Isothermal Compressibility and Mean Square Fluctuation

a) Derive the relation

where k_τ is the isothermal compressibility:

b) From (a), find the relation between k_τ and the mean square fluctuation of N in the grand canonical ensemble. How does this fluctuation depend on the number of particles?

SOLUTION

a) Let us use the Jacobian transformation for thermodynamic variables:

Since the chemical potential μ is expressed in P, τ and does not depend on N, we can write

(1)

where s = S/N and v = V/N are reduced entropy and volume respectively. Using the equation for the Gibbs free energy of a single-component system, G = μN, we can write

where we also used V = (∂G/∂P)_τ. But from (1),

So finally

b) By definition the average number of particles in the grand canonical ensemble is

where Ω = F – μN. Now, from (a),

where N is an average number of particles:

(2)

where we have used ⟨N⟩ = -∂Ω/∂μ. So

(3)

From (3)

and

Since V is proportional to ⟨N⟩,

The relative fluctuation is given by