Fluctuation correlations, definition

The presence of organisms (large or small) in proximity to corrosion by itself is not proof of biologically influenced corrosion, any more than a correlation of lunar phases with stock market fluctuations establishes a lunar-financial connection. It should be stressed vigorously that all evidence must be consistent with any single corrosion mode before a definitive diagnosis can be made (see Critical Factors above). Further, all alternative explanations must be carefully examined. 

The structure of the chapter is as follows. First, we start with a brief introduction of the important theoretical developments and relevant interesting experimental observations. In Sec. 2 we present fundamental relations of the liquid-state replica methodology. These include the definitions of the partition function and averaged grand thermodynamic potential, the fluctuations in the system and the correlation functions. In the second part of... 

The operators Fk(t) defined in Eq.(49) are taken as fluctuations based on the idea that at t=0 the initial values of the bath operators are uncertain. Ensemble averages over initial conditions allow for a definite specification of statistical properties. The statistical average of the stochastic forces Fk(t) is calculated over the solvent effective ensemble by taking the trace of the operator product pmFk (this is equivalent to sum over the diagonal matrix elements of this product), so that = Trace(pmFk) is identically zero (Fjk(t)=Fk(t) in this particular case). The non-zero correlation functions of the fluctuations are solvent statistical averages over products of operator forces,... 

Abstract The theoretical basis for the quantum time evolution of path integral centroid variables is described, as weU as the motivation for using these variables to study condensed phase quantum dynamics. The equihbrium centroid distribution is shown to be a well-defined distribution function in the canonical ensemble. A quantum mechanical quasi-density operator (QDO) can then be associated with each value of the distribution so that, upon the application of rigorous quantum mechanics, it can be used to provide an exact definition of both static and dynamical centroid variables. Various properties of the dynamical centroid variables can thus be defined and explored. Importantly, this perspective shows that the centroid constraint on the imaginary time paths introduces a non-stationarity in the equihbrium ensemble. This, in turn, can be proven to yield information on the correlations of spontaneous dynamical fluctuations. This exact formalism also leads to a derivation of Centroid Molecular Dynamics, as well as the basis for systematic improvements of that theory. 

According to this definition dilute solutions of long macromolecules are critical. The role of the correlation length is played by the radius of gyration Rg rsj Nu — oo N — oo, and by virtue of the chain structure a polymer coil shows density fluctuations on all scales r < Rg. Indeed, a blob of size r is just a correlated fluctuation of the density. 

One of the features that makes Equation (1) such a good starting point for our work is that it can be, in principal, exact. It is possible to show, without ever explicitly evaluating T and 37, that these crucial functions really do exist and are well defined (31). These formal definitions are rarely, if ever, useful in practical numerical calculations, but one can also work backwards from the exact dynamics x(t) (e.g., from a molecular dynamics simulation) to derive what the friction in particular must look like (32). The analysis tells us, moreover, that the exact T and are actually related to one another (28,31). The requirement that the relaxed system must be in equilibrium at some temperature T can be shown to set the magnitude and correlations of the fluctuating force ... 

Backbone fluctuations with correlation times in the nanosecond regime are revealed in variations of the R1 residue scaled mobility, Ms, along the sequence (see Section III,A,1 for definition). Figure 15A (see color insert) shows a plot of Ms versus sequence for C1-C3, H8, and adjacent sequences in the TM helices. Figure 15B shows the cytoplasmic surface of rhodopsin color-coded according to Ms values. In this figure, C3 was modeled from the SDSL data, as in Fig. 9. 

According to the general definitions of the coil and the globule241, the macromolecule is in the coil state, if the fluctuations of the monomer concentration within the macromolecule are of order of the monomer concentration itself and the correlation radius of the fluctuations of concentration is of order of the macromolecular dimensions while in the globular state the concentration fluctuations are small compared with the concentration and the correlation radius is considerably smaller than the globular dimensions. 

Hill, 1986). We have emphasized that fluctuation contributions, e.g. Eq. (4.71) p. 90, have a definite sign. This Debye-Hiickel theory treats correlations between ionic species, and here we observe again that treatment of correlations lowers this free energy. 

Conventionally, correlation is explained as dynamic fluctuations in the electronic density. By definition, it is the difference of the exact energy from the HF energy. Correlation energy can be separated into dynamical and static contributions. Notice that exchange repulsion occurs between electrons of equal spin. This implies some correlated motion between these electrons which is absent for electrons of opposite spin in HF. When this correlation is related to intramolecular interactions it is called dispersion, a more difficult quantity to calculate. 

The fluorescence intensity trajectories of the donor (/d(f)) and acceptor (/a(t)) give autocorrelation times (Fig. 24.2b) indistinguishable from fitting an exponential decay to the autocorrelation functions, (A/d (0) A/d (t)) and (A/a (0) A/a (t)), where A/d(t) is /d(t) — (Id), (Id) is the mean intensity of the overall trajectory of a donor, and A/a(t) has the same definition for an intensity trajectory of an acceptor. In contrast, the cross-correlation function between the donor and acceptor trajectories, (A/d (0) A/d (t)), is anticorrelated with the same decay time (Fig. 24.2b) which supports our assignment of anticorrelated fluctuations of the fluorescence intensities of the donor and acceptor to the spFRET process. 

The concept of softness is associated with polarizability. The larger the chemical system is, the softer it will be. This correlation of softness with polarizability can be found directly from a bond charge model [65 68] where softness is found to be proportional to the internuclear distance of a molecule [69-72]. To extend this definition (Eq. (28)) to open systems, the system is considered as a member of a grand canonical ensemble with bath parameters /t, v( T ), and temperature 0. This definition of S in such an ensemble can be written in terms of a number fluctuation formula [64] ... 

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