Jump diffusion model

Fig.4.19 Tseif(Q) obtained for a all the protons in PVE empty MD simulations,/ /// NSE, /=0.55) and b the main chain (filled circle, /=0.66) and the side group hydrogens (empty circle, /=0.51), both from the MDS. Dotted lines are expected Q-dependence from the Gaussian approximation in each case. Solid lines are description in terms of the anomalous jump diffusion model. Insets Chemical formula of PVE (a) and distribution functions obtained for the jump distances (b)...

Fig. 5.22 a Q-dependence of igeif obtained from KWW fits of 5seif(Q,t) of PId3 at 340 K ((3=0.57) (filled circle FZ-Jiilich, empty circle INI 1C) and 300 K (f=0.50) (empty diamond IN lie, filled diamond IN13). deduced for 340 K from the Rouse description of Schain(Q>0 [47] are also depicted (empty square). Dotted line shows the Rouse prediction, dashed lines the Gaussian extrapolations Tseif< Q" and solid lines the descriptions in terms of the anomalous jump diffusion model [9,154]. b Q-dependence of for 340 K symbols as above... 

A possible modification of this expression is presented elsewhere (82). The value of t, can be related to a diffusion coefficient (e.g., tj = l2/6D, where / is the jump distance), thereby making the Ar expressions qualitatively similar for continuous and jump diffusion. A point of major contrast, however, is the inclusion of anisotropic effects in the jump diffusion model (85). That is, jumps perpendicular to the y-ray direction do not broaden the y-ray resonance. This diffusive anisotropy will be reflected in the Mossbauer effect in a manner analogous to that for the anisotropic recoil-free fraction, i.e., for single-crystal systems and for randomly oriented samples through the angular dependence of the nuclear transition probabilities (78). In this case, the various components of the Mossbauer spectrum are broadened to different extents, while for an anisotropic recoil-free fraction the relative intensities of these peaks were affected. 

An alternative approach is based on the jump diffusion model adopted by Karis and Tyndall [90,91]. 

The F values were analysed on the basis of a jump diffusion model ... 

Table 3. Parameter values for the jump diffusion model for the aqueous solution of LiCl.S.OHjO at various temperatures.

An alternative approach to describe thin film spreading is the jump diffusion model developed by Karis and Tyndall. In their analysis, the flow rate is calculated by integrating the velocity v throughout the depth of the film... 

The dependence of the inverse of the characteristic times for water translation does not show exclusively the linear behavior of continuous diffusion (Equation 3.1) or exclusively the shape of a simple jump diffusion model (Equation 3.2), but is very well represented (lines) by considering that both mechanisms contribute to the relaxation (Equation 3.3). The symbols correspond to the simulation data at different temperatures 365 K (circles), 335 K (squares), and 310 K (triangles). 

Another model of rotational reorientation is the jump-diffusion model first described by Ivanov (1964). In this model the molecule reorients by a series of discontinuous jumps (with an arbitrary distribution of jump angles). This should be contrasted with the Debye model, which involves infinitesimal jumps, and the Gordon model, which involves continuous free rotations between collisions. This model is probably applicable to the situation where the molecular orientation is frozen until a volume fluctuation occurs, at which time the molecular orientation jumps to a new frozen value. We present our own version of the jump model here. It is assumed that (a) the jump takes place instantaneously, (b) successive jumps are uncorrelated in time with an average time tv between jumps, and (c) the dihedral angle between the two planes defined by the orientation vector u in two successive jumps is randomized. 

In recent molecular dynamics (MD) studies of propane in Na-Y zeolite, the HWHM obtained from the simulations have been compared with jump diffusion models and with the experiment [19]. Figure 4 shows fits of different models to the MD data, at three temperatures. The error bars on the MD points is too large to select the best model. However, the oscillatory behavior expected for the CE model does not seem to be present either in the MD data or in the experimental QENS broadenings (Fig. 5). 

From the NSE data obtained at various momentum transfers ranging from 0.08 to 0.3 A transport diffusivities could be determined using a jump diffusion model to take into account the curvature of D(Q). A value of 2 X 10 m s was derived at 490 K, which is sevenfold larger than the self-diffusivity previously obtained in Na-ZSM-5. Part of this difference is due... 

