Self-consistent Einstein model

In Sect.5,4.2, we demonstrate the basic physical principles with the help of the diatomic molecule. This is done in a rather intuitive way without any detailed derivations. In Sect.5.4.3, we discuss the SCHA for a Bravais crystal and in Sect.5.4.4, we treat the self-consistent Einstein model. 

Einstein reportedly said What really interests me is whether God had any choice in the creation of the world . What he meant by this informal remark was whether the physical universe must necessarily exist as it is or whether it could have been otherwise (or could have not existed at all). Today, almost all scientists believe that the universe could indeed have been otherwise no logical reason exists why it has to be as it is. In fact, it is the job of the experimental scientist to determine which universe actually exists, from among the many universes that might possibly exist. And it is the job of the theoretician to construct altemative models of physical reality, perhaps to simplify or isolate a particular feature of interest. To be credible, these models must be mathematically and logically self-consistent. In other words, they represent possible worlds. 

Clearly, a general theory able to naturally include other solvent modes in order to simulate a dissipative solute dynamics is still lacking. Our aim is not so ambitious, and we believe that an effective working theory, based on a self-consistent set of hypotheses of microscopic nature is still far off. Nevertheless, a mesoscopic approach in which one is not limited to the one-body model, can be very fruitful in providing a fairly accurate description of the experimental data, provided that a clever choice of the reduced set of coordinates is made, and careful analytical and computational treatments of the improved model are attained. In this paper, it is our purpose to consider a description of rotational relaxation in the formal context of a many-body Fokker-Planck-Kramers equation (MFPKE). We shall devote Section I to the analysis of the formal properties of multivariate FPK operators, with particular emphasis on systematic procedures to eliminate the non-essential parts of the collective modes in order to obtain manageable models. Detailed computation of correlation functions is reserved for Section II. A preliminary account of our approach has recently been presented in two Letters which address the specific questions of (1) the Hubbard-Einstein relation in a mesoscopic context [39] and (2) bifurcations in the rotational relaxation of viscous liquids [40]. 

The characteristic frequency, previously determined ( 2.6.4) as 34 cm is here found to be 38 cm, whilst the calculated effective mass of the bifluoride is 44 amu, close to the molecular mass of 39 amu. The two calculations are clearly self-consistent, as they must be since the phonon wings are simply the start of the molecular recoil in the lattice. However, the extreme naivety of the Einstein model is not usually successful at modelling the lattice dynamics of even simple systems. 

Dynamics. There are calculations in which the metal is modeled as an Einstein solid with harmonic vibrations[33j. When surface molecules and ions are strongly adsorbed molecular dynamics becomes an inefEcient way to study surface processes due to the slow exchange between surface and solution. In this case it is possible to use umbrella sampling to compute distribution profiles[34, 35]. Recently the idea underlying Car-Parrinello was used for macroion dynamics[36, 37] in which the solvent surrounding charged macroions is treated as a continuum in a self consistent scheme for the potential controlling ion dynamics. Dynamical corrections from the solvent can be added. There is a need to develop statistical methods to treat the dynamics of complex objects that evolve on several different time scales. 

In order to illustrate the SCHA, we consider the simplest possible model, the self-consistent isotropic Einstein model [5.461. In the Einstein model, each atom is bound by a spring with force constant (j> to its equilibrium position. Starting from the general relation (3.12), we disregard all coupling coefficients other than the diagonal term (00) write for the force acting on the atom 0 in the direction a... 

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