Space-like correlations

In this brief review of dynamics in condensed phases, we have considered dense systems in various situations. First, we considered systems in equilibrium and gave an overview of how the space-time correlations, arising from the themial fluctuations of slowly varying physical variables like density, can be computed and experimentally probed. We also considered capillary waves in an inliomogeneous system with a planar interface for two cases an equilibrium system and a NESS system under a small temperature gradient. 

The first experiments to analyze EPR correlations used polarized light beams rather than electronic spin systems. The results obtained by Aspect [44] are especially relevant since the systems for study were prepared to be separated space-like. Aspect analyzed the polarization of pairs of photons emitted by a single source toward separate detectors. Measured independently, the polarization of each set of photons fluctuated in a seemingly random way. However, when two sets of measurements were compared, they displayed an agreement stronger than could be accounted for by any local realistic theory. 

Short-range interactions (knowledge-based or semiemperical) must account for the polypeptide chain s conformational stiffness [20,42]. In other words, protein-like correlations enforced by the potential should extend over several residues. This would considerably narrow the available conformational space. 

No data are reported for s < 4 in Table 1 for a reason. This reason is connected with the tight correlation of the sets of g direct lattice vectors and k reciprocal space points selected in the calculation, when using a local basis set. Iterative Fourier transforms of matrices from direct to reciprocal space, like in Eq. [36], and vice versa (Eq. [38]), are the price to be paid for the already mentioned advantage of determining the extent of the interparticle interactions to be evaluated in direct space on the basis of simple criteria of distance. Consequently, the sets of the selected g vectors and k points must be well balanced. The energy values reported in Table 1 were all obtained for a particular set of g vectors, corresponding to the selection of those AOs in the lattice with an overlap of at least 10 with the AOs in the 0-cell. This process determines the g vectors for which F , S , and the I matrices (Eqs. [34] and [38]) need to be calculated, and if the number of k points included in the calculation is too small compared with the number of the direct lattice vectors, the determination of the matrix elements is poor and numerical instabilities occur. 

Fig. 1. Schematic representation of (a) nematic, (b) smectic and (c) cholesteric (or chiral nematic) liquid crystalline phases. In the nematic phase only orientational correlations are present with a mean alignment in the direction of the director n. In the smectic phase there are additional layer-like correlations between the molecules in planes perpendicular to the director. The planes, drawn as broken lines, are in reality due to density variations in the direction of the director. The interplane separation then corresponds to the period of these density waves. In the cholesteric phase the molecules lie in planes (defined by broken lines) twisted with respect to each other. Since the molecules in one plane exhibit nematic-like order with a mean alignment defined by the director n, the director traces out a right- or left-handed helix on translation through the cholesteric medium in a direction perpendicular to the planes. When the period of this helix is of the order of the wavelength of light, the cholesteric phase exhibits bright Bragg-like reflections. In these illustrations the space between the molecules (drawn as ellipsoids for simplicity) will be filled with the alkyl chains, etc., to give a fairly high packing...

This form neglects the self-scattering term appropriate for the ktr— regime, but which is irrelevant in a continuum-of-sites description. Equation (3.1) very accurately describes the exact Gaussian continuum model. In particular, it exactly reproduces the )k = 0 value and the self-similar intermediate scaling regime, d> k) = 2 k(Ty for R k (r. In real space, this selfsimilar behavior corresponds to power law, or critical-like, correlations, a (r) acr". This is a polymeric effect associated with the ideal random... 

The space-time correlation function like the radial distribution function has a simple physical meaning which is most helpful in suggesting simple models for the construction of this function. As we wiD show below the construction of this function is the first step in caloilating the frequency distribution of the scattered light. 

In a study in 1982, Luken and Culberson analyzed the change of the Fermi hole shape with respect to the position of reference electron to gain information about the spatial localizatirai of electrons [36], The Fermi hole density is derived from the same-spin pair density and describes the probability density to find an electron at given position, when another same-spin electron is localized at the reference position with all the other electros located somewhere in the space. Like in Sect. 2.2, it shows how the electronic motion of electrons creating a same-spin pair is correlated. For a closed-shell Hartree-Fock wave function, the so-called Fermi hole mobility function F(r) ... 

