Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 344 / 1 / c

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- Solution c

Solution

Apply the Jensen inequality to the convex exponential function under the trial equilibrium measure:

⟨ e^{- (H - H_{0})} ⟩_{0} \geq e^{- ⟨ H - H_{0} ⟩_{0}} .

The given identity therefore implies

e^{- F} \geq e^{- F_{0} - ⟨ H - H_{0} ⟩_{0}}

. Taking minus the logarithm reverses the inequality and proves the Feynman-Bogoliubov inequality

F \leq F_{0} + ⟨ H - H_{0} ⟩_{0} = F_{0} - ⟨ H_{0} ⟩_{0} + ⟨ H ⟩_{0} .

The units

β = 1

make

H

and

F

dimensionless here. Equality holds when

H - H_{0}

is constant on the support of the trial measure. Optimizing the trial parameters tightens an upper bound, which is the basis of the Gaussian variational approximation.

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