Thermal fluctuation method

Procedures for the reliable calibration of normal forces (i.e., /iN) are well established. Several methods can be used, such as the thermal fluctuation method [11-13], the reference lever [14], or the added mass technique [15]. We will mention in the following the thermal noise and the reference lever methods in some detail. 

One of the simplest methods is the comparison of the initial structure of the macromolecule to that throughout the trajectory via a distance measure such as the root mean square deviation (RMSD). This method is most informative for a system like a folded protein under native conditions, where the molecule is expected to spend the vast majority of the time in conformations quite similar to the crystal structure. If one computes the RMSD time series against the crystal structure, one expects to see a rapid rise due to thermal fluctuations, followed by a long plateau or fluctuations about a mean at longer timescales. If the RMSD... 

Detailed analysis of x-ray diffractometry reveals the local structure of semiconductor clusters. Bawendi et al. (21) applied the method to CdSe nanoclusters of 3.5-4.0 nm. They concluded that these clusters have a mixture of crystalline structures intermediate between zincblende and wurtzite, on the basis of detailed simulation studies taking the thermal fluctuation into account. Conventional diffractometry could overlook the coexistence of wurtzite component. 

As was shown in Section 2.1, in some cases thermal fluctuations of reactant densities affect the reaction kinetics. However, the equations of the formal chemical kinetics are not suited well enough to describe these fluctuations in fact they are introduced ad hoc through the initial conditions to equations. The role of fluctuations and different methods for incorporating them into formal kinetics equations were discussed more than once. 

The role of thermal fluctuations for membranes interacting via arbitrary potentials, which constitutes a problem of general interest, is however still unsolved. Earlier treatments G-7 coupled the fluctuations and the interaction potential and revealed that the fluctuation pressure has a different functional dependence on the intermembrane separation than that predicted by Helfrich for rigid-wall interactions. The calculations were refined later by using variational methods.3 8 The first of them employed a symmetric functional form for the distribution of the membrane positions as the solution of a diffusion equation in an infinite well.3 However, recent Monte Carlo simulations of stacks of lipid bilayers interacting via realistic potentials indicated that the distribution of the intermembrane distances is asymmetric 9 the root-mean-square fluctuations obtained from experiment were also shown to be in disagreement with this theory.10... 

The variational method proposed earlier by the authors relied on Monte Carlo simulations to select an intermembrane distance distribution function.8 The purpose of this paper is to present a new approach, in which the interaction between two membranes, in the presence of thermal fluctuations, is calculated directly by employing a suitable approximate partition function. Thus, the asymmetry of the distance distributions results in a natural manner from the calculation. 

To conclude, we presented a new method to account for the effect of the thermal fluctuations on the interactions between elastic membranes, based on a predicted intermembrane separation distribution. It was shown that for a typical potential, the distribution function is asymmetric, with an asymmetry dependent on the applied pressure and on the interaction potential between membranes. Equations for the pressure, root-mean-square fluctuation, and asymmetry as functions of the average distance (and the parameters of the interacting membranes) were derived. While no experimental data are available for two interacting lipid bilayers, a comparison with experimental data for multilayers of lipid bilayer/water was provided. The values of the parameters, determined from the fit of experimental data, were found within the ranges determined from other experiments. 

III.B. The Role of Thermal Fluctuations on the Transition from Common Black Films to Newton Black Films. The method described in the previous section will be now applied to thin films with fluctuating interfaces, with the interaction energy calculated as in section II.G. For low values ofthe external pressure, the enthalpy has two metastable minima at Zk and 2c, and a stable one at 2 - 0 (the former two correspond to the Newton and to the common black films, respectively, and the latter implies the rupture of the film), separated by two maxima located at Z and 22 (see Figure 7a). At metastable equilibrium the distances between the surfaces are distributed between 21 and 22 for the Newton black film and between z2 and 2 —°° for the common black film. The stability of the metastable states depends on the chance for a small area S of the interface to reach the... 

X 10" K" and ap(1000 K) = 10.4 x 10 K". As can be seen in Table 4, these are in much better agreement with the experimental data than are the fluctuation formula results. Particularly striking is the fact that the ratios of both the calculated and the experimental values at the two temperatures are now nearly the same, 1.4 (calc.) vs 1.36 (exp.). (With the fluctuation formula, this ratio is 2.2.) It should also be mentioned that when the coefficient of thermal expansion is obtained from its definition, which involves the volume and not its fluctuation, the convergence is much more rapid than with the fluctuation method. Figure 12 shows that the final volume is reached within a few picoseconds. 

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