Statistical pressure for a canonical ensemble

[Eulersheep]

So the pressure for a canonical ensemble is:

P = k_bT dZ/dV

P = pressure

P = -∑p_i dE_i/dV
Z = ∑e^-βE_i

p_i is the probability of being in microstate i
E_i is the energy of state i
β = 1/k_bT

<E> = U = average energy

U = -1/Z dZ/dβ = -d(Ln(Z))/dβ

How can the pressure (given above) be derived in terms of the partition function?

 

[LightHero]

The pressure can be derived from the partition function by taking the derivative of the logarithm of the partition function with respect to the inverse temperature, β: P = -1/Z dZ/dβ = -d(Ln(Z))/dβ