Stochastic phases

We adopt a modified-phase diffusion model for the partially coherent laser source in which the phase SA is allowed to be time dependent and random. Thus, the molecule-laser interaction is modeled by the interaction with an ensemble of lasers, each of different phase. This ensemble is described by a Gaussian correlation for the stochastic phases with a decorrelation time scale rxc ... 

The effect of laser phase fluctuations on PIER4 has been considered by Agarwal, and detailed line shapes were calculated. In a recent publication we have derived (within the phase diffusion model) equations of motion in the limit of short correlation times. The effect of the stochastic phase fluctuations was shown to be similar to T2 dephasing processes, and a procedure was given for the inclusion of this similarity in many nonlinear processes. In particular, two predictions were made ... 

For a laser with a fluctuating phase, the simulation procedure is similar to that of the collision. Several models may be assumed for the stochastic phase fluctuations. In this calculation we have assumed a random jump between 0 and 2ir with no memory of the previous phase. This is a model of Markovian "hard" phase jumps, which is different than the phase diffusion model assumed in our analytic work. This model may be applicable for actual lasers suffering from acoustic mirror jitter or other mundane laboratory noises. 

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

A second type of relaxation mechanism, the spin-spm relaxation, will cause a decay of the phase coherence of the spin motion introduced by the coherent excitation of tire spins by the MW radiation. The mechanism involves slight perturbations of the Lannor frequency by stochastically fluctuating magnetic dipoles, for example those arising from nearby magnetic nuclei. Due to the randomization of spin directions and the concomitant loss of phase coherence, the spin system approaches a state of maximum entropy. The spin-spin relaxation disturbing the phase coherence is characterized by T. ... 

The simplest scheme that accounts for the destruction of phase coherence is the so-called stochastic interruption model [Nikitin and Korst 1965 Simonius 1978 Silbey and Harris 1989]. Suppose the process of free tunneling is interrupted by a sequence of collisions separated by time periods vo = to do After each collision the system forgets its initial phase, i.e., the off-diagonal matrix elements of the density matrix p go to zero, resulting in the density matrix p ... 

Huller and Baetz [1988] have undertaken a numerical study of the role played by shaking vibrations. The vibration was supposed to change the phase of the rotational potential V (p — a(t)). The phase a(t) was a stochastic classical variable subject to the Langevin equation... 

Fig. 59. Time dependence of phases /1a and for a realization of stochastic force at T = 7 c- Also shown are the straight lines of the zero-temperature behavior of /I (solid line) and A (dashed line). Time is measured in units 2I/h.

Before trying to solve the master equation for growth processes by direct stochastic simulation it is usually advisable to first try some analytical approximation. The mean-field approximation often gives very good results for questions of first-order phase transitions, and at least it provides a qualitative understanding for the interplay of the various model parameters. 

It is known that the interaction of the reactants with the medium plays an important role in the processes occurring in the condensed phase. This interaction may be separated into two parts (1) the interaction with the degrees of freedom of the medium which, together with the intramolecular degrees of freedom, represent the reactive modes of the system, and (2) the interaction between the reactive and nonreactive modes. The latter play the role of the thermal bath. The interaction with the thermal bath leads to the relaxation of the energy in the reaction system. Furthermore, as a result of this interaction, the motion along the reactive modes is a complicated function of time and, on average, has stochastic character. 

The brief review of the newest results in the theory of elementary chemical processes in the condensed phase given in this chapter shows that great progress has been achieved in this field during recent years, concerning the description of both the interaction of electrons with the polar medium and with the intramolecular vibrations and the interaction of the intramolecular vibrations and other reactive modes with each other and with the dissipative subsystem (thermal bath). The rapid development of the theory of the adiabatic reactions of the transfer of heavy particles with due account of the fluctuational character of the motion of the medium in the framework of both dynamic and stochastic approaches should be mentioned. The stochastic approach is described only briefly in this chapter. The number of papers in this field is so great that their detailed review would require a separate article. 

Figure 8.40. Computer-simulated IDFs gi (u) of ID two-phase structure formed by the iterated stochastic structure formation process. tt is the thickness of the transition layer at the phase boundary. o> is the standard deviation of a Gaussian crystallite thickness distribution... 

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