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Bogoliubov variational bound (Peierls–Bogoliubov inequality)

Upper-bound the free energy of an interacting system by comparing to a solvable reference Hamiltonian.
Bogoliubov variational bound (Peierls–Bogoliubov inequality)

The Bogoliubov variational bound is a widely used inequality that turns free-energy estimation into an optimization problem over tractable reference models. It underlies many “mean-field-like” approximations and provides controlled upper bounds on the .

Statement

Let HH be the true Hamiltonian and fix . Choose a reference Hamiltonian H0H_0 with partition function Z0Z_0 and free energy

F0=−1βlog⁡Z0. F_0 = -\frac{1}{\beta}\log Z_0.

Let ⟨⋅⟩0\langle\cdot\rangle_0 denote expectation in the reference defined by H0H_0.

Then the Bogoliubov (Peierls–Bogoliubov) inequality states

F≤F0+⟨H−H0⟩0, F \le F_0 + \langle H - H_0\rangle_0,

where F=−(1/β)log⁡ZF=-(1/\beta)\log Z is the true free energy.

Derivation idea (Jensen / convexity)

Write the partition function using a (for classical systems, this trace is the sum/integral over microstates):

Z=Tr exp⁡(−βH)=Tr exp⁡(−βH0) exp⁡ ⁣(−β(H−H0))=Z0 ⟨exp⁡ ⁣(−β(H−H0))⟩0. Z = \mathrm{Tr}\,\exp(-\beta H) = \mathrm{Tr}\,\exp(-\beta H_0)\,\exp\!\big(-\beta(H-H_0)\big) = Z_0\,\Big\langle \exp\!\big(-\beta(H-H_0)\big)\Big\rangle_0.

Because the exponential is convex, implies

⟨exp⁡ ⁣(−β(H−H0))⟩0≥exp⁡ ⁣(−β⟨H−H0⟩0). \Big\langle \exp\!\big(-\beta(H-H_0)\big)\Big\rangle_0 \ge \exp\!\big(-\beta\langle H-H_0\rangle_0\big).

Taking −β−1log⁡-\beta^{-1}\log of both sides gives the stated upper bound on FF.

The same inequality can also be viewed as a consequence of nonnegativity of between the trial Gibbs state generated by H0H_0 and the true Gibbs state.

Variational use

If H0H_0 depends on variational parameters θ\theta (e.g. effective fields), then

F≤inf⁡θ[ F0(θ)+⟨H−H0(θ)⟩0,θ]. F \le \inf_{\theta}\Big[\,F_0(\theta) + \langle H - H_0(\theta)\rangle_{0,\theta}\Big].

Optimizing the right-hand side yields the tightest bound within the chosen class of reference Hamiltonians.

A standard choice is to take H0H_0 to be a sum of independent one-site terms; optimizing then produces self-consistency equations that coincide with the .

Physical interpretation

The bound says: “an interacting system is no less stable (no lower free energy) than a solvable trial system plus the average interaction correction measured in the trial ensemble.” Practically, it converts difficult interacting thermodynamics into an optimization problem, while guaranteeing that the resulting approximate free energy is an upper bound on the exact one.