Mean-squared fluctuations

The microcanonical ensemble is a certain model for the repetition of experiments in every repetition, the system has exactly the same energy, Wand F but otherwise there is no experimental control over its microstate. Because the microcanonical ensemble distribution depends only on the total energy, which is a constant of motion, it is time independent and mean values calculated with it are also time independent. This is as it should be for an equilibrium system. Besides the ensemble average value (il), another coimnonly used average is the most probable value, which is the value of tS(p, q) that is possessed by the largest number of systems in the ensemble. The ensemble average and the most probable value are nearly equal if the mean square fluctuation is small, i.e. if... 

If this condition is not satisfied, there is no unique way of calculating the observed value of ff, and the validity of the statistical mechanics should be questioned. In all physical examples, the mean square fluctuations are of the order of 1/Wand vanish in the thennodynamic limit. 

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... 

Noise. So fat, as indicated at the beginning of this section on semiconductor statistics, equihbtium statistics have been considered. Actually, there ate fluctuations about equihbtium values, AN = N— < N >. For electrons, the mean-square fluctuation is given by < ANf >=< N > 1- ) where (Ai(D)) is the Fermi-Dirac distribution. This mean-square fluctuation has a maximum of one-fourth when E = E-. These statistical fluctuations act as electrical noise and limit minimum signal levels. 

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-... 

In the case shown in Fig. 5.132, the initial mean square fluctuations would be... 

The constrained-junction model was formulated in order to explain the decrease of the elastic moduli of networks upon stretching. It was first introduced by Ronca and Allegra [39], and Flory [40]. The model assumes that the fluctuations of junctions are diminished below those of the phantom network because of the presence of entanglements and that stretching increases the range of fluctuations back to those of the phantom network. As indicated by the second part of Equation (26), the fluctuations in a phantom network are substantial. For a tetrafunctional network, the mean-square fluctuations of junctions amount to as much as half of the mean-square end-to-end vector of the network chains. The strength of the constraints on these fluctuations is measured by a parameter k, defined as... 

The intensity of scattering of light by a swollen network is related to two factors. One of these is the mean square fluctuation in refractive index which is proportional to the mean squared fluctuation in Mc, with a proportionality constant determined by the difference in refractive index between the rubber and the diluent. The second is the... 

Let (V2) be the mean square value of the fluctuating force, and (Q2) the mean square value of Q. Suppose that the system is known to be in the nth eigenstate. Since Hq is hermitian the expectation value of Q, (En Q En) vanishes and the mean square fluctuation of Q is given by the expectation value of Q2, i.e. 

Usually it is assumed that tc is the only temperature-dependent variable in Eq. 9. This might be the case for an order-disorder type rigid lattice model, where the only motion is the intra-bond hopping of the protons, since the hopping distance is assumed to be constant and therefore also A and A2 are constant. This holds, however, only for symmetric bonds. Below Tc the hydrogen bonds become asymmetric and the mean square fluctuation amplitudes are reduced by the so-called depopulation factor (l - and become in this way temperature-dependent also. The temperature dependence of tc in this model is given by Eq. 8, i.e. r would be zero at Tc, proportional to (T - Tc) above Tc and proportional to (Tc - T) below Tc. 

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. 

The mean square fluctuations are — 1, which is less than observed. Furry 0 improved the model by taking rn = 0, gn = yn and found... 

Adjust the constant r so that the stationary solution reproduces the correct mean square fluctuations as known from statistical mechanics, or find r from other considerations, see, e.g., (5.1) below. 

Here FMF is the mean field free energy, defined for the lam hex and BCC phases in eqns 2.13-2.15. The term a = q 2/4ji 0 occurs in the expression for the mean square fluctuation... 

It is typical in the study of cosmological perturbations to use not P k) and Ct themselves, but rather the quantities A2 k) = k3P(k)/(2ir2) and 8T2 = ( + 1 )CtJ(27r). These dimensionless quantities give the contribution to the total variance in density or temperature from a given 3D or spherical wavenumber, or even more heuristically, the mean-square fluctuation at wavelength A 27x/k or angular scale 0 180°/ . In addition, the Sachs-Wolfe effect has ( + l)Ct/(2ir) = const at low . 

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