Coupling to the environment

Unlike in the case of the gas-phase measurements, no tunneling has been detected in the IR spectra of the malonaldehyde molecule in the noble matrices at 15-30K [Firth et al. 1989], The lack of tunneling is caused by detuning of the potential as a result of weak antisymmetric coupling to the environment. 

The first term on the right-hand side corresponds to Eq. (2), whereas the second term describes dissipative effects that are induced in the system due to its coupling to the environment. The latter is modeled, as usual [32, 33], as the thermal (temperature T) ensemble of harmonic oscillators, with nonlinear coupling A Qiq) F( thermal bath, expressed in terms of nonlinear molecular and linear environment coupling operators Q(q) and F( qk )- As shown in Ref. 15, it is important to describe the dissipative term in Eq. (10) by making use of the non-Markovian expression... 

In the 2-level limit a perturbative approach has been used in two famous problems the Marcus model in chemistry and the small polaron model in physics. Both models describe hopping of an electron that drags the polarization cloud that it is formed because of its electrostatic coupling to the environment. This environment is the solvent in the Marcus model and the crystal vibrations (phonons) in the small polaron problem. The details of the coupling and of the polarization are different in these problems, but the Hamiltonian formulation is very similar.15... 

A theoretical Langevin equation is generally constructed from a Hamiltonian model for a simple dynamical system coupled to the environment. The equations of motion for the coupled system are manipulated so as to eliminate the degrees... 

Note that the form of Eq. (8.148) expresses the possibility that the different degrees of freedom Xj are coupled to each other not only via their interaction potential V, but in principle also through their mutual coupling to the environment. 

The reaction rate of proton transfer in condensed phases depends on several parameters temperature, potential barrier height, transfer distance, reactant frequency, strength of coupling to the environment. For different values of these parameters, different physical mechanisms dominate which have been described by different theoretical models, in the chronological order they were studied, the parameter regions were ... 

For certain crystal cuts the motion is of the thickness-shear type. Since the motion at the crystal surface is then in the surface plane, these modes do not emit longitudinal sound (or at least not very much of it). The weak acoustic coupling to the environment increases the Q factor of the resonances to rather exceptional levels. The bandwidth is orders of magnitude smaller than the resonance frequency, which greatly simplifies the data analysis. 

The transpiration ratio is, as defined here, the ratio of T to P. A main problem in interpretation, however, is that T is tightly coupled to the environment through A. Hence, a graph of T versus A would be linear if the diffusion resistances are constant. Similarly, a graph of T P versus A would be linear and hence, the transpiration ratio (T P) will vary with the evaporative demand of the environment. 

In principle, there is a close analogy between QCM and surface plasmon resonance (SPR) spectroscopy. Both types of resonators couple to the environment by rapidly deca3dng waves, and the... 

After the system has been coupled to the environment for a sufficiently long time, it will, in general, settle down to a steady state. To describe this state, we have to find the stationary solution, i.e. 9 p = 0, of (19). It turns out that this is. 

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