Collective variables

Trosset J-Y, Scheraga HA (1999) Flexible docking simulations scaled collective variable Monte Carlo minimization approach using Bezier splines, and comparison with a standard Monte Carlo algorithm. J Comp Chem 20 244-252... 

Sklenar, H., Wiistner, D., Rohs, R. Using internal and collective variables in Monte Carlo simulations of nucleic acid structures chain breakage/closure algorithm and associated Jacobians. J. Comput. Chem. 2006, 27, 309-15. 

Golani I, Kafkafi N, Drai D (1999) Phenotyping stereotypic behavior collective variables, range of variation and predictability. Appl Anim Behav Sci 65 191-220 Graeff FG (2002) On serotonin and experimental anxiety. Psychopharmacology (Berl) 163 467-476... 

Within this section, it will be assiuned that the LRA holds in the forms of both Eqs. (8) and (10). Thus no distinction will be made between Cfif) and Cl (i) and the subseript denoting the solute electronic state will be dropped. The dynamical variable AC that describes the solvation response is collective, dependent on the relative distances and orientations of the solute and of all V solvent moleeules in the system. In order to umavel how different types of dynam-ies eontribute to the time correlation of a collective variable such as 8AC, it is useful to introduce the corresponding velocity time correlation, which in this case is the solvation velocity ... 

In principle, we already have in our disposal the SRPA formalism for description of the collective motion in space of collective variables. Indeed, Eqs. (11), (12), (18), and (19) deliver one-body operators and strength matrices we need for the separable expansion of the two-body interaction. The number K of the collective variables qk(t) and pk(t) and separable terms depends on how precisely we want to describe the collecive motion (see discussion in Section 4). For K = 1, SRPA converges to the sum rule approach with a one collective mode [6]. For K > 1, we have a system of K coupled oscillators and SRPA is reduced to the local RPA [6,24] suitable for a rough description of several modes and or main gross-structure efects. However, SRPA is still not ready to describe the Landau fragmentation. For this aim, we should consider the detailed Iph space. This will be done in the next subsection. 

T-odd Pk(t) collective variables can be also specified as harmonic oscilla-... 

I think Dr. Williams avowed pessimism is inevitable if one is trying to explain all biological phenomena in terms of molecular, microscopic concepts. But molecular description can be a Procrustean bed when dealing with complex, intrinsically macroscopic phenomena, because simple interpretability may not be feasible. In fact, the selection of a few essential macroscopic variables from a vast number of microscopic variables (or their combinations) is crucial not only for understanding via simplification, but also because collective variables (order parameters) tend to obey qualitatively different rules or laws that are not obvious in,... 

The bottleneck of this approach is obvious the expression for transition probabilities through collective variables of a whole system (total number of particles) means that rather rare fluctuations are taken into account only, whereas their spatial correlations are neglected (i.e., different parts of a system interact being separated by a distinctive distance - the correlation length). 

To treat the stochastic Lotka and Lotka-Volterra models, we have now to extend the formalism presented in Section 2.2.2, where collective variables-numbers of particles iVA and Vg were used to describe reactions. The point is that this approach neglects local density fluctuations in small element volumes. To incorporate both these fluctuations and their correlations due to diffusive conjunction, we are in position now to reformulate these models in terms of the diffusion-controlled processes - in contrast to the rather primitive birth-death formalism used in Section 2.2.2. It permits also to demonstrate in the non-trivial way a role of diffusion in the autowave processes. The main results of this Chapter are published in [21, 25]. 

Let us outline briefly a possible way to calculate the normal modes of a molecule, and the relation between the positions of individual atoms and collective variables. We assume, that the atomic configuration of a system is determined mainly by the elastic forces, which are insensitive to the transport electrons. The dynamics of this system is determined by the atomic Hamiltonian... 

Simulations Scaled Collective Variable Monte Carlo Minimization Approach Using Bezier Splines, and Comparison with a Standard Monte Carlo Algorithm. 

Hayward S, Go N. Collective variable description of native protein dynamics. Ann Rev Phys Chem 1995 46 223-250. 

Let us discuss the physical meaning of this formal result. The form of Eq. (300) suggests that 8 is a control parameter generating cooperative and noncooperative properties. We see that the choice 8=1 corresponds to creating a collective motion that is totally insensitive to the frequencies of the bath oscillators and, consequently, yields no cooperation. The collective variable i becomes equivalent to white noise. 

Hamiltonian dynamics show that classical mechanics is invariant to ( t) and (t). In a macroscopic description of dissipative systems, we use collective variables of temperature, pressure, concentration, and convection velocity to define an instantaneous state. The evolution equations of the collective variables are not invariant under time reversal... 

In chemical reactions, orbits on stable and unstable manifolds of NHIMs describe movements of reaction coordinates. In some cases, these reaction coordinates are those degrees of freedom describing the behavior of individual nuclei such as a bond length between a pair of atoms. In other cases such as protein folding, reaction coordinates describe collective behavior where multiple nuclei participate. In either case, the processes of leaving a NHIM and approaching another one involve reformulation of reaction coordinates. In particular, when reaction coordinates are collective variables, reformulation processes themselves are of interest. We think that the study of intersections is crucial to understand how a certain collective movement is replaced by another one. 

Raman cross sections are commonly determined by quantitatively comparing the Raman signal for an unknown to that for a standard with known cross section. The standard cross sections were determined by comparison to some radiometric standard, with painstaking attention to collection variables and geometry. Table 2.2 lists several cross sections determined for some common materials. When available, the frequency-independent cross section (crj) defined by Eq. (2.9) is listed. The cross sections in Table 2.1 show good agreement among several labs and are considered very reliable, provided the conditions are fixed. 

Figure 3.1. Schematic of the collection variables for I SC backscattering. L, indicates specific intensity, C is collection function.

Table 6. Dependence of Effective Path Length and KSId on Sample and Collection Variables, 180° Geometry...

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