3.3 Problems for Chapter 3

Tutors' page

EXERCISES:

3.1 Geometry of $n$-spheres I: Calculate the volumes and surfaces of spheres with $r=1.0$ in 3,6,12 and 100 dimensions. In evaluating $100!$ use the Stirling approximation $n! \approx \sqrt{2 \pi n} (n/e)^{n}$.

3.2 Geometry of $n$-spheres II: For the various $n$-spheres of the foregoing example, compute the fraction of the total volume contained in a shell between $r=0.95$ and $r=1.0$. Comment the result.

3.3 Approximate formula for $\ln V_{n}(z_{0})$: Choose your own example to verify the validity of the approximation 3.30. Put the meaning of this equation in words; demonstrate the geometrical meaning for a three-dimensional sphere - even if the approximation is not yet valid in this case. Make a few runs with Applet Entropy1 and comment on your findings.

3.4 Sackur-Tetrode equation: Calculate the entropy of an ideal gas of noble gas atoms; choose your own density, energy per particle, and particle mass (keeping in mind that the gas is to be near ideal); use the mimimal grid size in phase space, $g=g_{min} \equiv h/m$.




TEST YOUR UNDERSTANDING OF CHAPTER 3:

3.1 Geometry of phase space: Explain the concepts phase space, energy surface, energy shell.

3.2 Entropy and geometry: What is the relation between the entropy of a system and the geometry of the phase space?

3.3 Geometry of $n$-spheres: Name a property of high-dimensional spheres that simplifies the entropy of an ideal gas. (Hint: shell volume?)