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,
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 . 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 and number of particles . 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
and they are also in equilibrium with respect to exchange of particles
with a ``particle'' reservoir where the temperature is
and the chemical potential of the particles is .
This ensemble is useful for comparisons to measurements on systems with
Exercise 1: Compute the ensemble average associated with the incoherent superposition of states introduced by Eq. (9) and verify that such an average coincides with Eq. (7).