Einstein frequency dependence

Note that the above expression is known as the generalized Einstein equation and that the memory function, ((z), is the frequency-dependent friction. 

The time-dependent VACF is obtained by numerically Laplace inverting the frequency-dependent VACF, which is related to the frequency-dependent friction through the following generalized Einstein relation given by Eq. (81) ... 

The frequency (z)-dependent velocity correlation function Cv(z) is related to the frequency dependent friction by the well-known generalized Einstein relation,... 

As displayed by the out-of-equilibrium generalized Stokes-Einstein relation (203), independent measurements of the particle mean-square displacement and frequency-dependent mobility in an aging medium give access, once Ax2(z) and p(z) = p(co = iz) are determined, to T (z) and to T (co) = T (z = — iffi). Then, the identity (189) yields the effective temperature ... 

P+ and P are the probabilities for absorption and emission, respectively B+ and B are the coefficients of absorption and of induced emission, respectively A- is the coefficient of spontaneous emission and p v) is the density of radiation at the frequency that induces the transition. Einstein showed that B+ = B, while A frequency dependence, spontaneous emission (fluorescence), which usually dominates in the visible region of the spectrum, is an extremely improbable process in the rf region and may be disregarded. Thus the net probability of absorption of rf energy, which is proportional to the strength of the NMR signal, is... 

Einstein distribution function, r is the frequency-dependent phonon relaxation time, v is the frequency-dependent phonon group velocity and s is the unit direction vector. Next, we discuss several approaches to solve Eq. (2.1). 

For a large particle in a fluid at liquid densities, there are collective hydro-dynamic contributions to the solvent viscosity r, such that the Stokes-Einstein friction at zero frequency is In Section III.E the model is extended to yield the frequency-dependent friction. At high bath densities the model gives the results in terms of the force power spectrum of two and three center interactions and the frequency-dependent flux across the transition state, and at low bath densities the binary collisional friction discussed in Section III C and D is recovered. However, at sufficiently high frequencies, the binary collisional friction term is recovered. In Section III G the mass dependence of diffusion is studied, and the encounter theory at high density exhibits the weak mass dependence. 

In the previous sections a model of the frequency-dependent collisional friction has been derived. Because the zero-frequency friction for a spherical particle in a dense fluid is well modeled by the Stokes-Einstein result, even for particles of similar size as the bath particles, there has been considerable interest in generalizing the hydrodynamic approach used to derive this result into the frequency domain in order to derive a frequency-dependent friction that takes into account collective bath motions. The theory of Zwanzig and Bixon, corrected by Metiu, Oxtoby, and Freed, has been invoked to explain deviation from the Kramers theory for unimolec-ular chemical reactions. The hydrodynamic friction can be used as input in the Grote-Hynes theory [Eq. (2.35)] to determine the reactive frequency and hence the barrier crossing rate of the molecular reaction. However, the use of sharp boundary conditions leads to an unphysical nonzero high-frequency limit to Ib(s). which compromises its utility. 

To make the elements of this model more concrete, we consider the way in which the transition temperature depends upon the difference in the Einstein frequencies of the two structural competitors. To do so, it is convenient to choose the variables a> = (cob + coa)/2 and Aco = cob — coa) 2, where u>a and cob are the Einstein frequencies of the two structural competitors. If we now further measure the difference in the two frequencies, Aco, in units of the mean frequency m according to the relation Aco = fco, then the difference in free energy between the two phases may be written as... 

Fig. 6.15. Dependence of the critical temperature, here reported as /3c = 1 /kTc, on the difference in Einstein frequencies of the competing phases.

Figure 5 For H20 (a) wideband absorption frequency dependence (b) total loss spectrum (c) partial absorption contributions (1-4) (d) partial loss contributions (1-4) (e) is similar to (c) but for a reduced, domain, and on a linear ordinate scale (f) is similar to (d) but on a reduced ordinate scale, (g) comprises the partial contributions to the Raman spectra in the R (v) representation (h) in the Bose-Einstein representation.

The frequency dependence of the gain coefficient a(v) is related to the line profile g(y — vq) of the amplifying transition. Without saturation effects (i.e., jfor small intensities), a(v) directly reflects this line shape, for homogeneous as well as for inhomogeneous profiles. According to (2.83) we obtain with the Einstein coefficienct Bik... 

In Table 8.4 we have collected values of Cl quadrupole coupling constants and correlation times for a number of protein-chloride complexes. In a few cases, has been obtained experimentally from a comparison of T and T2 or from the frequency dependence of relaxation. In order to make possible a comparison between different proteins we have also calculated from experimental line width data using correlation times estimated by means of the Debye-Stokes-Einstein relation for spherical molecules. We then proceeded similarly as described in Ref. [4Z1],... 

In the Smoluchowski limit, one usually assumes that the Stokes-Einstein relation (Dq//r7)a = C holds, which fonns the basis of taking the solvent viscosity as a measure for the zero-frequency friction coefficient appearing in Kramers expressions. Here C is a constant whose exact value depends on the type of boundary conditions used in deriving Stokes law. It follows that the diffiision coefficient ratio is given by ... 

Even though the Einstein and Debye models are not exact, these simple one-parameter models illustrate the properties of crystals and should give reliable estimates of the volume dependence of the vibrational entropy [15]. The entropy is given by the characteristic vibrational frequency and is thus related to some kind of mean interatomic distance or simpler, the volume of a compound. If the unit cell volume is expanded, the average interatomic distance becomes larger and the... 

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