Free energy explicit functionals

In particular, we have yj = 1 for one-dimensional association because we can neglect the surface free energy. The function u (x) becomes i<(x) = l/(l—x), and hence we can express x explicitly in terms of the polymer number density c = kcp/n as... 

It is possible to go beyond the SASA/PB approximation and develop better approximations to current implicit solvent representations with sophisticated statistical mechanical models based on distribution functions or integral equations (see Section V.A). An alternative intermediate approach consists in including a small number of explicit solvent molecules near the solute while the influence of the remain bulk solvent molecules is taken into account implicitly (see Section V.B). On the other hand, in some cases it is necessary to use a treatment that is markedly simpler than SASA/PB to carry out extensive conformational searches. In such situations, it possible to use empirical models that describe the entire solvation free energy on the basis of the SASA (see Section V.C). An even simpler class of approximations consists in using infonnation-based potentials constructed to mimic and reproduce the statistical trends observed in macromolecular structures (see Section V.D). Although the microscopic basis of these approximations is not yet formally linked to a statistical mechanical formulation of implicit solvent, full SASA models and empirical information-based potentials may be very effective for particular problems. 

In the CVM, the free energy of a given alloy is approximated in terms of probabihties for a selected set of finite clusters. The largest cluster explicitly considered in the free energy functional specifies the level of the approximation. The common practice for an fcc-based system is the tetrahedron approximation [26] in which nearest neighbor tetrahedron cluster is taken as the largest cluster. Hence, within the tetrahedron approximation, the free energy expression, F,is symbolically expressed as... 

For the PPM, corresponding to the free energy of the CVM is the Path Probability Function (hereafter PPF), P t t -t- At), which is an explicit function of time and is defined as the product of three factors Pj, P2 and P3. Each factor is provided in the following in the logarithmic expression. 

Recently, much attention has been paid to the investigation of the role of this interaction in relation to the calculations for adiabatic reactions. For steady-state nonadiabatic reactions where the initial thermal equilibrium is not disturbed by the reaction, the coupling constants describing the interaction with the thermal bath do not enter explicitly into the expressions for the transition probabilities. The role of the thermal bath in this case is reduced to that the activation factor is determined by the free energy in the transitional configuration, and for the calculation of the transition probabilities, it is sufficient to know the free energy surfaces of the system as functions of the coordinates of the reactive modes. 

As we pointed out earlier, calculating the derivative of the free energy appears to require a full set of generalized coordinates. However, this may seem quite surprising. Assume that we want to define the PMF as a function of the distance between two molecules. This force is clearly independent of the particular choice of generalized coordinates made to calculate it. In fact, we are now going to prove that an equation can be derived which does not require an explicit definition of generalized coordinates other than . 

The construction of the phase diagram of a heteropolymer liquid in the framework of the WSL theory is based on the procedure of minimization of the Landau free energy T presented as a truncated functional series in powers of the order parameter with components i a(r) proportional to Apa(r). The coefficients of this series, known as vertex functions, are governed by the chemical structure of heteropolymer molecules. More precisely, the values of these coefficients are entirely specified by the generating functions of the chemical correlators. Hence, before constructing the phase diagram of the specimen of a heteropolymer liquid, one is supposed to preliminarily find these statistical characteristics of the chemical structure of this specimen. Here a pronounced interplay of the statistical physics and statistical chemistry of polymers is explicitly manifested. 

Equation (3.5) also shows that the activation free energy at the peak, AGj, is an increasing function of temperature, taking into account the explicit presence of T and also the variation of k, [equation (1.34)] and Dh. Thus, increasing scan rate and decreasing temperature favor the transition between concerted and stepwise mechanisms, and vice versa. 

The description of the chain dynamics in terms of the Rouse model is not only limited by local stiffness effects but also by local dissipative relaxation processes like jumps over the barrier in the rotational potential. Thus, in order to extend the range of description, a combination of the modified Rouse model with a simple description of the rotational jump processes is asked for. Allegra et al. [213,214] introduced an internal viscosity as a force which arises due to a transient departure from configurational equilibrium, that relaxes by reorientational jumps. Thereby, the rotational relaxation processes are described by one single relaxation rate Tj. From an expression for the difference in free energy due to small excursions from equilibrium an explicit expression for the internal viscosity force in terms of a memory function is derived. The internal viscosity force acting on the k-th backbone atom becomes ... 

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