How to prove the Gibbs-Bogoliubov inequality

[evantop]

Hi

I need to prove the Gibbs-Bogoliubov inequality in two stages.
First I need to prove that if I have a canonical partition function so:

Q(N,V,T)>=Sigma(exp(-beta*<fi|H|fi>) by using the ritz variational principle
fi = set of orthonormal functions in the hilbert space.
Then... by using this inequality I need to prove that:
A1=< A2+<H1-H2> which is the Gibbs-Bogoliubov inequality. (A is the free helmholtz energy, and H is hamiltonian)

Anyone can help me or give me a link to a paper with the solution?

Thanks

Evan

 

[Jhero]

Hello Evan,

The Gibbs-Bogoliubov inequality is a fundamental result in statistical mechanics, which relates the free energy and the expectation value of the Hamiltonian in a canonical ensemble. I will provide an outline of the proof for you to follow:

Part 1: Proving the inequality using the ritz variational principle

1. Begin by writing the partition function for a canonical ensemble as: Q(N,V,T) = Tr(exp(-βH)) where β = 1/kT, k is the Boltzmann constant, and H is the Hamiltonian.

2. Using the ritz variational principle, we can write the partition function as a minimum of the trial partition function, Q_T(N,V,T) = Tr(exp(-βH_T)), over a set of orthonormal functions {ϕ_i} in the Hilbert space.

3. By definition, the trial partition function is always greater than or equal to the true partition function, i.e. Q_T(N,V,T) ≥ Q(N,V,T). This is because the trial partition function is a minimum of the true partition function over a restricted set of functions.

4. Now, we can expand the trial partition function using the orthonormality of the trial functions: Q_T(N,V,T) = Σ_i exp(-βE_i) <ϕ_i|ϕ_i> where E_i is the energy corresponding to the state ϕ_i.

5. Using the inequality from step 3, we can write: Σ_i exp(-βE_i) <ϕ_i|ϕ_i> ≥ Σ_i exp(-βE_i).

6. Since the true partition function is the sum of all possible energy states, we can write: Q(N,V,T) = Σ_i exp(-βE_i).

7. Combining steps 5 and 6, we get: Q_T(N,V,T) ≥ Q(N,V,T) which proves the first part of the Gibbs-Bogoliubov inequality.

Part 2: Proving A1 = A2 + <H1-H2> using the Gibbs-Bogoliubov inequality

1. The free energy can be written as: A = -kTlnQ. Using this, we can write the Gibbs-Bogoliubov inequality as: A1 ≥ A2 + <H1-H2>.

2. Substituting for the partition function, we get