Phenomenological relaxation rates

After transformation into the interaction picture and application of the rotating-wave approximation [46, SO, 54] the population dynamics can be calculated numerically by solving the time-dependent three-level Schrodinger equation or (if phenomenological relaxation rates are considered) by solving the density matrix equation (3) for the molecular system. The density matrix equation is given by... 

To describe the impact of the x-values distribution type on the relative precision of relaxation rate estimates, we shall use a phenomenological factor fd. We expect it to be independent of all the other factors, but dependent upon the type of relaxation rate quantity to be determined (for example, the fastest- or the slowest-relaxing component in a multi-component mixture). 

The electronic contribution to the relaxation rate of protons in TMQ has been arrived at after a careful sustrac-tion of the methyl group rotation contribution, Fig. 5. The pressure dependence of the room temperature T in TQ and TMQ is shown in Fig. 6. A log-log plot of ( TiTV,t versus reveals a phenomenological relation (T,T) %, with an... 

With the inclusion of phenomenological T, To in equations (2), and by setting (fi = 0, one is left with the standard Bloch Equations. In this notation " 2 no proper dephasing and the population relaxation rate T is twice the spontaneous emission rate. 

Fig. 17. Relaxation rate F (.=hlTf) ( ) of the fast process evaluated from a simple phenomenological model (Fig. 16) and Ae boson peak frequency ft) , (O) and the friction term y ( ) evaluated from the VR model (Fig. 19).(Reprinted with permission from [39]. Copyright 1998 American Institute of Physics, New York)...

The paper is organized in five sections. In Section 4.2 we describe the phenomenological theory for the heat release, based on the tunneling model and the influence of different relaxation rates, and the cooling process. In Section 4.3 we briefly describe the experimental procedures and samples. In Section 4.4 we show and discuss the experimental results. Conclusions are drawn in Section 4.5. 

Phillies, et a/. (69) present results confirming Streletzky and Phillies s(64) prior interpretation that HPC solutions have a dominant, concentration-independent characteristic dynamic length scale, namely the radius of a polymer chain, which for this species is i 50 nm. In particular (i) There are distinct small-probe and large-probe phenomenologies, with the division between small and large probes being about 50 nm, the same at all polymer concentrations, (ii) For small probes, the relative amplitude of the sharp and broad modes depends markedly on scattering vector q with a crossover near q 70 nm. (iii) The mean relaxation rate of the small-probe broad mode increases markedly near 50 nm. (iv) The probe intermedi-... 

From the analysis of the TR MFE curves, a number of distinctive conclusions can be made. First of all, when a high field relaxation rate is phenomenologically treated using a value of Ti = 220 ns, the decay in the TR MFE curve is not accelerated by... 

W. B. Davis, M. R. Wasidewski, M. A. Ratner, V. Mujica, A. Nitzan, Electron Transfer Rates in Bridged Molecular Systems - a Phenomenological Approach to Relaxation , J. Phys. Chem. 1997,101, 6158-6164. 

The plan of this chapter is the following. Section II gives a summary of the phenomenology of irreversible processes and set up the stage for the results of nonequilibrium statistical mechanics to follow. In Section III, it is explained that time asymmetry is compatible with microreversibility. In Section IV, the concept of Pollicott-Ruelle resonance is presented and shown to break the time-reversal symmetry in the statistical description of the time evolution of nonequilibrium relaxation toward the state of thermodynamic equilibrium. This concept is applied in Section V to the construction of the hydrodynamic modes of diffusion at the microscopic level of description in the phase space of Newton s equations. This framework allows us to derive ab initio entropy production as shown in Section VI. In Section VII, the concept of Pollicott-Ruelle resonance is also used to obtain the different transport coefficients, as well as the rates of various kinetic processes in the framework of the escape-rate theory. The time asymmetry in the dynamical randomness of nonequilibrium systems and the fluctuation theorem for the currents are presented in Section VIII. Conclusions and perspectives in biology are discussed in Section IX. 

In the present phenomenological model, only regions that have not yet equilibrated (i.e., of size s > s t)) can release stress energy in the form of a net amount of heat to the surroundings. This means that only transitions with AF < 0 contribute to the overall relaxation toward equihbrium. Therefore, the rate of energy dissipated by the system can be written as... 

Remarkably, when our general ME is applied to either AN or PN in Section 4.4, the resulting dynamically controlled relaxation or decoherence rates obey analogous formulae provided the corresponding density matrix (generalized Bloch) equations are written in the appropriate basis. This underscores the universality of our treatment. It allows us to present a PN treatment that does not describe noise phenomenologically, but rather dynamically, starting from the ubiquitous spin-boson Hamiltonian. 

In sharp contrast to the large number of experimental and computer simulation studies reported in literature, there have been relatively few analytical or model dependent studies on the dynamics of protein hydration layer. A simple phenomenological model, proposed earlier by Nandi and Bagchi [4] explains the observed slow relaxation in the hydration layer in terms of a dynamic equilibrium between the bound and the free states of water molecules within the layer. The slow time scale is the inverse of the rate of bound to free transition. In this model, the transition between the free and bound states occurs by rotation. Recently Mukherjee and Bagchi [14] have numerically solved the space dependent reaction-diffusion model to obtain the probability distribution and the time dependent mean-square displacement (MSD). The model predicts a transition from sub-diffusive to super-diffusive translational behaviour, before it attains a diffusive nature in the long time. However, a microscopic theory of hydration layer dynamics is yet to be fully developed. 

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