHAPTER 1:

1.1 Thermodynamic concepts:
- What does the entropy balance tell us about the reversibility/irreversibility of a process? Demonstrate, using a specific example.
- Describe the process of thermal interaction between two bodies. When will the energy flow stop?
- Which thermodynamic potential is suited for the description of isothermal-isochoric systems?

1.2 Model systems:
- Describe 2-3 model systems of statistical mechanics.
- What quantities are needed to completely describe the momentary state of a classical ideal gas of $N$ particles?
- What quantities are needed for a complete specification of the state of a quantum ideal gas?

1.3 Statistical concepts:
- Explain the concepts ``distribution function'' and ``distribution density''; give two examples.
- What are the moments of a distribution? Give a physically relevant example.

1.4 Equal a priori probability: Explain the concept and its significance for statistical mechanics.