Weak-coupling limit

Moreover, in this linear-response (weak-coupling) limit any reservoir may be thought of as an infinite number of oscillators qj with an appropriately chosen spectral density, each coupled linearly in qj to the particle coordinates. The coordinates qj may not have a direct physical sense they may be just unobservable variables whose role is to provide the correct response properties of the reservoir. In a chemical reaction the role of a particle is played by the reaction complex, which itself includes many degrees of freedom. Therefore the separation of reservoir and particle does not suffice to make the problem manageable, and a subsequent reduction of the internal degrees of freedom in the reaction complex is required. The possible ways to arrive at such a reduction are summarized in table 1. 

The equation of motion for the expectation < in the weak-coupling limit has a Langevin-like form... 

Our conclusions about the case of large /tls have a rather speculative character, and pursue merely an illustrative goal, since (2.41) and (2.42) are obtained in the weak-coupling limit. 

In Ref. [4], the soliton lattice configuration and energy within the SSH model were found numerically. Analytical expressions for these quantities can be obtained in the weak-coupling limit, when the gap 2A() is much smaller than the width of the re-electron band 4/0. At this point it is useful to define the lattice correlation length ... 

In the weak-coupling limit unit cell a (, 0 7a for fra/u-polyacetylene) and the Peierls gap has a strong effect only on the electron states close to the Fermi energy eF-0, i.e., stales with wave vectors close to . The interaction of these electronic states with the lattice may then be described by a continuum, model [5, 6]. In this description, the electron Hamiltonian (Eq. (3.3)) takes the form ... 

On the other hand, after the phase transition, in the weak coupling limit (A quantum decoherence and classical correlation are given approximately by... 

Near the critical point coefficients of (1) can be expanded in t = (T — Tc)/Tc. In the weak coupling limit they render ... 

Fluctuations dominate for T > For typical values fiq (350-F500) MeV and for Tc > (50 A- 70) MeV in the weak coupling limit from (26), (22) we estimate Tq< (0.6 A- 0.8)TC. If we took into account the suppression factor / of the mean field term oc e A /T, a decrease of the mass m due to the fluctuation contribution (cf. (11)), and the pseudo-Goldstone contribution (25), we would get still smaller value of T < (< 0.5TC). We see that fluctuations start to contribute at temperatures when one can still use approximate expressions (22), (20) valid in the low temperature limit. Thus the time (frequency) dependence of the fluctuating fields is important in case of CSC. 

The Landau-Zener expression is calculated in a time-dependent semiclassical manner from the diabatic surfaces (those depicted in Fig. 1) exactly because these surfaces, which describe the failure to react, are the appropriate zeroth order description for the long-range electron transfer case. As can be seen, in the very weak coupling limit (small A) the k l factor and hence the electron transfer rate constant become proportional to the absolute square of A ... 

To keep the calculation tractable, we will only consider the weak-coupling limit Ti,T2 t), in this case the advanced (and retarded) Green functions do not depend on the Ts, and for the model defined by equation (6) they are given as... 

No divergences and dependence on the contact parameters Ti 2 remain in the form for r. It shows the transmittance function (at least in the weak-coupling limit) is indeed a well-defined molecular quantity. We can rewrite equation (38), taking into account the definition of 6 (see equation (35)) and the definition of the Chebyshev polynomials of the second kind U (cos 6) — sin[(n +l)0]/sin 6 as... 

In the weak coupling limit with Eq. (32), this reduces to the well-known kinetic equation for the average number of the excited particle obtained by the X t approximation. 

In the weak coupling limit [160, 163] the transfer rate constant is given by... 

The resulting expression is especially simple in the weak coupling case. In this case, the two propagators in Eq. (12) can be approximated by their first order (i.e, single hop) terms. (The zeroth order term makes no contribution of Kjf as long as i 5 f) In this weak coupling limit, the expression for Pif (t) can be expressed as ... 

In the weak coupling limit, the effect of ZFS must be accounted for in the effective Hamiltonian ... 

Analyzing thermodynamic data and phonon densities of states Hilscher and Michor (1999) concluded that for 1 2 2 1 borocarbides the BCS weak-coupling limit is not... 

From the point of view of thermodynamics we have now a microscopic model of entropy (see Eq. (52)). Therefore, we can verify that it leads to the basic expressions of thermodynamics of irreversible processes in the neighborhood of equilibrium.29 These expressions were derived until recently in the weakly coupled limit, or for dilute gases. 

The process has been treated theoretically in terms of simplified models.14 58 The quantum mechanics is one of formulating the probability of crossing from an excited to a ground state, summed over all vibrational levels. For coordination compounds, the weak coupling limit is presumably the important approximation. Here, the transition is from low lying vibrational levels of the excited state to very high vibrational levels of the ground state. 

For the case of typical ionic crystals aP 1-10, and the weak coupling limit is applicable. The most important conclusion from this treatment is that the weak coupling limit leads to a perturbed Bloch type wave function characterized by equal probability for finding the electron at any point of the medium. Thus, in the case of the ionic crystals, the current description of the polaron is that of a mobile electron followed by lattice polarization. 

In the weak coupling limit, as is the case for most molecular systems, each molecule can be treated as an independent source of nonrlinear optical effects. Then the macroscopic susceptibilities X are derived from the microscopic nonlinearities 3 and Y by simple orientationally-averaged site sums using appropriate local field correction factors which relate the applied field to the local field at the molecular site. Therefore (1,3)... 

This model, called the spin-boson Hamiltonian, is probably the only problem (except maybe for some very artificial ones) whose full solution can be obtained without any additional approximations. The equation of motion for the expectation value (crz) in the weak coupling limit has a... 

Our conclusions about the case for large pTLS are speculative in nature, and are meant to be merely illustrative, because (2.43) and (2.44) are obtained only in the weak coupling limit. 

To sum up, we have developed a general non-perturbative method that allows one to calculate the rate of relaxation processes in conditions when perturbation theory is not applicable. Theories describing non-radiative electronic transitions and multiphonon relaxation of a local mode, caused by a high-order anharmonic interaction have been developed on the basis of this method. In the weak coupling limit the obtained results agree with the predictions of the standard perturbation theory. 

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