Statistical Mixture of States

The collection of a large number N of independently prepared replicas of the system is called an ensemble. An ensemble of N replicas of systems is in a statistical mixture of states $\vert\phi _k>$, with probabilities $p_k$, when $n_k$ members of the ensemble are in state $\vert\phi _k>$, with $p_k=n_k/N$. Note that each member of the ensemble is in a specific state $\vert\phi _k>$, not in a coherent superposition of states as described by Eq. (2). Therefore, the ensemble averages associated with the observables $\hat{o}$ and $\hat{A}$ are

$$A = \sum_k p_k <\phi_k\vert\hat{A}\vert\phi_k>,$$ (7)

and

$$o = \sum_k p_k <\phi_k\vert\hat{o}\vert\phi_k> =\sum_k p_k o_k,$$ (8)

respectively. Note that the ensemble average $o$, introduced by Eq. (8), coincides with the ensemble average of the pure state described by Eq.(5). However, the ensemble average $A$, introduced by Eq. (7), does not coincide with the corresponding ensemble average of the pure state, introduced by Eq. (6). As a matter of fact, it coincides only with the first term of Eq. (6) since the second term of the r.h.s. of Eq. (6) is missing in Eq. (7). Therefore, in a statistical mixture there are no contributions to the ensemble average coming from interferences between different states (e.g., interferences between states $\vert\psi_k \rangle$ and $\vert\psi_j \rangle$).

The statistical mixture introduced in this section, is also equivalent to an ensemble of N replicas of the system in incoherent superposition of states represented as follows,

$$\mid \psi(\xi) \rangle = \sum_k \sqrt{p_k} e^{i \theta_k(\xi)}\vert\phi_k>,$$ (9)

where the phases $\theta_k(\xi)$ are distributed among the different members $\xi$ of the ensemble according to a uniform and random distribution.

In the remaining of this section we introduce the most important types of ensembles by considering systems with only one species of molecules. Additional details for multicomponent systems are considered later.
In the canonical ensemble all of the replica systems are in thermal equilibrium with a heat reservoir whose temperature is $T$. This ensemble is useful for comparisons of the ensemble averages with measurements on systems with specified number of particles N, volume V and temperature T. It is central to Monte Carlo simulations, an important approximation method of Statistical Mechanics.

In the microcanonical ensemble all of the replica systems have the same energy $E$ and number of particles $N$. This ensemble is no very simply applicable to comparisons with systems we usually study in the laboratory, since those are in thermal equilibrium with their surroundings. However, the microcanonical ensemble is centrally involved in Molecular Dynamics simulations which is one of the most important approximation methods of Statistical Mechanics.

In the grand canonical ensemble all of the replica systems are in thermal equilibrium with a heat reservoir whose temperature is $T$ and they are also in equilibrium with respect to exchange of particles with a ``particle'' reservoir where the temperature is $T$ and the chemical potential of the particles is $\mu$. This ensemble is useful for comparisons to measurements on systems with specified $\mu$, $T$ and $V$.
Exercise 1: Compute the ensemble average $\bar{A}$ associated with the incoherent superposition of states introduced by Eq. (9) and verify that such an average coincides with Eq. (7).