FIELDS AND INTERNAL DEGREES OF FREEDOM

As a consequence of the collective motion of the neutral system across the homogeneous magnetic field, a motional Stark term with a constant electric field arises. This Stark term inherently couples the center of mass and internal degrees of freedom and hence any change of the internal dynamics leaves its fingerprints on the dynamics of the center of mass. In particular the transition from regularity to chaos in the classical dynamics of the internal motion is accompanied in the center of mass motion by a transition from bounded oscillations to an unbounded diffusional motion. Since these observations are based on classical dynamics, it is a priori not clear whether the observed classical diffusion will survive quantization. From both the theoretical as well as experimental point of view a challenging question is therefore whether quantum interference effects will lead to a suppression of the diffusional motion, i.e. to quantum localization, or not. 

The bond additivity approximation (BAA) appears to work for polymers dissolved in isotropically polarizable nonpolar solvents. However in the gas phase, BAA has been shown to be incorrect by Ward and coworkers (11). It has been speculated that the solvent provides a symmetrical environment in which local electric fields at a given bond caused by adjoining bonds, are cancelled by fields due to solvent molecules. Thus assuming the correctness of the RIS and BAA models, the configurational average over all internal degrees of freedom r is given by... 

Here D, D , and Dr are, respectively, the longitudinal, transverse, and rotational diffusion coefficients of the chain averaged over the internal degree of freedom, h an external field, and v and angular velocity of the chain induced by a flow field in the solution. Furthermore, I is the unit tensor and 91 is the rotational operator defined by... 

This means Eq. (4.38) is the interaction between the atoms in dipole approximation. Thus, (4.37) turns out to be the average potential of a single atom in the field of the other atoms. In this sense, Eqs. (4.34) and (4.36) represent a system for the description of composites (atoms with internal degrees of freedom) and their interaction. 

Equations (4.329) for a solid assembly and (4.332) for a magnetic suspension are solved by expanding W with respect to the appropriate sets of functions. Convenient as such are the spherical harmonics defined by Eq. (4.318). In this context, the internal spherical harmonics used for solving Eq. (4.329) are written Xf (e, n). In the case of a magnetic fluid on this basis, a set of external harmonics is added, which are built on the angles of e with h as the polar axis. Application of a field couples [see the kinetic equation (4.332)] the internal and external degrees of freedom of the particle so that the dynamic variables become inseparable. With regard to this fact, the solution of equation (4.332) is constructed in the functional space that is a direct product of the internal and external harmonics ... 

In an exposition which aims to encompass general systems and ensembles, it is appropriate to make use of the Hamiltonian version of dynamics. In this view forces do not appear explicitly and the dynamics of the system evolve so as to keep the Hamiltonian function constant. In Newtonian dynamics forces appear explicitly and molecules move as a response to the forces they experience. For our purposes, the Newtonian view is sufficient since we will illustrate the large scale computational aspects with simplest possible particles, atoms with spherical, central force fields. The same principles hold for molecules with internal degrees of freedom as well. 

We thus conclude the section on the numerical implementation of SLLOD dynamics for two very important and useful ensembles. However, our work is not yet complete. The use of periodic boundary conditions in the presence of a shear field must be reconsidered. This is explained in detail in the next section. Furthermore, one could imagine a situation in which SLLOD dynamics is executed in conjunction with constraint algorithms for the internal degrees of freedom and electrostatic interactions. An immediate application of this extension would be the simulation of polar fluids (e.g., water) under shear. This extension has been performed, and the integrator is discussed in detail in Ref. 42. 

In Eq. (177) in the presence of nonzero external fields, the spur operation over the internal degrees of freedom and the integration over the angles cannot be performed separately, since p depends on the angles fl and (l0 and is simultaneously the matrix in the basis of the vibronic states. The analysis of the density matrix p is given in the next section. 

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