= 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
$$
F\leq F_0+\langle H-H_0\rangle_0,
$$
where $F_0=-\log Z_0$ and the expectation uses the trial Gibbs distribution $Z_0^{-1}e^{-H_0}$.
The identity $e^{-F}=e^{-F_0}\langle e^{-(H-H_0)}\rangle_0$ and the <Jensen inequality> for the convex <exponential function> give $\log\langle e^{-(H-H_0)}\rangle_0\geq-\langle H-H_0\rangle_0$. Taking minus the logarithm proves the bound.
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