Gaussian free energy function

A Gaussian random field 0(r) is defined by a probability distribution exp(-.Fo[0]) with a quadratic free energy functional. T ol I ]- This applies in particular to the functional defined in Eq. (15). 

Polymer chains in solution form a loose coil such that two coils can interpenetrate. The resulting effective interaction potential O(lrl) is repulsive and is well approximated by a Gaussian potential of strength Oq and with a range ro which is proportional to the radius of gyration Rg. The excess free energy functional for such soft systems can be approximated by a quadratic form... 

In most cases the distribution function is Gaussian (or approximately so) and the corresponding free energy function can be approximated by a simple parabolic equation (Figure 12.2.15). In this case, we assiune a Hookean energy (which is correct at least for the region around the maximum of the distribution curve), so we have ... 

If we assume composition fluctuations to be Gaussian, we can write down a free energy functional compatible with Eq. 91. 

Note that this free energy functional is Gaussian in the Fomier coefficients of the composition, and, hence, the critical behavior still is of mean-field type. Transforming back from Fourier expansion for the spatial dependence to real space, we obtain for the free energy functional ... 

The polymer melt is modeled as a compressible system, consisting of Gaussian chain molecules in a mean field environment. The free energy functional for copolymer melts has a form that is similar to the free energy that was used before ... 

The equilibrium structures predicted by SCMFT correspond to the solutions obtained at the extrema of the free-energy functional of the system. These solutions do not necessarily ensure the minimization of the free-energy functional. The mean-field solution may, for example, correspond to a saddle point. In order to investigate the stability of the ordered phases, we have to consider the effect of the higher-order contributions to the free-energy functional. In particular, the Gaussian fluctuation contributions derived above can be used to predict the stability of any ordered structure. In what follows we formulate the theory of Gaussian fluctuations in ordered phases [15,27,31,32]. 

Amadei, A., Apol, M. E. F., Di Nola, A., Berendsen, H. J. C. The quasi-Gaussian entropy theory Free energy calculations based on the potential energy distribution function. J. Chem. Phys. 104 (1996) 1560-1574... 

Flere, A (T) is a function of temperature alone. Equation (6) represents the elastic free energy of a Gaussian chain with ends fixed at a separation of r. The average force required to keep the two ends at this separation is obtained from the thermodynamic expression [28]... 

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... 

These considerations raise a question how can we determine the optimal value of n and the coefficients i < n in (2.54) and (2.56) Clearly, if the expansion is truncated too early, some terms that contribute importantly to Po(AU) will be lost. On the other hand, terms above some threshold carry no information, and, instead, only add statistical noise to the probability distribution. One solution to this problem is to use physical intuition [40]. Perhaps a better approach is that based on the maximum likelihood (ML) method, in which we determine the maximum number of terms supported by the provided information. For the expansion in (2.54), calculating the number of Gaussian functions, their mean values and variances using ML is a standard problem solved in many textbooks on Bayesian inference [43]. For the expansion in (2.56), the ML solution for n and o, also exists, lust like in the case of the multistate Gaussian model, this equation appears to improve the free energy estimates considerably when P0(AU) is a broad function. 

Nanda, H. Lu, N. Kofke, D. A., Using non-Gaussian density functional fits to improve relative free energy calculations, J. Chem. Phys. 2005,122, 134110 1-8... 

ABF shares some similarities with the technique of Laio et al. [30-34], in which potential energy terms in the form of Gaussian functions are added to the system in order to escape from energy minima and accelerate the sampling of the system. However, this approach is not based on an analytical expression for the derivative of the free energy but rather on importance sampling. 

Fig. 6.9. When one of the probability distribution functions f(W) and g(W) is a Gaussian, the other must also is a Gaussian with the same variance (a ). These two density functions peak at A A + and A A - f3a y, respectively. Their crossing point gives the free energy...

The cumulants [26] are simple functions of the moments of the probability distribution of 5V-.C2 = (V- V))2),C3 = (V- V)f),C4 = ((]/-(]/))4) 3C22,etc. Truncation of the expansion at order two corresponds to a linear-response approximation (see later), and is equivalent to assuming V is Gaussian (with zero moments and cumulants beyond order two). To this order, the mean and width of the distribution determine the free energy to higher orders, the detailed shape of the distribution contributes. 

If AU( 1) has Gaussian fluctuations, this means that the free energy is also a parabolic function of AU(1) itself. AU( 1) is known as the energy gap [52-54] we will denote it by p... 

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