Statistical Physics: Ensembles

Ensembles in Chemical Sciences and Statistical Physics are a collection of a number of macroscopically identical but essentially independent systems.

The System, here, is defined by the collection of a large number of particles and the term macroscopically identical means that each of the systems constituting an ensemble satisfies the same macroscopic conditions, like Volume, Energy, Pressure, Temperature and the total number of particles, etc.

Here again, the term essentially independent means the system (in the ensemble) is mutually non-interacting to others, i.e., the systems differ in microscopic conditions like parity, symmetry, quantum states etc.

Types of Ensembles

There are three types of ensembles:

  1. Micro-canonical Ensemble
  2. Canonical Ensemble
  3. Grand Canonical Ensemble

Micro-canonical Ensemble

It is the collection of a large number of essentially independent systems having the same energy E, volume V and total number of particles N.

The systems of a micro-canonical ensemble are separated by rigid impermeable and insulated walls, such that the values of E, V & N are not affected by the mutual pressure of other systems.

Micro-canonical ensemble is as shown in the figure below.

Here all the borders are impermeable and insulated.

Micro-canonical Ensembles

Canonical Ensemble

It's the collection of a large number of essentially independent systems having the same temperature T, volume V and the number of particles N.

The equality of temperature of all the systems can be achieved by bringing all the systems in thermal contact. Hence, in this ensemble, the systems are separated by rigid, impermeable but conducting walls, the outer walls of the ensemble are perfectly insulated and impermeable though.

This ensemble is as shown in the figure:

Canonical Ensembles

Here, the borders in bold shade are both insulated and impermeable, while the borders in light shade are conducting and impermeable.

Grand Canonical Ensemble

It is the collection of a large number of essentially independent systems having the same temperature T, volume V & chemical potential μ.

The systems of a grand canonical ensemble are separated by rigid permeable and conducting walls. This ensemble is as shown in the figure:

Grand Canonical Ensembles

Here inner borders are rigid, permeable and conducting, while outer borders are impermeable as well as insulated.

As the inner separating walls are conducting and permeable, the exchange of heat energy as well as that of particles between the system takes place, in such a way that all the systems achieve the same common temperature T and chemical potential μ.

Ensemble Average

Every statistical quantity has not an exact but an approximate value. The average of a statistical quantity during motion is equal to its ensemble average.

Let $R(x)$ be a statistical quantity along the x-axis and $N(x)$ be the number of phase points in phase space, then the ensemble average of the statistical quantity $R$ is defined as,

$ \bar{R} := \dfrac{\int_{-\infty}^{\infty} R(x) N(x) \mathrm{d} x}{\int_{-\infty}^{\infty} N(x) \mathrm{d} x}$

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