Root mean square fluctuations

Both (E) and Cy are extensive quantities and proportional to N or the system size. The root mean square fluctuation m energy is therefore proportional to A7 -, and the relative fluctuation in energy is... 

The average value and root mean square fluctuations in volume Vof the T-P ensemble system can be computed from the partition fiinction Y(T, P, N) ... 

Since 5(A /5 j. (N), tlie fractional root mean square fluctuation in N is... 

Figure 14 Measures of disorder m the acyl chains from an MD simulation of a fluid phase DPPC bilayer, (a) Order parameter profile of the C—H bonds (b) root-mean-square fluctuation of the H atoms averaged over 100 ps.

To make contact with the diffusion-in-a-sphere model, we have defined the spherical radius as the root-mean-square fluctuation of the protons averaged over 100 ps. The varia-... 

Bond Lengths and Their Root Mean Square Fluctuations for the Decaniobate Ion in Figure 4 as Predicted From Molecular Dynamics Calculations... 

It has recently been pointed out by Gordon1 that the root-mean-square fluctuations in the sampled values of the autocorrelation function of a dynamical variable do not necessarily relax to their equilibrium values at the same rate as the autocorrelation function itself relaxes. It is the purpose of this paper to investigate the relative rates of relaxation of autocorrelation functions and their fluctuations in certain systems that can be described by Smoluchowski equations,2 i.e., Fokker-Planck equations in coordinate space. We exhibit the fluctuation and autocorrelation functions for several simple systems, and show that they usually relax at different rates. 

Figure 4. Root-mean-square fluctuation a vs the separation distance a, for (1) eq 15 and (2-4) a calculated for truncated Gaussian distribution with various asymmetrycoefficients (2) a = 1 (3) a = 1.4 (4) a = 2.0 (5) HelMch proportionality relation, a2=tta2 (ii = 0.183) (6) approximate solution for small distances, eq 16.

The dependence of the interaction force between two undulating phospholipid bilayers and of the root-mean-square fluctuation of their separation distances on the average separation can be determined once the distribution of the intermembrane separation is known as a function of the applied pressure. However, most of the present theories for interacting membranes start by assuming that the distance distribution is symmetric, a hypothesis invalidated by Monte Carlo simulations. Here we present an approach to calculate the distribution of the intermembrane separation for any arbitrary interaction potential and applied pressure. The procedure is applied to a realistic interaction potential between neutral lipid bilayers in water, involving the hydration repulsion and van der Waals attraction. A comparison with existing experiments is provided. 

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 average separation distance and its root-mean-square fluctuation were recently experimentally determined as functions of the applied pressure, for lipid bilayers/water multilayers,10 and a disagreement between the existing theories and experiment was noted.12 No comparable experimental results are yet available for two lipid bilayers, and it is difficult to extend the present theory to multilayers. For this reason, we will employ the present procedure to estimate the interaction parameters of eq 10 from the experimental results for multilayers.10... 

Figure 3. Fit of the experimental data of ref 10 for (a) the root-mean-square fluctuation a and (b) the applied pressure p as functions of the average separation distance (z), for EPC/ water (circles/continuous line) and DMPC/water (squares/dotted line) multilayers.

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. 

A. The Thermal Fluctuations of the Interfaces for Arbitrary Interactions. After the Helfrich initial theory,18 Helfrich and Servuss17 suggested an alternate derivation of the entropic repulsion due to the confinement of a membrane between rigid walls, by considering the lipid bilayer composed of many independent pieces , whose area is related to the root mean square fluctuations of the positions of the undulatingbilayer. As shown below, this representation can be extended to interfaces interacting via arbitrary potentials. 

Using eqs 29—31, one can calculate the average thickness and the root mean square fluctuation of the intersurfaces separation as a function ofthe applied pressure, once the interaction potential is known. For illustration purposes, let us first apply the method to the simple interaction potential... 

In Figure 6, the pressure (a) and the root mean square fluctuation (b) are plotted as functions of the average thickness ofthe film, for Kc = 10 x ICC19 J and for (1) the anharmonic (eq 32) and (2) the harmonic (eq 23) interaction potentials. The spring constant for the second case was obtained from the harmonic approximation of eq 32 around its minimum, at p = 0. 

Sachs (Si) attempted to establish the extent of the turbulent fluctuations present. He reports root-mean-square fluctuating velocities ranging up to values on the order of 40% of the mean velocities. Although their quantitative significance may be doubtful, these data do represent... 

If a flow in the tank is turbulent, either because of high power levels or low viscosity, then a typical velocity pattern at a point would be illustrated by Fig. 3. The velocity fluctuation i can be changed into a root mean square value (RMS), which has great utility in estimating the intensity of turbulence at a point. So in addition to the definitions above, based on average velocity point, we also have the same quantities based on the root mean square fluctuations at a point. We re interested in this value at various rates of power dissipation, since energy dissipation is one of the major contributors to a particular value of RMS v. ... 

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