Grand canonical ensemble | Statistical mechanics | L-9

Grand canonical ensemble

Grand canonical ensemble

Microcanonical ensemble is a collection of independent assemblies in which energy (E), volume (V), and number of particles (N) remain constant.
Canonical ensemble is a collection of independent assemblies in which temperature (T), volume (V), and number of particles (N) remain constant.
It mean a microcanonical ensemble ⟶ Canonical ensemble, if we ignore the condition E = constant.
Therefore the energy exchange takes place in this ensemble.
Actually in chemical process, the number of particles N varies and it is very difficult to keep the number of particles constant in various phenomenon like radioactive decay process.
Thus grand canonical ensemble is an ensemble in which the exchange of energy as well as the number of particles takes place with the heat reservoir.
The grand canonical ensemble is a collection of essentially independent assemblies having the same temperature T, volume V, and a chemical potential µ.

Thus grand canonical ensemble is a situation in which we know both the average energy and the average number of particles in assembly, otherwise we don’t know the state of the system.
The density function in Гspace for grand canonical ensemble = ρ (p, q, N)
Here p is momenta, q is coordinate and N is number of particles.

To find ρ (p, q, N), we consider the canonical ensemble for a system with particles N, volume V and temperature T.
Here we can not take exactly V = constant, so our focus is on a small sub-volume V₁ of the system.
Let N₁ particles be in volume V₁, and N₂ particles be in volume V₂
N₁ + N₂ = N ⇒ N₂ = N - N₁ and V₁ + V₂ = V ⇒ V₂ = V - V₁
Also we assume that V₂ >> V₁ and N₂ >> N₁
If H₁ and H₂ be the Hamiltonian of the system then
The total Hamiltonian of the composite system
H (p, q, N) = H₁ (p₁, q₁, N₁) + H₂ (p₂, q₂, N₂)

Partition function of the system

The partition function of the total system

Here A (N, V, T) is the Helmholtz free energy.

In the last term we consider that the system under consideration to become infinite in size or 0 ≤ N < ∞.
For thermodynamic function, we define grand partition function

Average number of particle

By ensemble average, the average number of particles

Internal energy

To know more about this topic please click the link https://youtu.be/1dpbmImt8UE

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