# Grand Cannonical Ensemble

Each system in the microcannonical ensemble has the same fixed energy and the same amount of particles (N, V, E). This ensemble's system is a closed, isolated thermodynamic system (because its energy remains same). The practical applicability of this kind of ensemble is limited.

In a canonical ensemble, we permit the system to interchange energy with the other components of the system but not particles. This enables the interchange of energy between the systems, which is a common occurrence in many thermodynamically related systems.

There are various procedures where the particle count fluctuates. These system are open system and can exchange both heat and matter with its surrounding and therefore both the energy and particle number will fluctuate such that ensemble µ, V and T fixed and E and N fluctuating is called Grand cannonical ensemble. Grand Cannonical ensemble direct access to the equation of state.

Let us consider a cannonical ensemble for a system with particles N, volume V and temperature. Let us focus on a small sub volume V1 has N1 particles such that V2 = V-V1 and N2 = N- N1

Hamiltonian of the system after neglecting molecular interactions the surface separating V1 and V2 may be written as

H(p,q,N) = H1(p1,q1, N1) + H2(p2,q2, N2).................1

The density function ρ1(p1,q1,N1) is proportional to the probability that in volume V1 there are N1 particles with coordinates (p1,q1). We can obtain ρ(p1,q1,N1) by interacting over the phase space of the system β i.e

ρ(p1,q1,N1) ∝ QN2 (V2,T)

This note is taken from Statistical Mechanics, MSC physics, Nepal.

This note is a part of the Physics Repository.