(a) Solve the integral 3 N 0 3 N i = 1 | x...

(a) Solve the integral

$${\int }_{\begin{array}{c}3N\\ 0\le {\sum }_{i=1}^{3N}\left|x_i\right|\le R\end{array}}(d{x}_1\dots d{x}_{3N})$$

and use it to determine the "volume" of the relevant region of the phase space of an extreme relativistic gas $(\epsilon =pc$ ) of $3N$ particles moving in one dimension. Determine, as well, the number of ways of distributing a given energy $E$ among this system of particles and show that, asymptotically, ${\omega }_0=h^{3N}$.

(b) Compare the thermodynamics of this system with that of the system considered in Problem 2.8.

Data From Problem 2.8

Following the method of Appendix C, replacing equation (C.4) by the integral

$${\int }_0^{\infty }{e}^{-r}{r}^{2}dr=2$$

show that

$$V_{3N}=\int \underset{{\int }_{0\le 1}^N{r}_i\le R}{}\dots \prod _{i=1}^N\left(4\pi r_i^{2}d{r}_i\right)={\left(8\pi R^3\right)}^N/\left(3N\right)!$$

Using this result, compute the "volume" of the relevant region of the phase space of an extreme relativistic gas $(\epsilon =pc)$ of $N$ particles moving in three dimensions. Hence, derive expressions for the various thermodynamic properties of this system and compare your results with those of Problem 1.7.

Data From Problem 1.7

Study the statistical mechanics of an extreme relativisitic gas characterized by the single-particle energy states

\[
\varepsilon\left(n_{x}, n_{y}, n_{z}ight)=\frac{h c}{2 L}\left(n_{x}^{2}+n_{y}^{2}+n_{z}^{2}ight)^{1 / 2}
\]

instead of (1.4.5), along the lines followed in Section 1.4. Show that the ratio \(C_{P} / C_{V}\) in this case is \(4 / 3\), instead of \(5 / 3\).