Correlation statistical theory

This chapter considers the distribution of spin Hamiltonian parameters and their relation to conformational distribution of biomolecular structure. Distribution of a g-value or g-strain leads to an inhomogeneous broadening of the resonance line. Just like the g-value, also the linewidth, W, in general, turns out to be anisotropic, and this has important consequences for powder patterns, that is, for the shape of EPR spectra from randomly oriented molecules. A statistical theory of g-strain is developed, and it is subsequently found that a special case of this theory (the case of full correlation between strain parameters) turns out to properly describe broadening in bioEPR. The possible cause and nature of strain in paramagnetic proteins is discussed. 

Our finding that linewidth anisotropy in biomolecular EPR spectra can be described by a statistical theory in which the random variables that cause the broadening are fully correlated, does not only make analysis by simulation practical it also holds a message on the nature of the ultimate source of the broadening if the three principal elements of the p-tensor are fully correlated, then they should find their cause in a single, scalar quantity. 

Extension of the Peturbed Hard Chain Correlation (Statistical Mechanical Theory of Fluids)" (2, 5). Extend the PHC program under development to include additional compounds including water. This work is an attempt to combine good correlations for phase equilibrium, enthalpy, entropy, and density into a single model. 

Abstract—This paper is an analysis of measurements of the thermodynamic properties of aqueous solutions of non-electrolytes, which has been made in order to establish both the relative strength of different kinds of hydrogen bonds in such solutions and the correlation between bond-strengths and the phase-behaviour of the solutions. The thermodynamic properties are compared with the results of statistical theories of solutions and with the properties of more simple solutions. 

Barnwell, J.D., Loeser, J.G. and Herschbach, D.R. (1983). Angular correlations in chemical reactions. Statistical theory for four-vector correlations, J. Phys. Chem., 87, 2781-2786. 

There are general relationships of transport phenomena based on phenomenological theory, i.e., on the correlations between macroscopically measurable quantities. The molecular theories explain the mechanism of transport processes taking into account the molecular structure of the given medium, applying the kinetic-statistical theory of matter. The hydrodynamic theories are also applied especially to describe - convection. 

The advantage of the kinetic theory over the statistical branching theory rests in its adherence to the kinetically controlled chemical process while the statistical theory working with units does not take into consideration the coimections between units developed in time (stochastic correlations). The greater mathematical complexity and impossibility to get information on the internal structure of the molecules and gel are the disadvantages of the kinetic theory. 

Kirkwood, in his statistical theory of dipolar dielectrics, derived the following angular correlation parameter ... 

In this section we develop a statistical theory of EELS lineshapes for binary systems. This treatment will be developed around the correlation expansion. 

A statistical theory of turbulence which is applicable to continuous movements and which satisfies the equations of motion was introduced by Taylor [159, 160, 161] and [162, 163], and further developed by von Karman [178, 179]. Most of the fundamental ideas and concepts of the statistical turbulence theory were presented in the series of papers published by Taylor in 1935. The two-point correlation function is a central mathematical tool in this theory. Considering the statistics of continuous random functions the complexity of the probability density functions needed in a generalized flow situations was found not tractable in practice. An idealized flow based on the assumption of... 

Equations of state developed from perturbation theory are composed of a reference fluid equation to which are added perturbation terms. The reference fluid expresses the high repulsive energy that determines the molecular correlation and hence the structure of the fluid. The attractive energies that are relatively weak are treated as perturbations. Perturbation theory opens the door to the separate development of statistical theories for various intermolecular energies hard-sphere-fluid... 

For vapor-liquid equilibrium calculations up to moderate pressures, the B equation is suitable and convenient for the vapor phase for its applicability and simple form. Formulas have been derived from statistical theory for the calculation of virial coefficients, including B, from intermo-lecular potential energy functions, but intermolecular energy functions are hardly known quantitatively for real molecules. B is found for practical calculations by correlating experimental B values. Pitzer [1] correlated B of normal flnids in a generalized form with acentric factor to as the third parameter. 

The most effective approach to describe the optical properties of the arrays with short-range ordering is the statistical theory of multiple scattering of waves (STMSW) [5]. This approach considers electrodynamic coupling of spatially correlated scatterers as interference summation and subsequent averaging of... 

Goodness of Fit. The fitted model with Intercept for Case I is seen to have a correlation coefficient of 0.9996 which would often be interpreted to mean that the equation fits the data very well. However, we shall see from the Case II data set that the correlation coefficient Is not a sensitive method of evaluating curve fit. Hunter ( ) notes that in statistical theory, correlation Is a measure of the relationship between two random (dependent) variables. In a calibration problem, however, it is assumed that there Is a definite functional relationship between the dependent and Independent variables. Correlation, in Its strict statistical sense, does not exist. Van Arendonk et al. (21) point out that the correlation coefficient Is an insensitive tool for use In evaluating the quality of the fitted equation, and its use In such a manner may lead to erroneous conclusions. 

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