Equilibrium calculations approximation approach steps

Fig.12. Computation by Monte Carlo methods of the first four order parameters of an ensemble of 1000 chromophores (of dipole moment 13 Debye) existing in a medium of uniform dielectric constant. At the beginning of the calculation, the chromophores are randomly ordered thus, ==O. During the first 400 Monte Carlo steps, an electric poling field (600 V/micron) is on but the chromophore number density (=10 7 molecules/cc) is so small that intermolecular electrostatic interactions are unimportant. The order parameters quickly evolve to well-known equilibrium values obtained analytically from statistical mechanics (black dots in figure also see text). During steps 400-800 the chromophore number density is increased to 5xl020 and intermolecular electrostatic interactions act to decrease order parameters consistent with the results of equilibrium statistical mechanical calculations discussed in the text. Although Monte Carlo and equilibrium statistical mechanical approaches described in the text are based on different approximations and mathematical methods, they lead to the same result (i.e., are in quantitative agreement)...

The Sabatier analysis can in principle be performed for any reaction. In case that there are more than two reaction steps, the Sabatier volcano could be constructed in analogy to Equation (7.16) by assuming that all intermediates that go in the forward direction have optimal coverages and by calculating the approach to equihbrium for each forward rate from the given approach to equilibrium for the overall reaction (as done in Eq. 7.8 for the reaction earher) under the assumption that all other partial reactions are in equihbrium. This wiU give a first approximation to each forward rate of the reaction. The Sabatier volcano provides an upper limit to the total rate by setting this rate equal to the minimum of all forward rates ... 

The non-equilibrium particle distribution is clearly observed through the joint correlation functions plotted in Fig. 6.47. Note that under the linear approximation [74] the correlation function for the dissimilar defects Y (r, t) increases monotonically with r from zero to the asymptotic value of unity Y(r —y oo,t) = 1. In contrast, curve 1 in Fig. 6.47 (f = 101) demonstrates a maximum which could be interpreted as an enriched concentration of dissimilar pairs, AB, near the boundary of the recombination sphere, r tq. With increasing time this maximum disappears and Y(r, t) assumes the usual smoothed-step form. The calculations show that such a maximum in Y(r, t) takes place within a wide range of the initial defect concentrations and for a random initial distribution of both similar and dissimilar particles used in our calculations X (r, 0) = Y(r > 1,0) = 1. The mutual Coulomb repulsion of similar particles results in a rapid disappearance of close AA (BB) pairs separated by a distance r < L (seen in Fig. 6.47 as a decay of X (r, t) at short r with time). On the other hand, it stimulates strongly the mutual approach (aggregation) of dissimilar particles leading to the maximum for Y(r, t) at intermediate distances observed in Fig. 6.47. 

An alternative approach to the dynamics of a protein or one of its constituent elements (e.g., an a-helix) is to assume that the harmonic approximation is valid. Early attempts to examine dynamical properties of proteins or their fragments used the harmonic approximation. They were motivated by vibrational spectroscopic studies [24], where the calculation of normal mode frequencies from empirical potential functions has long been a standard step in the assignment of infrared spectra [25]. In calculating the normal vibrational modes of a molecule, one assumes that the vibrational displacements of the atoms from their equilibrium positions are... 

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