Phase fluctuations stochastic

In many physical systems the situation becomes simpler due to the inherent stochastic nature of the driving field itself. To see the possible significance of this effect, consider a conventional COj-laser pulse with 10-ns duration and a bandwidth of 1 cm incident on a diatomic molecule characterized by an environment-induced energy relaxation time of 100 ns. The laster pulse is obviously not uncertainty limited, and its width is associated with the random fluctuations in its phase and/or amplitude. For simplicity we consider random phase fluctuations, whence the external field is... 

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 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. 

The present investigation applies deterministic methods of continuous mechanics of multiphase flows to determine the mean values of parameters of the gaseous phase. It also applies stochastic methods to describe the evolution of polydispersed particles and fluctuations of parameters [4]. Thus the influence of chaotic pulsations on the rate of energy release and mean values of flow parameters can be estimated. The transport of kinetic energy of turbulent pulsations obeys the deterministic laws. 

To model the experimental data we used a global-fit procedure to simulate EPS, integrated TG, heterodyne-detected TG, and the linear absorption spectrum simultaneously. The pulse shape and phase were explicitly taken into account, which is of paramount importance for the adequate description of the experimental data. We applied a stochastic modulation model with a bi-exponential frequency fluctuation correlation function of the following form ... 

Thermal fluctuations cause a group of guest (CO2) molecules to be arranged in a configuration similar to that in the clathrate hydrate phase. The structure of water molecules around locally ordered guest molecules is perturbed compared to that in the bulk. The thermodynamic perturbation of the liquid phase is due to the finite temperature of the system. This process is stochastic. 

Let us now consider stochastic motion in an OB system. In general, noise in an OB system may result from fluctuations of the incident field, or from thermal and quantum fluctuations in the system itself. We shall consider the former. The fluctuations of the intensities of the input or reference signals give rise respectively to either multiplicative or additive noise driving the phase. Both types of fluctuations can be considered within the same approach [108], Here we discuss only the effects of zero-mean white Gaussian noise in the reference signal ... 

A rep < 1, Des < 1, the nucleation dynamics is stochastic in nature as a critical fluctuation in one, or more, order parameters is required for the development of a nucleus. For DeYep > 1, Des < 1 the chains become more uniformly oriented in the flow direction but the conformation remains unaffected. Hence a thermally activated fluctuation in the conformation can be sufficient for the development of a nucleus. For a number of polymers, for example PET and PEEK, the Kuhn length is larger than the distance between two entanglements. For this class of polymers, the nucleation dynamics is very similar to the phase transition observed in liquid crystalline polymers under quiescent [8], and flow conditions [21]. In fast flows, Derep > 1, Des > 1, A > A (T), one reaches the condition where the chains are fully oriented and the chain conformation becomes similar to that of the crystalline state. Critical fluctuations in the orientation and conformation of the chain are therefore no longer needed, as these requirements are fulfilled, in a more deterministic manner, by the applied flow field. Hence, an increase of the parameters Deiep, Des and A results into a shift of the nucleation dynamics from a stochastic to a more deterministic process, resulting into an increase of the nucleation rate. 

Figure 11.8 An example of the stochastic trajectory from Monte Carlo simulation according to the Gillespie algorithm for reaction system given in Equation (11.19) and corresponding master equation graph given in Figure 11.4. Here we set Ns = 100 and Nes = 0 at time zero and total enzyme number Ne = 10. (A) The fluctuating numbers of S and ES molecules as functions of time. (B) The stochastic trajectory in the phase space of (m, n).

An explanation of the enhancement and other anomalies of the catalytic reaction rates on metals and certain insulators associated with the large fluctuations of the internal degrees of freedom that occur near a phase transition or by alloying has been attempted by d Agliano, Schaich, Kumar, and Suhl within the framework of stochastic theories. 

Furthermore, it has recently been found that the discrete nature of a molecule population leads to qualitatively different behavior than in the continuum case in a simple autocatalytic reaction network [29]. In a simple autocatalytic reaction system with a small number of molecules, a novel steady state is found when the number of molecules is small, which is not described by a continuum rate equation of chemical concentrations. This novel state is first found by stochastic particle simulations. The mechanism is now understood in terms of fluctuation and discreteness in molecular numbers. Indeed, some state with extinction of specific molecule species shows a qualitatively different behavior from that with very low concentration of the molecule. This difference leads to a transition to a novel state, referred to as discreteness-induced transition. This phase transition appears by decreasing the system size or flow to the system, and it is analyzed from the stochastic process, where a single-molecule switch changes the distributions of molecules drastically. 

Can chaotic systems be differentiated from random fluctuations Yes, even though the dynamics are complex and resemble a stochastic system, they can be differentiated from a truly stochastic system. Figure 11.15 compares the plots of N = 10,001 and a selection of points chosen randomly from 13,000 to 0. Note that after approximately 10 time intervals, the dynamics of both are quite wild and it would be difficult to distinguish one from another as far as one is deterministic and the other chaotic. However, there is a simple way to differentiate these two alternatives the phase-space plot. 

The internal spin interaction Hamiltonian Hmt can be decomposed into spatial Tm[ ua(l) ] and spin Sm degrees of freedom Hin (t) = 2mTm[ ua(t) ]Sm. The spatial contribution, hereafter an NMR interaction rank-2 tensor T, is a stochastic function of time Tm[ ua(t) because it depends on generalized coordinates < ( ) of the system (atomic and molecular positions, electronic or ionic charge density, etc.) that are themselves stochastic variables. To clarify the role of these coordinates in the NMR features, a simple model is developed below.19,20 At least one physical quantity should distinguish the parent and the descendant phase after a phase transition. For simplicity, we suppose that the components of the interaction tensor only depend on one scalar variable u(t) whose averaged value is modified from m to m + ( at a phase transition. To take into account the time fluctuations, this variable is written as the sum of three terms, i.e. u(t) — m I I 8us(t). The last term is a stationary stochastic process such that — 0, where <.) denotes a... 

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