Coordinate, collective

First we consider static properties of concentrated solutions in equilibrium. Let Ran be the position of the n-th segment of the a-th chain. The equilibrium distribution function for Ran is written as a natural generalization of eqn (2.95) 

Although the starting point is clear, theoretical development is not easy since calculation of any average from eqn (5.9) involves a highly nonseparable integration which is common to many body problems. However, under certain conditions we can progress by using collective coordinates.  

Instead of describing the problem in terms of R n, we focus our attention on the local segment density c(r) defined by 

The component cq is equal to the average concentration c, and does not fluctuate at all, while with denotes the fluctuation of the local concentration. 

A technical remark has to be made here. In the representation of c, not all are independent of each other since and c j are related as 

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Percus J K and Yevick G J 1958 Analysis of classical statistical mechanics by means of collective coordinates Phys. Rev. 110 1... 

Bruce A D 1981 Probability density functions for collective coordinates in Ising-like systems J. Phys. O Soiid State Phys. 14 3667-88... 

Step 3 The eigenvectors of C define 3N - 6 collective coordinates (quasi... 

The OPMBS featured a 2-day organizational meeting between study management, including the Study Director and sample collection coordinator, and representatives... 

The sampling plan for the OPMBS required that no organic or hydroponically grown produce be obtained, that the precise commodity, i.e., head, not leaf lettuce, be collected, and that only two representative single-serving samples of each commodity be collected on a given date by each of 20 shoppers. The plan also specified two back-up (secondary) stores for each primary store and instructed that the collection coordinator be notified if samples could not be collected on the specified date so that the sample collection could be rescheduled. 

In the OPMBS, shoppers were selected from a list maintained by the sample collection coordinator and were for the most part trained dietitians. Each shopper received at least three in-depth telephone contacts prior to sample collection. Each shopper also received written instructions. 

Contact with the shoppers was restricted to field phase management study personnel, for two reasons. First, clearly defined lines of communication had to be maintained. Second, in order to ensure that the identity of the stores remained blind (i.e., unknown to everyone downstream from sample collection), in compliance with one of the design criteria, communication with the shoppers had to be restricted. Overall, limiting contact with shoppers to one entity and using modern technology, such as facsimiles and e-mail to facilitate and document communications between shoppers and the collection coordinator, were essential factors in the successful conduct of the sampling phase of the study. 

This steep wire-length dependence is in striking contrast with the more rigorous PMF calculations of the same proton transfer process [14, 102], In the PMF calculations, a collective coordinate [103] is used to monitor the progress of the proton transfer without enforcing specific sequence of events involving individual protons along the wire the use of a collective coordinate is important because this allows... 

In general, collective-coordinate approaches separate AG, (x) into two... 

Next problem is to average Detiow over instanton collective coordinates to get quark partition function Z[V,m. ... 

The remaining problem is to average the quark determinant over collective coordinates . It is a rather simple procedure, since the low density of the instanton medium (n2 ( )4 0.1) allows us to average over positions and orientations of the instantons independently. Then... 

Since the pioneer work of Mayer, many methods have become available for obtaining the equilibrium properties of plasmas and electrolytes from the general formulation of statistical mechanics. Let us cite, apart from the well-known cluster expansion 22 the collective coordinates approach, the dielectric constant method (for an excellent summary of these two methods see Ref. 4), and the nodal expansion method.23... 

Sz(i) being the total angular momentum of the i spins. The linear combination 5 (l)/a1 + Sz(2)jx2 is now coupled to the dipole-dipole heat bath, whereas the difference Sz(l)/x1 — Sz(2)jx2 remains constant. It may be shown that the chemical equilibrium condition (31) is then equivalent to an equalization of the temperatures of the collective coordinate Sz(l)jx1 + Sz(2)/x2 and of the dipole-dipole heat bath. 

The second step of the evolution towards equilibrium is the Zeeman dipole-dipole relaxation. Hartmann and Anderson estimated this time using the hypothesis that p at any time is of the form (22). As a consequence of the shortness of the dipole-dipole relaxation time we may assume that the dipole-dipole system always remains in equilibrium we are thus led to treat the evolution of the Zeeman system as the Brownian motion of a collective coordinate in the dipole-dipole heat bath. We assume that the diagonal elements of p have the form... 

