Correlation correlated motion

The question then is, to what degree can the microscopic motions influence the macroscopic ones is there a flow of infonnation between them [66] Biological systems appear to be nonconservative par excellence and present at least the possibility that random thermal motions are continuously injecting new infonnation into the macroscales. There is certainly no shortage of biological molecular machines for turning heat into correlated motion (e.g. [67] and section C2.14.5 note also [16]). 

IlyperChem can either use initial velocilies gen eraled in a previous simulation or assign a Gaussian distribution of initial velocities derived from a random n iim her generator. Random numbers avoid introducing correlated motion at the beginn ing of a sim illation. ... 

Restricted diffusion, correlated motion of spins, or any deviation from a free behavior of the molecules will result in a propagator shape different from a Gaussian one. A wide range of studies have dealt with such problems during the last two decades and NMR has turned out to be the method of choice for quantifying restricted diffusion phenomena such as for liquids in porous materials or dynamics of entangled polymer molecules. 

Figure 5.17 Correlated motion during vacancy diffusion (a) vacancy can jump to any surrounding position and its motion follows a random walk (b, c) the motion of a tracer atom is correlated, as a jump into a vacancy (b) is most likely to be followed by a jump back again (c).

The correlation factor, for any mechanism, is given by the ratio of the values of the mean square displacement of the atom (often the tracer) moving in a correlated motion to that of the atom (or vacancy) moving by a random-walk process. If the number of jumps considered is large, the correlation factor/can be written as... 

Figure 5.18 Correlated motion during intersticialcy diffusion (a, b) an interstitial can ...

Growing Range of Correlated Motion in a Polymer Melt on Cooling Towards the Glass Transition. 

Polymer-Specific Effects of Bulk Relaxation and Stringlike Correlated Motion in the Dynamics of a Supercooled Polymer Melt. 

To obtain a more complete description, we need to find an analytic expression for the pre-exponential factor Dq of the diffusion coefficient by considering the microscopic mechanism of diffusion. The most straightforward approach, which neglects correlated motion between the ions, is given by the random-walk theory. In this model, an individual ion of charge q reacts to a uniform electric field along the x-axis supplied, in this case, by reversible nonblocking electrodes such that dCj(x)/dx = 0. Since two... 

In the random-walk model, the individual ions are assumed to move independently of one another. However, long-range electrostatic interactions between the mobile ions make such an assumption unrealistic unless n is quite small. Although corrections to account for correlated motions of the mobile ions at higher values of n may be expected to alter only the factor y of the pre-exponential factor Aj., there are at least two situations where correlated ionic motions must be considered explicitly. The first occurs in stoichiometric compounds having an = 1. but a low AH for a cluster rotation the second occurs for the situation illustrated in Fig. 3.6(c). 

Fig. 11 Schematic representation of the temperature dependence of the spin-lattice relaxation Ti for thermally activated motion (a) according to Eq. 4 and for correlated motion connected with a soft lattice mode (b)...

Figure 14 Mechanism of oxygen elimination from the structure of BajYC Oy 0. (a) As an effect of temperature increase, atom A may jump into position A (b) As a consequence of this shift, atom B may jump into position B, C into C etc., thus causing a correlated motion of the oxygen atoms terminating with the expulsion of half oxygen atom from the structure (c) Atom C now may jump into positions C or C" generating a second cascade...

Perhaps enzyme-substrate recognition and interaction are facilitated by an oscillating active site Its correlated motion could position more readily and reliably the catalytically essential groups of atoms. Recognition of the substrate could be visualized as a nonlinear resonance phenomenon, perhaps providing the mechanism of energy transfer from the entatic active site region to the substrate. An off-resonance condition could characterize an enzyme-inhibitor interaction. 

Up to now, we have been discussing many-particle molecular systems entirely in the abstract. In fact, accurate wave functions for such systems are extremely difficult to express because of the correlated motions of particles. That is, the Hamiltonian in Eq. (4.3) contains pairwise attraction and repulsion tenns, implying that no particle is moving independently of all of the others (the term correlation is used to describe this interdependency). In order to simplify the problem somewhat, we may invoke the so-called Born-Oppenheimer approximation. This approximation is described with more rigor in Section 15.5, but at this point we present the conceptual aspects without delving deeply into the mathematical details. 

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