Fixed axis rotation model

In the fixed axis rotation model of dielectric relaxation of polar molecules a typical member of the assembly is a rigid dipole of moment p rotating about a fixed axis through its center. The dipole is specified by the angular coordinate < ) (the azimuth) so that it constitutes a system of 1 (rotational) degree of freedom. The fractional diffusion equation for the time evolution of the probability density function W(4>, t) in configuration space is given by Eq. (52) which we write here as... 

Furthermore, just as in the one-degree-of-freedom fixed-axis rotation model, in the high damping limit ((3 > 1), Eq. (282) can be simplified yielding the generalization to fractional dynamics of the Rocard [44] equation, namely,... 

In order to demonstrate how the anomalous relaxation behavior described by the hitherto empirical Eqs. (9)—(11) may be obtained from our fractional generalizations of the Fokker-Planck equation in configuration space (in effect, fractional Smoluchowski equations), Eq. (101), we first consider the fractional rotational motion of a fixed axis rotator [1], which for the normal diffusion is the first Debye model (see Section II.C). The orientation of the dipole is specified by the angular coordinate 4> (the azimuth) constituting a system of one rotational degree of freedom. Electrical interactions between the dipoles are ignored. 

This potential has two potential minima on the sites at <(> = 0 and = n as well as two energy barriers located at < ) = jt/2 and <[) = 3n/2. This model has been treated in detail for normal diffusion in Refs. 8,61, and 62. Here we consider the fractional Fokker-Planck equation [Eq. (55)] for a fixed axis rotator with dipole moment p moving in a potential [Eq. (163)]. 

The behavior of the dielectric spectra for the two-rotational-degree-of-ffeedom (needle) model is similar but not identical to that for fixed-axis rotators (one-rotational-degree-of-fireedom model). Here, the two- and one-rotational-degree-of-freedom models (fractional or normal) can predict dielectric parameters, which may considerably differ from each other. The differences in the results predicted by these two models are summarized in Table I. It is apparent that the model of rotational Brownian motion of a fixed-axis rotator treated in Section IV.B only qualitatively reproduces the principal features (return to optical transparency, etc.) of dielectric relaxation of dipolar molecules in space for example, the dielectric relaxation time obtained in the context of these models differs by a factor 2. 

One possible such mechanism for fixing a pattern is to have a phase transition. For example, if the pattern is in terms of a distribution of large molecules on the outer membrane surface, as in the Fucus-like models discussed here, then a membrane phase transition from a more liquid-like to a more crystal-like state of the membrane could essentially immobilize the membrane bound species and freeze in the pattern. In fact several hours after fertilization in Fucus the lability (rotatability) of the polar axis significantly decreases. Indeed this freezing of the Fucus patterning is not easily explained in terms of a Turing mechanism since the rotational symmetry of the Fucus egg, as discussed previously, implies that the electrical polarity is not stable (or more precisely is marginally stable) to polar axis rotation. 

We shall now demonstrate how the CTRW in the diffusion limit may be used to justify the fractional diffusion equation. We consider an assembly of permanent dipoles constrained to rotate about a fixed axis (the dipole is specified by the angular coordinate unit circle with fixed angular spacing A. We note that A may not necessarily be fixed for example, if we have a Gaussian distribution of jumps, the standard deviation of A serves as a fixed quantity. A typical dipole may remain in a fixed orientation at a given site for an arbitrary long waiting time. It may then reorient to another discrete orientation site. This is the discrete orientation model. 

Anomalous rotational diffusion in a potential may be treated by using the fractional equivalent of the diffusion equation in a potential [7], This diffusion equation allows one to include explicitly in Frohlich s model as generalized to fractional dynamics (i) the influence of the dissipative coupling to the heat bath on the Arrhenius (overbarrier) process and (ii) the influence of the fast (high-frequency) intrawell relaxation modes on the relaxation process. The fractional translational diffusion in a potential is discussed in detail in Refs. 7 and 31. Here, just as the fractional translational diffusion treated in Refs. 7 and 31, we consider fractional rotational subdiffusion (0rotation about fixed axis in a potential Vo(< >)- We suppose that a uniform field Fi (having been applied to the assembly of dipoles at a time t = oo so that equilibrium conditions prevail by the time t = 0) is switched off at t = 0. In addition, we suppose that the field is weak (i.e., pFj linear response condition). 

