Weakly coupled system

MSN.12. R. Brout and I. Prigogine, Statistical mechanics of irreversible processes. Part VII General theory of weakly coupled systems, Physica 22, 621-636 (1956). 

MSN. 19.1. Prigogine and R. Brout, Irreversible processes in weakly coupled systems. Proceedings International Symposium on Transport Processes in Statistical Mechanics, part 11, Brussels, 1958, Interscience Publishers, New York, 1958, p. 25-32. 

Fig. 5 Tunnel and lattice mode frequencies as a function of temperature with parameters 7/ 2 = 18.2, g2/g4 = -0.8 and = 0.46. Solid lines refer to the uncoupled system (C = 0), dashed linesto the weakly coupled system (C = 1 x 10 K/A) and dash-dotted line to the strongly coupled system (C = 5.7 x lO K/A)...

Therefore we may eliminate the correlations in many ways. A different definition of the physical quantity 0R corresponds to each %. Equation (51) already looks like the average, Eq. (47), taken in a weakly coupled system with p0 replaced by p and O by 0R. There still remains a difference. We have as yet no relation between p and the entropy. If we could find a p such that the quantity of Boltzmann would be given by... 

Weakly coupled systems of nonlinear elliptic boundary value problems (with K. Zygour-akis). Nonlin. Anal. 6, 555-569 (1982). 

The frequencies assigned to the individual coherences are different for any two conformers present in dynamic spin systems. The product functions, that is, the states the terms were assigned to, are identical. The single spin vector model of the previous section can be extended to weakly coupled systems by changing the precession frequencies of the single quantum coherences during the exchanges of the nuclei instead of... 

For a weakly coupled system of JV nuclei of spin /j, we can easily obtain explicit expressions for the exponential operators in Eq. 11.27 by applying Eq. 11.25 and the expansion ofEq. 11.20 ... 

Evolution based on spin coupling is indicated by the appropriate operator and time. Because the product operator treatment applies solely to weakly coupled systems, the operator is JisIzSz, with only the ar components considered. As we saw in Section 9.6, we must consider both in-phase and antiphase components of a magnetization (or coherence) as it precesses in the xy plane. In line with Eq. 9.4, we can now use the product operator terminology to note that, for example,... 

Several studies of bimetallic complexes in which the donor and acceptor are linked across aliphatic chains have demonstrated that these are generally weakly coupled systems. " Studies of complexes linked by l,2-bis(2,2 bipyridyl-4-yl)ethane (bb see Figure 5), indicate that these are good models of the precursor complexes for outer-sphere electron-transfer reactions of tris-bipyridyl complexes. A careful comparison of kinetic and spectroscopic data with computational studies has led to an estimate of //rp = 20cm for the [Fe(bb)3pe] + self-exchange electron transfer. In a related cross-reaction, the Ru/bpy MLCT excited state of [(bpy)2Ru(bb)Co(bpy)2] + is efficiently quenched by electron transfer to the cobalt center in several resolved steps, equations (57) and (58). ... 

The interaction between the atoms is represented by a time-dependent potential W. This leads to a time-dependent Schrodinger equation describing the entire weakly coupled system ... 

In the linked systems studied by Wegewis and Verhoeven [14], electron-transfer rates were estimated by the same methods—quenching of the acceptor fluorescence and linewidth measurements. The rates were found to vary over many orders of magnitude in the weakly coupled systems they are 10 -10 s (for the reduced forms lr-3r), increasing to lO" s for the double bond conjugated forms 1-3, and... 

One of the important features of Arnold diffusion is that it is supposed to be quite slow. This is shown in the original estimate of Arnold [1], Since Arnold diffusion is an essential process for relaxation in nearly integrable (i.e., weakly coupled) systems with many degrees of freedom, most relaxations can proceed in this very slow time scale. The slow relaxation is acceptable when the system we are concerned is the solar system or some other celestial object. But, does Arnold diffusion really occur in molecular systems in the course of relaxation or reaction Do the systems really show slow relaxation or not What is the phase-space structure and how does the system move in there Answers to all these questions are still unknown. 

Results and Predictions. Detailed close coupling calculations for "real" Av<0 vibrational predissociation of weak-coupling systems such as the hydrogen-inert gas complexes are more difficult and computationally more expensive than those for predissociation by internal rotation. The computational expense arises simply from the very large increase in the nvmber of channels which must be included in order to obtain converged results. The difficulty, on the other hand, arises from the fact that these resonances have very small widths, usually 10 cm , %jhich makes them very difficult to find. 

Table IV. Adlabatlcally quantized semlclasslcal results for energies of doubly excited states of HOD, In a two degrees of freedom model, as obtained by Skodje and Reinhardt (27) using the technique of Refs. (24), (25). All of these states are above the classical dissociation threshold. AE Is the uncertainty In the adiabatic quantization, as Illustrated In Fig. 6. Eo Is the "unperturbed energy, and E the result of the quantization procedure In this weakly coupled system.

In this paper a weakly coupled system of metallic chains will be investigated. The model we use was introduced by Gorkov and Dzyaloshinskii (1) and also investigated by Mihdly and Solyom (2 ) with the result that for attractive interactions a charge density wave (CDW) type phase transition occurs in the system at a finite temperature. Our aim is to study the effect of the ID fluctuations on the phase transition. It is found that the one dimensional fluctuations can strongly increase the critical temperature of the 3D phase transition provided the character of the ID fluctuations matches that of the 3D phase transition (i. e. the same type of response functions are enhanced and singular, respectively). 

Sampling the polarizations created by the CIDNP effect with a continuous wave NMR spectrometer is straightforward because with that detection scheme the line intensity of a transition L(r5) between nuclear spin levels r and s is proportional to the population difference (Pr — Ps) of these levels only. However, with pulsed detection, as is now universally employed, L(rs) depends on the population differences across all parallel transitions between levels t and u including the one between r and s, and also in a complicated manner on the tip angle 9 of the observation pulse. For a weakly coupled system of N spins, one has [37]... 

In the weak coupling case, where the excitation energy is localized, a different treatment is needed. A generalized formula for the excitation energy transfer of weakly coupled systems was derived by Forster. He wrote equations describing the rate of a transition from a localized excited state, ipa M, to another localized excited state, VWi- When the distance between the two molecules, Rab, is not too short, the resonance integral can be approximated by the interaction energy between the transition dipole moments, Pa and Pb,... 

For many of the model molecules studied by the trajectory simulations, the decay of P t) was exponential with a decay constant equal to the RRKM rate constant. However, for some models with widely disparate vibrational frequencies and/or masses, decay was either nonexponential or exponential with a decay constant larger than k E) determined from the intercept of P(f). This behavior occurs when some of the molecule s vibrational states are inaccessible or only weakly coupled. Thus, a micro-canonical ensemble is not maintained during the molecule s decomposition. These studies were a harbinger for what is known now regarding inelficient intramolecular vibrational energy redistribution (IVR) in weakly coupled systems such as van der Waals molecules and mode-specific unimolecular dynamics. 

The free induction decay acquired is thus the second half of a spin echo, the amplitude of which is modulated as a function of tj by scalar couplings. In weakly coupled systems, then, the effect of the 180 pulse is to suppress the effects of chemical shifts (and indeed... 

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