• It is a collection of essentially independent assemblies having the same temperature T, volume V, and number of identical particles N.
• To assure that all the assemblies have the same temperature, we could bring all the assemblies in thermal contact with each other.

• In figure a canonical ensemble is shown in which all the individual assemblies are separated by rigid, impermeable, but diathermic walls.
• Since energy can be exchanged between the assemblies therefore they will reach a common temperature.
• Thus in canonical ensemble, assemblies can exchange energy, but not particle.

• Let an isolated system be made up of two subsystems such that H₁ (p₁, q₁) and H₂ (p₂, q₂) be the Hamiltonian of the subsystems, and N₁ and N₂ be the number of particles in the subsystems.
• Also N₂ >> N₁, but N₁ and N₂ are macroscopically large.
• Now, we consider a microcanonical ensemble of the composite system having total energy between E and E + 2ΔE.
• Also the energies of the subsystems E₁ and E₂ satisfy, E < E₁ + E₂ < E + 2ΔE.
• Since the analysis of microcanonical ensemble shows that only one set of values of E₁ (bar) or E₂ (bar) is important.

         Let E₂ (bar) >> E₁ (bar)

• Also Γ₂ (E₂) is the volume occupied by system 2 in its own Γ space.
• The probability of finding the system 1 in a state within dp₁dq₁ of (p₁, q₁) regardless of the state of system 2 is directly proportional to dp₁dq₁ Γ₂(E₂)

∴       P₁ ∝ dp₁dq₁ Γ₂(E₂)                                 Where E₂ = E – E₁

∴        The density in Γspace for system 1 up to a probability constant.

• exp [S₂(E)/k] is independent of E₁ so it is constant for a small subsystem.
• Also E₁ = H₁ (p₁, q₁)

∴      The ensemble density for a small subsystem, ρ (p, q) ∝ exp [ –H (p, q) / kT]

• The ensemble defined by this equation is known as canonical ensemble and the larger subsystem behaves as a heat reservoir in thermodynamics.
• The volume in Γspace occupied by the canonical ensemble (partition function)

• Here h = dpdq is a constant, we use it to make Q_N  dimensionless.
• Also 1/N! appears from the rule of correct Boltzmann counting.
• The thermodynamics of the system can be obtained by,

         Q_N (V, T) = exp [–βA (V, T)],     here  A (V, T) = Helmholtz free energy

Justification of this equation

A is an extensive quantity

         Q_N (V, T) = ∫ [(d^3Np d^3Nq )/ (h^3N N!)] exp [–βH (p, q)]

• From above equation if the system is made up of two subsystems, whose mutual interaction can be neglected, then Q_N is a product of two factors.

• In this way we can say that A is an extensive quantity.

A is related to the internal energy and the entropy as

         U = < H > = A + TS

Proof

         From identity, 1 / (h^3N N!) ∫ dp dq exp [β { A (V, T) – H (p, q)}]  = 1

• Differentiating on either side with respect to β

• Other thermodynamic functions can be determined from A (V, T) by the Maxwell’s thermodynamic relation as