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. 

Presently, we consider a few of the details associated with the derivation of the Bogoliubov inequality. Our treatment mirrors that of Callen (1985) who, once again, we find gets to the point in the cleanest and most immediate fashion. As said above, we imagine that the problem of real interest is characterized by a Hamiltonian H which may be written in terms of some reference Hamiltonian Ho... 

Fig. 3.21. Schematic of the free energy as a function of the parameter X discussed in the context of the Bogoliubov inequality.

As mentioned above, in light of eqn (6.15) we now aim to exploit the Bogoliubov inequality to bound the free energy. Our bound on the liquid state free energy may be written as... 

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 ... 

The proof of this result is based on exact bounds on the asymptotic behavior of the interfacial correlation functions, obtainable from the Bogoliubov inequality, and uses a reductio ad absurdum a self-maintained interface is assumed to exist and it is then shown that this assumption leads to a contradiction. The key step in the demonstration consists in deriving and making use of the asymptotic behavior at large separations of the direct correlation function of Ornstein-Zernike, c(r, r ), defined in terms of the more familiar pair correlation function ft(r, r ) which measures the probability of having a molecule at point r given that there is one at point r ... 

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