To be historically fair, other people did observe the existence of jump motions in the rotation of water molecules in the liquid state but detailed analysis of the dynamics of an individual event was not carried out before. Given that perspective, the Laage-Hynes mechanism of water rotation by large-amplitude jumps is indeed a departure from conventional and prevailing wisdom that water rotation is Brownian that is, it occurs differently in water from in other liquids where motion by small steps dominates. Experimental verification of the jump diffusion model came from a beautiful study of the temperature-dependent rate of water rotation. However, both the experiments and the interpretation of results are quite involved. We shall discuss the results as simply as possible. 

All this suggests that asset-price behavior is more accurately described by nonstandard price processes, such as the jump diffusion model or a stochastic volatility, than by a model assuming constant volatility. For more-detailed discussion of the volatility smile and its implications, interested readers may consult the works listed in the References section. 

The dashed curves correspond to the Chudley-Elliott model with L = 1.5 (line b) and 1.8 A (line d) and the dolled curve to the Random Jump Diffusion model with = q.93 (line c) and 1.22 A (line e). 

Jump Diffusion Model with Damped Vibrations Mechanism of Conformational Transition. .. 

In this section we describe quasielastic neutron scattering studies [129,130] focusing attentions on conformational transitions of polymer chains. For this purpose we first summarize the results of the recent molecular dynamics (MD) simulations on conformational transitions and then discuss the conformational transition mechanism on the basis of neutron data analyzed in terms of a jump diffusion model with damped vibration which has a similar physical picture to that predicted by the MD simulations. 

Fig. 25. Schematic representation of the jump diffusion model with damped vibrations...

The jump diffusion model with damped vibrations was applied to the analysis of the dynamic scattering laws S(Q,co) of the E-process far above the glass transition temperature Tg, where the conformational transitions are mainly dominated by the intramolecular interactions in a polymer chain, so that it is not necessary to take into accounts intermolecular cooperativity effects on the conformational transitions. 

The theoretical equation of the jump diffusion model (Eq. 35) was fitted to the observed dependence of HWHM F of the Lorentzian function of the E-process. The results are shown in Fig. 26 for PB, PIB, PCP and PE. The agreements are very good, suggesting that the model well describes the E-process. In the fit the root mean square jump distance and the relaxation time of the ele-... 

Fig. 27. Temperature dependence of relaxation time for the conformational transitions evaluated using the jump diffusion model with damped vibrations. ( ) PB, ( ) PIB, ( ) PCP, (x) PE. (Reprinted with permission from [ 130]. Copyright 1999 American Chemical Society, Washington)...

When the relaxation mechanism is the modulation of the magnetic g and A components due to the rotational diffusion of the paramagnetic group (mainly for the nitroxide radicals, and for 5 = y paramagnetic ions), the analysis of the spectra in the fast-slow motion regime provides the correlation time for the rotational motion. An increase in the correlation time corresponds to a decrease in mobility of the paramagnetic probe or label. The evaluation of the correlation time for the rotational motion was performed by simple methods or by computation of the spectra. Different diffusion models can be considered, such as Brownian or jump diffusion models, and the rotational mobility may be considered isotropic or anisotropic. In this latter case, for nitroxide radicals, the main information was obtained from the perpendicular component of the correlation time. Furthermore, a shift of the main rotational axis accounts for the compression of the labels due to other molecules approaching the label at the dendrimer surface. 

A simplest transition rate R Cl Cl) is provided by the strong colhsion (or jump diffusion) model. 

In a second set of experiments [63], the same measurements were carried out on the isotopic molecule HD. The same jump diffusion model holds for this molecule, with Thd = 0 33 nm. As can be expected from simple arguments, the mean jump lengths are found to be in the ratio of the square root of their molecular masses ( Th2/Thd = V 3/2 ). On the other hand the temperature dependence of the diffusion coefficients (which are of the order of 10 m s ) points out an unexpected feature. As a matter of fact, one can represent each result on an Arrhenius coordinate system Ln(D) = f(l/T) but this leads to different Dq values and different activation energies, namely Eh2 = 240 K and Ehd = 337 K and one... 

To model these phenomena, different modification of the Black Scholes model have been proposed. The most two common models are the stochastic volatility model (see Hull White process (Hull White 1987)), CIR process (see (Cox Ross 1976)) and Ornstein-Uhmenbeck process (see (Heston 1993, Stein Stein 1991)) and the jump diffusion model. 

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