Relative mean-square fluctuation

The quantity in brackets is the isothermal compressibility, which is independent of Vo. We see, then, that the relative mean square fluctuation of the volume is inversely as the volume itself, becoming small for large volumes. This is in accordance with the behavior of the other fluctuations we have found. 

How important can fluctuations be for a nanoparticle Consider a nano droplet of water (in ambient conditions) suspended in oil. A droplet with the radius of 1 nm contains about 130 water molecules. Since the compressibility of water is lower than the compressibility of oil, the fluctuations (expansion/ contraction) of the water droplet will be controlled by the lower compressibility of water. If one neglects the mutual solubility of oil and water, the number of molecules in the droplet is constant, while the volume can fluctuate. The relative mean-square fluctuation of volume is given as... 

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... 

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. 

One of our objectives is to determine if the present model of copper can reproduce the variation of its properties with temperature. Since the simulations at 300 K and 1000 K that were discussed above were performed with different cutoffs, there is a potential element of inconsistency in comparing the two sets of results. We showed earlier that this is not significant for the radial distribution function and the mean square fluctuation, which are relatively insensitive to the cutoff being used. The coefficient of thermal expansion, on the other hand, was found to be much more dependent upon this parameter. Accordingly we repeated the 300 K simulation using the same cutoff, 5.40 A, as at 1000 K. 

Using this expression for SCeq together with Eq. (26a) permits us to write a simple expression for the expected relative root mean square fluctuation in equilibrium complex number due to stochastic effects in binding ... 

Fig. 7. Relative root mean square fluctuation in the equilibrium number of complexes on a cell. The plot shows the prediction of the probabilistic model for homogenous receptor/ligand binding. Note that fluctuations become more significant as the number of receptors or the ratio of L to KD decreases.

The results of these measurements are shown in Figure 3 as the dimensionless value ((8G)2)/(G)2. The uppermost solid straight line is drawn so that the relative mean square value of fluctuations is normalized to the total number of dissolved electrolyte molecules, NM that is,... 

This informative result shows that the mean square energy fluctuations are proportional to the size of the system and to the rate at which energy changes with temperature. The relative root mean square dispersion of the energy... 

Even though the Reynolds number gives some measure of turbulent phenomena, flow quantities characteristic of turbulence itself are of more direct relevance to modeling turbulent reacting systems. The turbulent kinetic energy q may be assigned a representative value <7o at a suitable reference point. The relative intensity of the turbulence is then characterized by either q()KH2 U2) or (77(7, where (/ = (2q0)m is a representative root-mean-square velocity fluctuation. Weak turbulence corresponds to U /U < 1 and intense turbulence has (77(7 of the order unity. 

This equation gives the mean-square value of the voltage appearing across the terminals of a capadtor filled with a dielectric of zero-frequency relative permittivity, s in terms of Co, the capacitance without a dielectric, and the absolute temperature, T. This noise voltage is caused by the thermally induced dipole-moment fluctuations which are themselves inextricably bound up with the dissipative processes. That this is so is indicated by the fact that equation (42) applied to aJ,to) leads to the relation... 

Very recently, experimental results that indicate a direct connection between the local Cu-O bond fluctuation and occurrence of superconductivity have been obtained (Fig. 12) [16]. The mean squared relative displacement (MSRD) of Cu-O bond lengths in the CUO2 plane, cr o > shows an anomalous increase below T and a sudden decrease around T. Ibe 5% substitution of Cu by magnetic atoms, Co or Ni, suppresses superconductivity completely, and so does the anomalous behavior... 

Experimentally, the excess noise is the difference between the mean-square voltage fluctuations < Sv > measured in the presence and absence of a quiet d.c. voltage VQ = IqR. We determine it using a four-probe technique designed to eliminate the relatively small excess contact noise. 

The effect of added foreign-phase dispersions on the excess noise in electrolytes was studied (21, 22). From relation 15 of Bezrukov et al. (21), it follows that the mean-square value of the relative conductance fluctuations that originate from nonconducting contaminants does not depend on the electrolyte concentration. Hence, to present the results of the excess noise measurements in the form of Hooge s formula, with the samples equally contaminated on the average, the parameter a must be taken to be proportional to the electrolyte concentration. 

Since Cy and E are both extensive properties (ocN), the root-mean-square energy fluctuations are smaller, by a factor 1/y, than t5q)ical average energies E. As the system size increases, the relative magnitude of... 

---