The first objective of this review is to describe a method of solution of the Langevin equations of motion of the itinerant oscillator model for rotation about a fixed axis in the massive cage limit, discarding the small oscillation approximation in the context of dielectric relaxation of polar molecules, this solution may be obtained using a matrix continued fraction method. The second... 

The simplest model for the rotation of a dumbbell molecule is planar reorientation about a fixed axis perpendicular to the midpoint of the H-H bond. The potential function has necessarily twofold symmetry and the model Hamiltonian for rotational dynamics can be written as ... 

The theoretical model of the IR linear dichroism suggests a single-axis orientation of a molecular assembly with axial (cylindrical) synunetry. This means that the molecules are located with their longest axis in the direction, which induces the anisotropy, determined by the director n, and the side groups are randomly fixed or rotate along the length of this axis. Under these conditions, the dipole moments of transition caused by the normal vibrations form conical surfaces with director n (Figure 1.2) [2,3,6-8]. 

The structural information derived from relaxation enhancement studies depends somewhat on the model chosen to describe the interaction of solvent protons with the protein molecules. For example even if the experiments indicated that the dispersion of Tfpr were essentially determined by the correlation time for rotational tumbling of the protein the tumbling of the hydration waters would not necessarily have to be restricted to that of the entire hydrated protein. Evidence was found that fast intramolecular tumbling about an axis fixed with respect to the surface of the hydrated species reduced the proton and O17 nuclear relaxation rates even in extremely stable aquocomplexes of Al3+ and other metal ions (Connick and Wiithrich (21)). The occurrence of similar... 

Minor variations of the backbone and glycosyl rotations from the fixed values used in the sample computations above produce a variety of theoretically acceptable double helices. As evident from the partial list of structures in Table I, these structures include several 10-fold duplexes similar to the B-DNA models from fiber diffraction studies as well as the larger 13-fgld complex. Despite the large fluctuations in h from 1.7 to 4.3 A, the bases associate at standard separation distances (3 ft < < 4 A and 2.8 X < < 3.0 A) and orientations (A < 30° and < 30°) in all cases. In order to avoid severe steric contacts at small values of h, the bases may tilt up to values of n = 45° with respect to the standard orientation (n = 90°) perpendicular to the helix axis. 

The planar symmetric pore consists of two parallel walls with the distance H between them which infinitely range into the x- and j/-direction of the pore-fixed coordinate system. The 2-axis stands perpendicularly on the i-j/-plane as the normale of both walls. The cylinder pore model places its j/-axis as the rotational axis. The z-axis stands perpendicularly on the pore wall as in slit-like pores and runs through the middle of the pore. Hence the x- differs from the y-axis inside the cylinder pore in opposite to the slit-like pore. This fact turns out to be important even for the adsorption of fluids which consists of non-spherical particles. 

Continuous annular chromatography (CAC) has been the subject of several recent experimental studies (120, 121), models (122, 123) and a brief review (124). The equipment is very similar to the CRAE (Figure 7). Feed, eluent, and regeneration solutions (if necessary) are fed to fixed points or arcs at the top of an annular packed bed which rotates slowly about its axis. As the chromatogram develops, the components separate... 

We first consider the A contribution. Equation 1, and an explanation in terms of a two-site model i.e., a model in which a water molecule exchanges between solution and sites (or class of sites) on or near a protein molecule such that at least one direction fixed in the water molecule is constrained to move rigidly with the protein molecule. In the simplest case, a water molecule attaches rigidly to the protein, moves with it for a while, and then leaves. In a somewhat more complex case, the attachment may be less rigid so that the water molecule is free to rotate about an axis fixed with respect to the protein. Additionally, a situation in which water molecules partially orient in the electric fields near the protein surface because of their electric dipole moments would also be a two-site model. Characteristic of a two site-model is that a time Tj, or a distribution of such times, can be defined that measures the mean lifetime of a water molecule in the protein-associated state. Moreover, such a time is in principle a measurable quantity, and its value must satisfy two criteria it must be at least comparable to if not longer than Tj, otherwise the nuclei of the bound water molecules could not sense the rotational motion of the protein molecules and it must be comparable to or shorter than the nuclear relaxation time of a bound water molecule, else it could not communi-... 

Fig. AIII.l. Coordinates used for the model of a rotationally restricted diatomic vibrator. For simplicity the center of mass of the nuclei is assumed to be fixed in space and the motion of the nuclei is assumed to be restricted to a plane perpendicular to the c-axis. No restrictions are imposed on the motions of the electrons. The positions of Mi and Ms should be exchanged in order to agree with the text.

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