I'm studying different ensembles and different statistics (M-B, B-E, F-D), and I have some ambiguities about which of these models are applicable to quantum systems and which are usable for classical systems or both.
(For example, the applicability of grand canonical ensemble to classical systems, M-B statistics to quantum mechanical systems, etc.)
So I arrived at the question of what are the key properties of classical and quantum statistical mechanical systems, and what are their differences?
Best Answer
Let's recall basics of classical and quantum mechanics for non-statistical systems.
In classical Hamiltonian mechanics, one models the non-statistical state of a system as a point in phase space $\mathcal P$. If the configuration space (space of spatial positions) of the system is $N$-dimensional, then the phase space is $2N$ dimensional because the state of the system is described both by its position and its momentum. The time evolution of the system is governed by the hamiltonian $H$, a real-valued function defined on phase space, in terms of which the dynamical equations of the system, Hamilton's equations, are written.
In quantum mechanics, one models a non-statistical state of a system as a point (vector) in a Hilbert space $\mathcal H$, a complex vector space that can be infinite-dimensional. The time evolution of the state of the system is government by the Hamiltonian $\hat H$, a self-adjoint operator on $\mathcal H$ in terms of which the dynamical equation of the system, the Shrodinger equation, is written.
When one moves to statistical mechanics, then the state of a classical mechanical system is no longer modeled as a point in phase space, but rather as a probability density $\rho$ on phase space. This probability density encodes the fact that one is ignorant about the exact states (positions and momenta) of individual particles in the system, and can be thought of in terms of ensembles of identically prepared systems, and one becomes primarily concerned with computing statistical quantities like ensemble averages of a given observables for a given phase density. The phase density will take different forms based on the ensemble (namely based on how the macroscopic state of the system in prepared), and for a given ensemble, one can define an object called the partition function which allows one to compute, for example, the ensemble average of any observable for a system in that ensemble.
For example, the partition function for $N$ identical particles in the canonical ensemble in classical mechanics will be the following phase space integral: \begin{align} Z(\beta) = \frac{1}{N!h^{3N}}\int d^{3N}pd^{3N}q\, e^{-\beta H(p,q)} \end{align} where $\beta = 1/kT$.
In quantum statistical mechanics, the state of the system is again no longer modeled in the same way (as a vector in Hilbert space), but rather as a non-negative self-adjoint operator $\hat \rho$ of unit trace called the density operator (or density matrix). As in the classical case, one uses this operator to determine statistical quantities such as ensemble averages of observables. In particular, one can again compute the partition function as in the classical case, but the expression will be different. Concretely, it its the Hilbert space trace of the density operator; \begin{align} Z = \mathrm{tr}\hat \rho = \sum_i \langle i|\hat \rho|i\rangle \end{align} where $\{|i\rangle\}$ is a basis for the Hilbert space.
In particular, for $N$ identical particles, one needs to careful to distinguish between the computation one does with fermions, and that performed for bosons. For $n$ identical fermions, the Hilbert space of the system will be restricted to the antisymmetric subspace of the $n$ particle Hilbert space. On the other hand, if the system consists of identical bosons, then the Hilbert space of the system is restricted to be the symmetric subspace of the $n$-particle Hilbert space. It follows that, for example, when one determines the partition function for such systems, one needs to trace over the appropriate Hilbert space. Since the antisymmetric and symmetric Hilbert spaces don't generally coincide, the partition functions for fermionic systems will generally be completely different than those of bosonic systems.