Classical Analogue

Microscopic states: Quantum statistical mechanics defines a microscopic state of a system in Hilbert space according to a well defined set of quantum numbers. Classical statistical mechanics, however, describes the microscopic state in phase space according to a well defined set of coordinates $(x_1, ...x_f)$ and momenta $(p_1, ..., p_f)$.
Ensembles: Quantum statistical mechanics describes an ensemble according to the density operator $\hat{\rho}$, introduced by Eq. (11). Classical statistical mechanics, however, describes an ensemble according to the density of states $\rho = \rho(x_1, ...x_f, p_1, ..., p_f)$.
Time-Evolution of Ensembles: Quantum statistical mechanics describes the time-evolution of ensembles according to Eq. (17), which can be regarded as the quantum mechanical analogue of the Liouville's theorem of classical statistical mechanics,

$$\frac{\partial \rho}{\partial t} = -\left( \rho , H \right),$$ (18)

Eq. (18) is the equation of motion for the classical density of states $\rho = \rho(x_1, ...x_f, p_1, ..., p_f)$. Thus the name density operator for $\rho$ appearing in Eq. (17). Note that the classical analog of the commutator $\frac{[G,F]}{i\hbar}$ is the Poisson bracket of $G$ and $F$,

$$\left( G, F \right) \equiv \sum_{j=1}^f \frac{\partial G}{\partia......}{\partial p_j}-\frac{\partial G}{\partial p_j}\frac{\partial F}{\partial q_j}.$$ (19)

Exercise 5: Prove Eq. (18) by using the fact that the state of a classical system is defined by the coordinates $(x_1, ...x_f)$ and momenta $(p_1, ..., p_f)$ which evolve in time according to Hamilton's equations, i.e.,

$$\frac{\partial p_j}{\partial t} = - \frac{\partial H}{\partial x_......hspace{1.cm} \frac{\partial x_j}{\partial t} = \frac{\partial H}{\partial p_j},$$ (20)

where $H = \sum_{j=1}^f p_j^2/(2 m_j) + V(x_1, ... x_f)$ is the classical Hamiltonian.
Ensemble Averages: Quantum statistical mechanics describes ensemble averages according to Eq. (10). Classical statistical mechanics, however, describes ensemble averages according to the classical analog of Eq. (10),

$$\bar{A} = \frac{\int d{\bf x} \int d{\bf p} \rho(x_1, ..., x_f, p_1, ..., p_f) A}{\int d{\bf x} \int d{\bf p} \rho(x_1, ..., x_f, p_1, ..., p_f)},$$ (21)

where $d{\bf x} d{\bf p}$ stands for a volume element in phase space.