= Gibbs--Bogoliubov--Feynman inequality
{c}
= Feynman-Bogoliubov inequality
{c}
{synonym}
For an exact dimensionless <Hamiltonian> $H$ and trial <Hamiltonian> $H_0$, the <free energies> obey
$$
F\le F_0+\langle H-H_0\rangle_0,
$$
where $F=-\ln Z$, $F_0=-\ln Z_0$, and the <expectation value> uses the normalized trial equilibrium state. A physical <Hamiltonian> is multiplied by $\beta=1/(k_BT)$ in these formulas; multiplying the bound by $k_BT$ restores physical <free energy> units.
For classical configurations, the identity $e^{-F}=e^{-F_0}\langle e^{-(H-H_0)}\rangle_0$ and the <Jensen inequality> for the convex <exponential function> give $\ln\langle e^{-(H-H_0)}\rangle_0\ge-\langle H-H_0\rangle_0$. Taking minus the logarithm proves the bound.
The same bound holds for finite-dimensional, noncommuting quantum <Hamiltonians>, although the preceding exponential-factorization identity cannot generally be used. Define the <density matrices> $\rho=e^{-H}/Z$ and $\rho_0=e^{-H_0}/Z_0$. With natural logarithms, <nonnegativity of quantum relative entropy> gives
$$
0\le D(\rho_0\|\rho)=\operatorname{Tr}[\rho_0(\ln\rho_0-\ln\rho)]=F_0-F+\operatorname{Tr}[\rho_0(H-H_0)].
$$
Rearrangement proves the quantum result without requiring $[H,H_0]=0$. Equality holds when $\rho_0=\rho$; in this finite-dimensional setting that means $H-H_0$ is a scalar multiple of the <identity operator>.<The Theory of Quantum Information by John Watrous>[Proposition 5.22, Klein's inequality; the displayed thermal-state substitution derives the variational free-energy bound, using natural rather than the book's base-two logarithms]{ref}
A tractable trial family therefore supplies upper bounds that can be minimized over its parameters. This variational minimum need not reproduce exact <correlations> or <phase transition> locations.
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