Langton further hypothesized that CAs able to perform complex computations will most likely be found in this regime, since complex computation in cellular automata requires sufficiently long transients and space-time correlation lengths. A review of this work that is skeptical about the relationships between A and dynamical and computational properties of CAs is given in Mitchell, Crutchfield, and Hraber (1994a). 

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. 

The second bullet point describes a special type of correlation that prevents two electrons of like spin being found at the same point in space, and it applies whenever the particles are fermions. For that reason, it is described as Fermi correlation. 

The second axiom, which is reminiscent of Mach s principle, also contains the seeds of Leibniz s Monads [reschQl]. All is process. That is to say, there is no thing in the universe. Things, objects, entities, are abstractions of what is relatively constant from a process of movement and transformation. They are like the shapes that children like to see in the clouds. The Einstein-Podolsky-Rosen correlations (see section 12.7.1) remind us that what we empirically accept as fundamental particles - electrons, atoms, molecules, etc. - actually never exist in total isolation. Moreover, recalling von Neumann s uniqueness theorem for canonical commutation relations (which asserts that for locally compact phase spaces all Hilbert-space representations of the canonical commutation relations are physically equivalent), we note that for systems with non-locally-compact phase spaces, the uniqueness theorem fails, and therefore there must be infinitely many physically inequivalent and... 

Usually we are only interested in mutual intensity suitably normalised to account for the magnitude of the helds, which is called the complex degree of coherence 712 (r). This quantity is complex valued with a magnitude between 0 and 1, and describes the degree of likeness of two e. m. waves at positions ri and C2 in space separated by a time difference r. A value of 0 represents complete decorrelation ( incoherence ) and a value of 1 represents complete eorrelation ( perfect coherence ) while the complex argument represents a difference in optical phase of the helds. Special cases are the complex degree of self coherence 7n(r) where a held is compared with itself at the same position but different times, and the complex coherence factor pi2 = 712(0) which refers to the case where a held is correlated at two posihons at the same time. 

Wave-like properties cause electrons to be smeared out rather than localized at an exact position. This smeared-out distribution can be described using the notion of electron density Where electrons are most likely to be found, there is high electron density. Low electron density correlates with regions where electrons are least likely to be found. Each electron, rather than being a point charge, is a three-dimensional particle-wave that is distributed over space in... 

In addition to sample rotation, a particular solid state NMR experiment is further characterized by the pulse sequence used. As in solution NMR, a multitude of such sequences exist for solids many exploit through-space dipolar couplings for either signal enhancement, spectral assignment, interauclear distance determination or full correlation of the spectra of different nuclei. The most commonly applied solid state NMR experiments are concerned with the measurement of spectra in which intensities relate to the numbers of spins in different environments and the resonance frequencies are dominated by isotropic chemical shifts, much like NMR spectra of solutions. Even so, there is considerable room for useful elaboration the observed signal may be obtained by direct excitation, cross polarization from other nuclei or other means, and irradiation may be applied during observation or in echo periods prior to... 

This quantity is of great importance, since it actually contains all information about electron correlation, as we will see presently. Like the density, the pair density is also a non-negative quantity. It is symmetric in the coordinates and normalized to the total number of non-distinct pairs, i. e., N(N-l).8 Obviously, if electrons were identical, classical particles that do not interact at all, such as for example billiard balls of one color, the probability of finding one electron at a particular point of coordinate-spin space would be completely independent of the position and spin of the second electron. Since in our model we view electrons as idealized mass points with no volume, this would even include the possibility that both electrons are simultaneously found in the same volume element. In this case the pair density would reduce to a simple product of the individual probabilities, i.e.,... 

This can only be true if p2 (xj, Xj) = 0. In other words, this result tells us that the probability of finding two electrons with the same spin at the same point in space is exactly zero. Hence, electrons of like spin do not move independently from each other. It is important to realize that this kind of correlation is in no way connected to the charge of the electrons but is a direct consequence of the Pauli principle. It applies equally well to neutral fermions and - also this is very important to keep in mind - does not hold if the two electrons have different spin. This effect is known as exchange or Fermi correlation. As we will show below, this kind of correlation is included in the Hartree-Fock approach due to the antisymmetry of a Slater determinant and therefore has nothing to do with the correlation energy E discussed in the previous chapter. 

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