Dynamics of correlations

The research was greatly facilitated by two important elements. The (formal, perturbative) solution of the Liouville equation is greatly simplified by a Fourier representation (see Appendix). The latter allows one to easily identify the various types of statistical correlations between the particles. The traditional dynamics thus becomes a dynamics of correlations. The latter is completed by... 

MSN. 186. 1. Prigogine, Dynamics of correlations—A formulation for both integrable and non-integrable dynamical systems. Appendix to the new edition of From Being to Becoming, 2001. 

The dynamics of correlations within the 5f band (spin fluctuations) was taken into account by Takaoka and Moriya (1983). If the 5f-electron states are hybridized with conduction-electron states, they contribute to the scattering. The thermal dehybridization reduces the scattering rate and even a maximum in the temperature dependence of p can develop. 

In summary, the strategy of these calculations is to explicitly consider products of phase-space density fields in the theory. These product fields describe the correlated motion of the solute and solvent molecules. An examination of the coupling of these higher order fields to the solute fields should then lead to a precise description of the effects on single-particle dynamics of correlated motion of many particles. 

Both for classical and quantum physics, elimination of interaction means existence of a unitary operator, which we call f/. We have shown that this operator U may be expressed in terms of the kinetic operators that I quoted above. Therefore our dynamics of correlation is equivalent to the various methods to obtain diagonalization of the Hamiltonian. In this case we also have the diagonalization of the Liouville operator for the density matrix. In short, Poincare integrability is equivalent to the integrability of the Liouville equation. But there is a much more interesting case, as there exists a class of non-integrable... 

PRIGOGINE I., Dynamics of correlations A formalism for both integrable and nonintegrable dynamical systems, proceedings of the XXIst Solvay Conference of Physics Dynamical systems and irreversibility , in Advances in Chemical Physics, vol. 122, pp. 261-275, Wiley, New York, 2002. 

Supposing radical pairs 1, 2 and 3,4 are spin correlated, then the spin-dynamics of correlated pair 1,2 can be described by wavefunction ijfn and correlated pair (3,4 by wavefunction t/r34. Each individual one-pair spin wavefunction evolves independently according to its Hamiltonian. Following a cross recombination, it is necessary to construct the total wavefunction V tot = V i2V 34 and collapse the wave-function t/f(ot onto either a singlet or triplet state depending on the spin-dynamics of the encountering pair. It is then necessary to extract the coefficients for the surviving radical pair and reconstruct the one pair wavefunction. From hereon, this is referred to as the decomposition method. 

The molecular beam and laser teclmiques described in this section, especially in combination with theoretical treatments using accurate PESs and a quantum mechanical description of the collisional event, have revealed considerable detail about the dynamics of chemical reactions. Several aspects of reactive scattering are currently drawing special attention. The measurement of vector correlations, for example as described in section B2.3.3.5. continue to be of particular interest, especially the interplay between the product angular distribution and rotational polarization. 

Furtlier details can be found elsewhere [20, 78, 82 and 84]. An approach to tire dynamics of nematics based on analysis of microscopic correlation fimctions has also been presented [85]. Various combinations of elements of tire viscosity tensor of a nematic define tire so-called Leslie coefficients [20, 84]. 

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

Since the stochastic Langevin force mimics collisions among solvent molecules and the biomolecule (the solute), the characteristic vibrational frequencies of a molecule in vacuum are dampened. In particular, the low-frequency vibrational modes are overdamped, and various correlation functions are smoothed (see Case [35] for a review and further references). The magnitude of such disturbances with respect to Newtonian behavior depends on 7, as can be seen from Fig. 8 showing computed spectral densities of the protein BPTI for three 7 values. Overall, this effect can certainly alter the dynamics of a system, and it remains to study these consequences in connection with biomolecular dynamics. 

There are two types of electron correlations static and dynamic. Static correlation refers to a near degeneracy of a given state a dynamic correlation refers to the in stantaneous avoidance of electrons with each other. 

Molecular modeling is an indispensable tool in the determination of macromolecular structures from NMR data and in the interpretation of the data. Thus, state-of-the-art molecular dynamics simulations can reproduce relaxation data well [9,96] and supply a model of the motion in atomic detail. Qualitative aspects of correlated backbone motions can be understood from NMR structure ensembles [63]. Additional data, in particular residual dipolar couplings, improve the precision and accuracy of NMR structures qualitatively [12]. 

Molecular dynamics simulations have also been used to interpret phase behavior of DNA as a function of temperature. From a series of simulations on a fully solvated DNA hex-amer duplex at temperatures ranging from 20 to 340 K, a glass transition was observed at 220-230 K in the dynamics of the DNA, as reflected in the RMS positional fluctuations of all the DNA atoms [88]. The effect was correlated with the number of hydrogen bonds between DNA and solvent, which had its maximum at the glass transition. Similar transitions have also been found in proteins. 

We discuss the rotational dynamics of water molecules in terms of the time correlation functions, Ciit) = (P [cos 0 (it)]) (/ = 1, 2), where Pi is the /th Legendre polynomial, cos 0 (it) = U (0) U (it), u [, Is a unit vector along the water dipole (HOH bisector), and U2 is a unit vector along an OH bond. Infrared spectroscopy probes Ci(it), and deuterium NMR probes According to the Debye model (Brownian rotational motion), both... 

By far the most common methods of studying aqueous interfaces by simulations are the Metropolis Monte Carlo (MC) technique and the classical molecular dynamics (MD) techniques. They will not be described here in detail, because several excellent textbooks and proceedings volumes (e.g., [2-8]) on the subject are available. In brief, the stochastic MC technique generates microscopic configurations of the system in the canonical (NYT) ensemble the deterministic MD method solves Newton s equations of motion and generates a time-correlated sequence of configurations in the microcanonical (NVE) ensemble. Structural and thermodynamic properties are accessible by both methods the MD method provides additional information about the microscopic dynamics of the system. 

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