We wanted to extend this approach to include dynamical effects on line shapes. As discussed earlier, for this approach one needs a trajectory co t) for the transition frequency for a single chromophore. One could extract a water cluster around the HOD molecule at every time step in an MD simulation and then perform an ab initio calculation, but this would entail millions of such calculations, which is not feasible. Within the Born Oppenheimer approximation the OH stretch potential is a functional of the nuclear coordinates of all the bath atoms, as is the OH transition frequency. Of course we do not know the functional. Suppose that the transition frequency is (approximately) a function of a one or more collective coordinates of these nuclear positions. A priori we do not know which collective coordinates to choose, or what the function is. We explored several such possibilities, and one collective coordinate that worked reasonably well was simply the electric field from all the bath atoms (assuming the point charges as assigned in the simulation potential) on the H atom of the HOD molecule, in the direction of the OH bond. 

Following the method of Ref. 59, we write the equation of motion for the collective coordinate of local segment density p(r). 

The overall objective of these studies is to unravel mechanisms of interfacial PT. This requires identification of collective coordinates (or reaction coordinates) and transition pathways of transferring protons. Differences in activation energies and rates of corresponding mechanism due to distinct polymer constituents, acid head groups, side chain lengths, side chain densities, and levels of hydration have to be examined. Comparison with experimental... 

In the presence of both order-disorder and displacive, as in the KDP family, the two dynamic concepts have somehow to be merged. It could well be that the damping constant Zs becomes somewhat critical too (at least in the over-damped regime of the soft mode), because of the bihnear coupling of r/ and p. It would, however, lead too far to discuss this here in more detail. The corresponding theory of NMR spin-lattice relaxation for the phase transitions in the KDP family has been worked out by Blinc et al. [19]. Calculation of the spectral density is here based on a collective coordinate representation of the hydrogen bond fluctuations connected with a soft lattice mode. Excellent and comprehensive reviews of the theoretical concepts, as well as of the experimental verifications can be found in [20,21]. 

VL = 98.163 MHz in D-RADP-15. The solid lines are fits based on a collective coordinate representation of the deuteron intra-bond fluctuations connected with a soft lattice mode... 

Collective coordinates, 35, 98 Collision theory, 528, 542 Comparative molecular field analysis, 308-310 Complete basis set, see Multilevel methods) Compressibility, 418, 446 Condensed-phase effects, see also Solvation... 

S. Mukamel In order to represent situations in which nuclear and electronic dynamics take place on the same time scale, one needs to incorporate nuclear degrees of freedom into the description. A frequency-dependent Redfield superoperator can capture some effects, but in general is very limited and may even yield negative probabilities. A method for decomposing a given spectral density into a few collective coordinates and identifying these coordinates was presented in Ref. 1. 

Thus neutrons probe the dynamics of density fluctuations. The properties pk(0 can be regarded as collective coordinates. 

In order to apply the memory function formalism to the collective coordinates of Eq. (367), it is necessary to define the dimensionless normalized collective coordinates,... 

It is necessary next to relate these small displacements to the collective coordinates g, from a Taylor expansion of the general potential energy U(Q). In the adiabatic approximation, the nuclear coordinates Q are free parameters and can be used as a basis for the Taylor expansion. Thus we write IJ(Q) in the general form (Ref. [2], Chapter 3) ... 

We have followed a phenomenological approach and used the cluster model [18]. In this model the eg-type distortion interacts more strongly with the electronic state of an octahedral coordinated Cr3+ ion than the distortions of t2g symmetry. According to Ham [19], we assume that the continuum of vibrational modes with eg character can be approximated by a single mode with an effective frequency o>, mass /r and coupling constant V. The collective coordinates of the eg mode are conventionally known as Qe x2 — y2) and Qs ( 3z2 — r2). The linear Jahn-Teller Hamiltonian in equation (1) for the X state is [18] ... 

Any linear function of 17 can serve equally well as a reaction coordinate as long as all the coordinates contributing to the collective coordinate 17 are globally harmonic (i.e., with the same force constant matrix for initial and final states, as in Equation (3.77)). A familiar alternative to 17 is the dimensionless progress parameter of Marcus, m, for the overall ET process [6], related to 17 (Equations (3.79) and (3.81)) by the following linear transformation... 

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