1
[Background information]
Consider a system consisting of N indistinguishable particles confined in a box of volume
V and completely isolated from its environment. Since the system is isolated, the total
energy E is conserved. Hence, only the phase points on the hyper surface specified by
$\mathcal{H}(q,p) = E$ are realizable, where $\mathcal{H}(q,p)$ is the Hamiltonian of the system. We assume
that each point on the surface is equally probable. Then, the probability density of finding
the system at a phase point $(q,p)$ is given by
$$
\rho(q,p) = \begin{cases}
\frac{1}{\Omega(E,V,N)}, & \mathcal{H}(q,p) = E \\
0, & \text{otherwise}
\end{cases}
$$
where $\Omega(E, V, N)$ is the "volume" of the hyper surface defined by
$$
\int_{\mathcal{H}(p,q)=E} d\Gamma = \Omega(E, V, N)
$$
with $d\Gamma = dq^{3N} dp^{3N} /h^{3N} N!$.
[Problem]
(a) Derive the Boltzmann's entropy $S(E, V, N) = k \ln \Omega(E, V, N)$ from the Gibbs' definition of entropy $S = -k \int \rho(q, p) \ln \rho(q, p) d\Gamma$.
(b) Using the Boltzmann's entropy, show that the temperature of the system is given by
$$
T = \frac{\Omega}{k \frac{\partial \Omega}{\partial E}}.
$$
