Saddle point approximation

The evaluation of Eq. 23 proceeds via the saddle-point expansion, where at the first level of approximation we look for an extremum 5(0,) of the bosonized action Seff with respect to the Bose and Lagrange multiplier fields 

The physically relevant saddle-point O, is determined to give the lowest free energy (per site) 

and The vector fields exhibit the same spatial vmation as the mag- 

If one substitutes Eq. 31 together with (obtained from the solution of the coupled self-consistency equations (Eq. 25)) into Eq. 29, the free energy of the t-t -J model is obtained as 

In particular, at the spatially uniform paramagnetic saddle-point, = e, Po, 1 0,0 0,0 0,0) the remaining bosonic fields 

For the sake of discussion, let us consider the smeared charge distribution so that Eq. (6.76) can be written as 

the Q s are the partition functions of individual components in the presence of a field. Explicitly, for the polyelectrolyte chain with smeared charge distribution along the backbone, single chain partition function is given by 

Similarly, the partition function for a solvent molecule is written as Qs = drexp[-iws(r)] and the partition function for the small ions of type j = c, +, — is given by 

As mentioned earlier, all the functional integrals over collective variables cannot be carried out exactly. One of the approximations used extensively in the literature to evaluate these functional integrals is called the saddle-point approximation [52, 55, 57, 58]. In this approximation, functional integrals over collective variables are approximated by the value of the integrand at the saddle point, that is, free energy is approximated to be 

Details of carrying out the functional derivatives [79] are presented elsewhere [53, 55]. The equations obtained after taking functional derivatives are presented here in the order presented in Eq. (73). 

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The total wavefunction r2,. . ., r is written as a product of single-particle functions (Hartree approximation). The various integrals are evaluated in tire saddle point approximation. A simple Gaussian fomr for tire trial one-particle wavefunction... 

A technical difference from other Gaussian wavepacket based methods is that the local hamionic approximation has not been used to evaluate any integrals, but instead Maiti nez et al. use what they term a saddle-point approximation. This uses the localization of the functions to evaluate the integrals by... 

A saddle point approximation to the above integral provides the definition for optimal trajectories. The computations of most probable trajectories were discussed at length [1]. We consider the optimization of a discrete version of the action. 

Since the integrand of the integral inside the square brackets is sharply peaked around the integral can be performed by the saddle point approximation, and the result is... 

The dots in (15) denote terms which are not relevant in the thermodynamic limit T —> oo. Using the saddle point approximation, one can evaluate (15) as ... 

In accord with an approach originally outlined by Jortner and coworkers,41 42 the influence of changing AG° upon the 180 KIE has been modeled using a saddle point approximation.43 At this stage, the experimental variations in 180 KIEs for reactions of O2 and O2" are yet to be determined. The vibronic model of Hammes-Schiffer, which has been used to model proton-coupled electron transfer in accord with a Bom-Oppenheimer separation of timescales, may also be applicable here.44 The objective is to account for the change in 0—0 vibrational frequency together with potential contributions from overlap of vibrationally excited states. The overlap factors involving these states are expected to become more important as AG° deviates from 0 kcal mol 1,39... 

For arbitrary potentials, given the low frequencies and high intensities employed in current experiments, for the numerical evaluation of the amplitude (4.1) in the form (4.4) the method of steepest descent [also known as the saddle-point approximation (SPA)] is the method of choice. Thus, we must determine the values of fc, //, and t for which the action Sp(t,t, k) is stationary, so that its partial derivatives with respect to these variables vanish. This condition gives the equations... 

The second approximation employed here is to consider that the configurations of the minimum energy provide most of the contributions to the average Boltzmann factor in eq 2b, the well-known saddle point approximation of statistical mechanics. 

The low-order cumulants may be utilized to give saddle-point approximations of the underlying distribution [385,386]. 

By applying the saddle point approximation to its formula, the semiclassical expression of the wave matrix is given as follows [22] ... 

Thus, in the saddle-point approximation, the absorption coefficient is the product of the averaged density of states (which is essentially the probability to find the necessary disorder fluctuation) and the oscillator strength of the optical transition between the two intragap levels ... 

This observation is most helpful for our asymptotical treatment of the atomic systems suggesting that the saddle point approximation (Mathews Walker, 1970), is suitable to fairly analytical perform the involved integrals. According with the saddle point method, to evaluate an integral of type (4.325) the intermediate form (4.326) is approximated by the saddle-point recipe (3.154) specialized here as (see also the Appendix of the present volume) ... 

Without going into details (Hassani, 1991), if one has to solve an integral of the (Al) type with a > 0, the saddle point approximation or the stationary phase method or the method of the steepest descendent requires its expansion around the point the solution of the extreme equation ... 

This demonstrates that the saddle point approximation in U enforces the incompressibility constraint only on average, but the literal fluctuations of the total density in the EP theory do not vanish. 

Unfortunately, the two methods for calculating composition fluctuations yield different results. Of course, after the saddle point approximation for the field U it is not a priori obvious to which extent the EP theory can correctly describe composition fluctuations, fii the following we shall utilize Eqs. 25 and 26 to calculate the thermal average of the composition and its fluctuations in the EP theory. As we have argued above, these expressions will converge to the exact result if the fluctuations in U become Gaussian. 

The saddle point approximation of the partition functions yield the SCF theory. Let us first make the saddle point approximation in the fields. like before, we denote the values of the fields at the saddle point by an asterisk. 

This functional can be used to investigate the dynamics of collective composition fluctuations (cf. Sect. 5). If we proceed to make a saddle point approximation for the collective density we arrive at... 

Generally, one can approximately relate the time evolution of the field W to the dynamic SCF theory [31 ]. The saddle point approximation in the external... 

Field theoretical simulations [74,75,80] avoid any saddle point approximation and provide a formally exact solution of the standard model of the self-consistent field theory. To this end one has to deal with a complex free energy functional as a fimction of the composition and density. This significantly increases the computational complexity. Moreover, for certain parameter regions, it is very difficult to obtain reliable results due to the sign problem that a complex weight imparts onto thermodynamical averages [80]. We have illustrated that for a dense binary blend the results of the field theoretical simulations and the EP theory agree quantitatively, i.e., density and composi-... 

This is essentially the saddle point approximation Edwards uses to obtain (6.56).] If we write Xd = (6.59) results in... 

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