Ensemble in Statistical Mechanics | Microcanonical | Canonical | Grand Canonical Ensemble

An ensemble is defined as collection of large number of microscopically but essentially independent systems. By the term macroscopically identical...

Ensembles

An ensemble is defined as collection of large number of microscopically but essentially independent systems. By the term macroscopically identical we mean that each of the systems constituting an ensemble satisfies the same macroscopic conditions e.g. volume, energy, pressure, total number of particle etc.

In an ensemble, the system plays the same role as the non-interacting molecules do in a gas. The macroscopic identity of the system constituting and assembly can be achieved by choosing the same values of same set of macroscopic parameters. These parameters uniquely determined the equilibrium state of the system.

Types of ensembles

Accordingly three types of ensembles that is, Micro canonical, Canonical and grand Canonical are most widely used. Their description is as follows.

  1. Microcanonical Ensemble:- The microcanonical assemble is a collection of essentially independent assemblies having the same energy E, volume V and number of systems N. The individual systems of a microcanonical ensemble are separated by rigid impermeable and we'll insulated walls such that the values of E, V and N for a particular system are not affected by the presence of other system.

  2. Microcanonical Ensemble
    Microcanonical Ensemble
  3. Canonical Ensemble:- The Canonical ensemble is a collection of essentially independent assemblies having the same temperature T volume V and number of identical particles N. The disparate systems of a canonical ensemble are separated by rigid, impermeable but conducting walls. Since energy can be exchange between the assemblies they will reach a common temperature. Thus in canonical ensemble can exchange energy but not particles. The quality of temperature of all the systems can be achieved by a bearing each in thermal contact with the large heat reservoir at constant temperature T.

  4. Canonical Ensemble
    Canonical Ensemble
  5. Grand Canonical Ensemble:- It is the collection of a large number of essentially independent systems having the same temperature T, volume V and chemical potential (μ). The individual system of grand canonical ensemble are separated by rigid, permeable and conducting walls. As the separating walls are conducting and permeable, the exchange of heat energy as well as that of particles between the systems takes place in such a way that all the system arrive at common temperature T and chemical potential (μ).

Grand Canonical Ensemble
Grand Canonical Ensemble

Uses of the Ensemble

In fact, the three ensembles which we have introduced are only examples of the infinite number of ensembles that can be considered. These three are particularly useful for two main reasons :

  • Firstly the corresponding approximately to the types of thermodynamics measurements most frequently made in practice.
  • Secondly, in large ensembles, it is useful to find that the values of thermodynamic quantities are not very sensitive to the method of measurement.

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