Fluctuations Around the Saddle Point

A number of attempts have been made to incorporate the effect of composition fluctuations [86-88] in theories involving neutral polymers. Here, we present systematic one-loop expansion to go beyond the saddle-point approximation described in the previous section. In order to carry out the loop expansion, it is advantageous to use Hubbard-Stratonovich transformation to get rid of redundant functional integrals over collective density variables (q in Eq. (6.85)) and use Eq. (6.81) as the starting point for the partition function with the explicitly known normalization constants except A,. Saddle-point approximation within this formalism now requires taking functional derivatives with respect to fields only. 

Let us say after summing over charge distributions, the partition function becomes (cf. Eq. (6.81)) 

Within saddle-point approximation, the partition function is approximated by Z q, where saddle-point values for the fields are to be obtained by 

Similar attempts to include fluctuations beyond one-loop in SC FT have also been exercised in the context of neutral polymers using field theoretical simulations [55,90] or by bridging SCFT with Monte Carlo techniques [91]. However, these techniques have not been applied for the case of polyelectrolytes with counterions and added salt ions due to very high computational cost. 

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Let us consider now processes where intermediate stationary Hamiltonians are mediating the interconversion. In these processes, there is implicit the assumption that direct couplings between the quantum states of the precursor and successor species are forbidden. All the information required to accomplish the reaction is embodied in the quantum states of the corresponding intermediate Hamiltonian. It is in this sense that the transient geometric fluctuation around the saddle point define an invariant property. 

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