(7) |

and

(8) |

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,

(9) |

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
specified ,
and .
**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).