Mean-field estimation

The resultant corrections to the SCF picture are therefore quite large when measured in kcal/mole. For example, the differences AE between the true (state-of-the-art quantum chemical calculation) energies of interaction among the four electrons in Be and the SCF mean-field estimates of these interactions are given in the table shown below in eV (recall that 1 eV = 23.06 kcal/mole). 

Although differences between mean-field estimates and exact limiting densities are to be expected because of the presence of correlations in actual evolutions, the mean-field approach nonetheless typically comes to within about 10 — 20% of the exact values for all elementary complex rules. 

Mean Field Estimate It is easy to predict the value of 7 from A. Consider a one-dimensional = 2, r = 1 CA evolving according to rule (j) for which (j>) = A. The probability that two randomly chosen (2r + 1) neighborhood blocks map to the same value is equal to Psame = A + (1 - A). The average left spreading rate of (j> is then given by... 

Mean Field Estimate H may be estimated from X (j>) in two steps First, we find the steady-state probability of a given site having value zero. Let po(t) be the... 

In fact, the mean-field estimate for H, given by equation 3.64, predicts a second-order transition. While abrupt changes in entropy characterize sudden transitions between regions of periodic and chaotic rules, smooth changes in entropy as A is increased instead suggest that the rule path sometimes passes through a region of... 

As with all statistical methods, the mean-field estimate will have statistical error due to the finite sample size (X ), and deterministic errors due to the finite grid size (S ) and feedback of error in the coefficients of the SDEs Ui,p). Since error control is an important consideration in transported PDF simulations, we will now consider a simple example to illustrate the tradeoffs that must be made to minimize statistical error and bias. The example that we will use corresponds to (6.198), where the exact solution141 to the SDEs has the form ... 

A mean-field estimate of this probability can be made for the general case of an ideal chain in cf-dimensional space by replacing a chain with an ideal gas of N monomers in the pervaded volume of a coil R. The probability of a given monomer to contact any other monomer within this mean-field approximation is simply the overlap volume fraction (j>, of a chain inside its pervaded volume, determined as the product of the monomer volume and the number density of monomers in the pervaded volume of the coil NjR ... 

A mean-field estimate of the free energy of adsorption and the thickness of the adsorbed chain can be made by assuming the monomers are uniformly distributed at different distances from the surface up to thickness, ads-Then the fraction of monomers in direct contact with the surface (within distance b from the surface) is / ads- The number of adsorbed monomers Nbj sids is multiplied by the adsorption energy per monomer-surface contact (—6kT) to calculate the energetic gain from the surface interaction ... 

Evaluate the mean-field estimate of the gel point p for bond percolation on the following lattices ... 

The error e in this expression must be the true error (i.e., all the systematic and statistical errors). Uncontrolled approximations cannot be allowed otherwise, the complexity problem is not well posed. Chemistry is unique in that, first, there is a well-tested, virtually exact theory (the Schrddinger equation), and second, the mean-field estimates of chemical energies are often surprisingly accurate. Unfortunately, very accurate estimates are required to provide input to real-world chemistry, since much of the interesting chemistry takes place at room temperature. Currently, the level of chemical accuracy is considered to be =1 kcal/... 

At the right-hand side of (10.8), the first five terms come from Floiy s semiflexibility treatment (8.55), the sixth term comes from the mean-field estimation for the pair interactions of parallel bonds (10.7), and the last term comes from the mean-field estimation for the mixing interactions between the chain units and the solvent molecules (8.21). According to the Boltzmann s relation F = —kTlnZ, the free energy of the solution system can be obtained as... 

The statistical thermodynamic theory makes a mean-field treatment to the crystallization energy Ep. However, in practice, thermal fluctuations for parallel ordering of local chains generally exist in the melt states of blends. Such an additional anisotropic contribution of thermal fluctuations will make the practical parallel-packing interactions larger than the mean-field estimation on the basis of the isotropic liquid, which leads to a deviation from the theoretical predictions of the phase separation binodals. 

In his original derivation of equations similar to eq. (V.25) Floiy assumed gaussian statistics for the chains plus a mean field estimate for the repulsive energies. His theory is successful the scientific community has... 

There are two main ways in which our lattice model is a simplification. First, the partition function Q should be a sum over all the possible states of the system. It should be a. sum of the number of states with small numbers of AB contacts, iuab, and states with large m.4B, rather than a mean-field estimate of the number of uniformly mixed conformations (see Figure 15.8). In this way, we have made the approximation that... 

Self-avoidance can then be introduced at the level of Flory theory as for linear polymers by balancing the elastic (entropic) free energy of the phantom object without self-avoidance with the mean-field estimate of the excluded volume interaction, " ... 

Substituting the saddle-point values, w+ and w, back into the free energy functional, we arrive at the mean field estimate for the Helmholtz free energy... 

Here the first term suppresses local density fluctuations [73,74] and the rest is an analog of the Ginzburg-Landau free energy associated with the instantaneous tensorial fields Q and B. Phenomenological parameters V,]i and A entering this functional are normally chosen such that the thermodynamic state of interest, e.g., a biaxial-nematic mesophase, is reproduced. In this respect, mean-field estimates can help to limit the physically adequate parameter ranges. Positive isothermal compressibility, for example, requires k >V + Ji + A [64]. 

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