Condensing trajectories

This is due to the fact that the relaxation rate at a narrow place, i.e. the region of "condensing trajectories [229], tends to zero more rapidly than the length of the trajectory where the relaxation is retarded. These properties, which can easily be obtained from the analysis of eqns. (27) and (28), make it possible to obtain the inequality [230]... 

Figure 12 Si02 versus MgO in coarse-grained CAIs from CV chondrites, compared with the condensation trajectory for a gas of solar composition at 10 atm. Note that type A CAIs lie along the condensation trajectory, but type B CAIs lie to the left of the trajectory. For more details on type A and B CAIs, see Chapter 1.08, by MacPherson. The CAI data have been corrected for nonrepresentative sampling of different minerals by correcting to the solar system Ca/Al ratio (sources Grossman et al., 2000 Simon et al., 2002).

Montgomery J A Jr, Chandler D and Berne B J 1979 Trajectory analysis of a kinetic theory for isomerization dynamics in condensed phases J. Chem. Phys. 70 4056... 

When desired, ihe machine can also be designed as an inductor to supply lagging reactive kVAr, with the help of AVR and QDC. It will serve little purpose, if used both as a motor and a condenser. Refer to Figure 16.3. illustrating the trajectory of the cm rent phasoi. /. For the current / at a p.f. cos 0 (leading). 

Chymotrypsin, 170,171, 172, 173 Classical partition functions, 42,44,77 Classical trajectories, 78, 81 Cobalt, as cofactor for carboxypeptidase A, 204-205. See also Enzyme cofactors Condensed-phase reactions, 42-46, 215 Configuration interaction treatment, 14,30 Conformational analysis, 111-117,209 Conjugated gradient methods, 115-116. See also Energy minimization methods Consistent force field approach, 113 Coulomb integrals, 16, 27 Coulomb interactions, in macromolecules, 109, 123-126... 

Finally, in Sect. 7.6, we have discussed how various free energy calculation methods can be applied to determine free energies of ensembles of pathways rather than ensembles of trajectories. In the transition path sampling framework such path free energies are related to the time correlation function from which rate constants can be extracted. Thus, free energy methods can be used to study the kinetics of rare transitions between stable states such as chemical reactions, phase transitions of condensed materials or biomolecular isomerizations. 

Reactions in solution proceed in a similar manner, by elementary steps, to those in the gas phase. Many of the concepts, such as reaction coordinates and energy barriers, are the same. The two theories for elementary reactions have also been extended to liquid-phase reactions. The TST naturally extends to the liquid phase, since the transition state is treated as a thermodynamic entity. Features not present in gas-phase reactions, such as solvent effects and activity coefficients of ionic species in polar media, are treated as for stable species. Molecules in a liquid are in an almost constant state of collision so that the collision-based rate theories require modification to be used quantitatively. The energy distributions in the jostling motion in a liquid are similar to those in gas-phase collisions, but any reaction trajectory is modified by interaction with neighboring molecules. Furthermore, the frequency with which reaction partners approach each other is governed by diffusion rather than by random collisions, and, once together, multiple encounters between a reactant pair occur in this molecular traffic jam. This can modify the rate constants for individual reaction steps significantly. Thus, several aspects of reaction in a condensed phase differ from those in the gas phase ... 

In the MQC mean-field trajectory scheme introduced above, all nuclear DoF are treated classically while a quantum mechanical description is retained only for the electronic DoF. This separation is used in most implementations of the mean-field trajectory method for electronically nonadiabatic dynamics. Another possibility to separate classical and quantum DoF is to include (in addition to the electronic DoF) some of the nuclear degrees of freedom (e.g., high frequency modes) into the quantum part of the calculation. This way, typically, an improved approximation of the overall dynamics can be obtained—albeit at a higher numerical cost. This idea is the basis of the recently proposed self-consistent hybrid method [201, 202], where the separation between classical and quantum DoF is systematically varied to improve the result for the overall quantum dynamics. For systems in the condensed phase with many nuclear DoF and a relatively smooth distribution of the electronic-vibrational coupling strength (e.g.. Model V), the separation between classical and quanmm can, in fact, be optimized to obtain numerically converged results for the overall quantum dynamics [202, 203]. 

AP composite propellants without aluminum particles are termed reduced-smoke propellants and are employed in tactical missiles to conceal their launch site and flight trajectory. No visible smoke is formed when the relative humidity of the atmosphere is less than about 40%. However, since high-frequency combustion oscillation tends to occur in the combustion chamber in the absence of solid particles that serve to absorb the oscillatory energy, a mass fraction of 0.01-0.05 of metallic particles is still required for the reduced-smoke propellants. These particles and/or their oxide particles generate thin smoke trails. The white smoke trail includes the white fog generated by the HCl molecules and the condensed water vapor of the humid atmosphere. 

Consider the example of condensed phase transitions between vibrational states, which have energies that are significantly drfferent compared with knT. The momentum on the initial surface before a hop and the final surface momentum after the hop are considerably drfferent for typical values of the initial momentum sampled Irom a canonical distribution. This causes the two branches of the combined trajectory to quickly diverge, and action for the combined trajectory to grow rapidly. The result is that the integrand converges very quickly as a function of x, particularly after the and Fj integrations have been performed. 

In order to test the small x assumptions in our calculations of condensed phase vibrational transition probabilities and rates, we have performed model calculations, - for a colinear system with one molecule moving between two solvent particles. The positions ofthe solvent particles are held fixed. The center of mass position of the solute molecule is the only slow variable coordinate in the system. This allows for the comparison of surface hopping calculations based on small X approximations with calculations without these approximations. In the model calculations discussed here, and in the calculations from many particle simulations reported in Table II, the approximations made for each trajectory are that the nonadiabatic coupling is constant that the slopes of the initial and final... 

It is weliknown that all static polarizations of a beam of radiation, as well as all static rotations of the axis of that beam, can be represented on a Poincare sphere [25] (Fig. la). A vector can be centered in the middle of the sphere and pointed to the underside of the surface of the sphere at a location on the surface that represents the instantaneous polarization and rotation angle of a beam. Causing that vector to trace a trajectory over time on the surface of the sphere represents a polarization modulated (and rotation modulated) beam (Fig. lb). If, then, the beam is sampled by a device at a rate that is less than the rate of modulation, the sampled output from the device will be a condensation of two components of the wave, which are continuously changing with respect to each other, into one snapshot of the wave, at one location on the surface of the sphere and one instantaneous polarization and axis rotation. Thus, from the viewpoint of a device sampling at a rate less than the modulation rate, a two-to-one mapping (over time) has occurred, which is the signature of an SU(2) field. 

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