Fluctuation-dominated motion

Case d) Fluctuation Dominated Motion. An initial distribution P x 0) concentrated at the potential minima x and x+ neighbouring the unstable point Xq will now be considered. In this case a slow equilibration process will take place by means of a probability flux between the modes until the stationary distribution Pst ( ) is established. Since the drift force K (jc) is directed towards the minima X- and x+, it cannot be primarily responsible for this process. In fact the motion... 

Case c) Fluctuation Initiated Motion. If motion is started from a distribution concentrated around an unstable stationary point Xq where K(xo) = 0 and K (xq) = y > 0, the expansion of (2.82, 84,85) fails since the initial fluctuations are enhanced exponentially, see (2.92) before the drift dominated motion sets in. The appropriate approximation here, developed by Haake [2.2, 3] and by Suzuki [2.4] essentially consists of two steps 1) Solve the Fokker-Planck equation for the first fluctuation dominated stage and 2) Find a smoothly fitting solution for the second drift dominated stage. 

The reason for this inconsistency is that long-range fluctuations dominate behavior near any spinodal limit, including a consolute point. When fluctuations of concentration and of fluid velocity couple, diffusion occurs. Under ordinary conditions, the concentration fluctuations are dominated by motion of single molecules, but near the critical point, these fluctuations exist even when the average fluid velocity is zero. The result is like a turbulent dispersion coefficient but without flow. 

The influence of solvent can be incorporated in an implicit fashion to yield so-called langevin modes. Although NMA has been applied to allosteric proteins previously, the predictive power of normal mode analysis is intrinsically limited to the regime of fast structural fluctuations. Slow conformational transitions are dominantly found in the regime of anharmonic protein motion. 

For liquids, the dominant relaxation mechanism is the nuclear-nuclear dipole interaction, in which simple motion of one nucleus with respect to the other is the most common source of relaxation [12, 27]. In the gas phase, however, the physical mechanism of relaxation is often quite different. For gases such as the ones listed above, the dominant mechanism is the spin-rotation interaction, in which molecular collisions alter the rotational state of the molecule, leading to rotation-induced magnetic fluctuations that cause relaxation [27]. The equation governing spin-rotation relaxation is given by... 

In real life, the parcels or blobs are also subjected to the turbulent fluctuations not resolved in the simulation. Depending on the type of simulation (DNS, LES, or RANS), the wide range of eddies of the turbulent-fluid-flow field is not necessarily calculated completely. Parcels released in a LES flow field feel both the resolved part of the fluid motion and the unresolved SGS part that, at best, is known in statistical terms only. It is desirable that the forces exerted by the fluid flow on the particles are dominated by the known, resolved part of the flow field. This issue is discussed in greater detail in the next section in the context of tracking real particles. With a RANS simulation, the turbulent velocity fluctuations remaining unresolved completely, the effect of the turbulence on the tracks is to be mimicked by some stochastic model. As a result, particle tracking in a RANS context produces less realistic results than in an LES-based flow field. 

In some practical processes, a high relative velocity may not exist and effects of turbulence on droplet breakup may become dominant. In such situations Kolmogorov, 280 and Hinze[27°l hypothesized that the turbulent fluctuations are responsible for droplet breakup, and the dynamic pressure forces of the turbulent motion determine the maximum stable droplet size. Using Clay s data, 2811 and assuming isotropic turbulence, an expression was derived for the critical Weber number 270 ... 

In general, fluctuations in any electron Hamiltonian terms, due to Brownian motions, can induce relaxation. Fluctuations of anisotropic g, ZFS, or anisotropic A tensors may provide relaxation mechanisms. The g tensor is in fact introduced to describe the interaction energy between the magnetic field and the electron spin, in the presence of spin orbit coupling, which also causes static ZFS in S > 1/2 systems. The A tensor describes the hyperfine coupling of the unpaired electron(s) with the metal nuclear-spin. Stochastic fluctuations can arise from molecular reorientation (with correlation time Tji) and/or from molecular distortions, e.g., due to collisions (with correlation time t ) (18), the latter mechanism being usually dominant. The electron relaxation time is obtained (15) as a function of the squared anisotropies of the tensors and of the correlation time, with a field dependence due to the term x /(l + x ). 

The processes observed in the depolarized Rayleigh spectrum correspond to internal modes of motion. Thus, they may have relaxation times which substantially exceed those obtained from the longitudinal or bulk relaxation alone. Nevertheless they are a part of the a relaxation process as it is normally observed in the creep compliance. All processes with the same shift factors make up the full a relaxation. In liquids with substantial depolarized Rayleigh scattering the slowly relaxing part of the W scattering is also dominated by the orientation fluctuations associated with the internal modes of motion. Each internal mode contributes some intensity, but it is believed that fairly short wavelength modes dominate the scattered intensity. 

Convection in Melt Growth. Convection in the melt is pervasive in all terrestrial melt growth systems. Sources for flows include buoyancy-driven convection caused by the solute and temperature dependence of the density surface tension gradients along melt-fluid menisci forced convection introduced by the motion of solid surfaces, such as crucible and crystal rotation in the CZ and FZ systems and the motion of the melt induced by the solidification of material. These flows are important causes of the convection of heat and species and can have a dominant influence on the temperature field in the system and on solute incorporation into the crystal. Moreover, flow transitions from steady laminar, to time-periodic, chaotic, and turbulent motions cause temporal nonuniformities at the growth interface. These fluctuations in temperature and concentration can cause the melt-crystal interface to melt and resolidify and can lead to solute striations (25) and to the formation of microdefects, which will be described later. 

Finally, higher-rate motions (106—109 Hz) can be examined by spin-lattice relaxation time studies. Spin-lattice relaxation (as other relaxation processes) relies on fluctuations in nuclear spin interactions induced by molecular motion. Thus, in cases where relaxation is dominated by one particular nuclear spin interaction, the spin-lattice relaxation times can be calculated for different motions and compared with experimentally derived values. 

The prevalence of structural compared with dynamical information arises from a scarcity of appropriate spectroscopic techniques. For example, the measurement process in NMR spectroscopy takes place on a millisecond timescale. The dominating part of the fluctuation correlation function responsible for dephasing processes, on the other hand, decays on the picosecond timescale, so that spin transitions are strongly in the motional... 

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