Gibbs-Bogoliubov inequality

The first tenn in the high-temperature expansion, is essentially the mean value of the perturbation averaged over the reference system. It provides a strict upper bound for the free energy called the Gibbs-Bogoliubov inequality. It follows from the observation that exp(-v)l-v which implies that ln(exp(-v)) hi(l -x) - (x). Hence... 

The Gibbs-Bogoliubov inequalities set bounds on A A of (AU)0 and (AU) which are easier a priori to estimate. These bounds are of considerable conceptual interest, but are rarely sufficiently tight to be helpful in practice. Equation (2.17) helps to explain why this is so. For distributions that are nearly Gaussian, the bounds are tight only if a is small enough. 

Free energy bounds can be established via the Gibbs-Bogoliubov inequality [72], which follows from Eq. (2.6) by considering the convexity of the exponential function... 

Typically, we formally remove the term (V)o from this relation by incorporating it into H(, the Gibbs-Bogoliubov inequality [Eq. (75)] is then simply A Ag. Also, in this way the final system-bath coupling is defined to affect the system only through fluctuations, that is, at second order. 

Earlier work by Melenkevitz et al. was based on the standard Gibbs-Bogoliubov inequality for the single-chain free energy ... 

One consequence of the positivity of a is that A A < (AU)0. If we repeat the same reasoning for the backwards transformation, in (2.9), we obtain A A > (AU)V These inequalities, known as the Gibbs-Bogoliubov bounds on free energy, hold not only for Gaussian distributions, but for any arbitrary probability distribution function. To derive these bounds, we consider two spatial probability distribution functions, F and G, on a space defined by N particles. First, we show that... 

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