Finite correlation time

Near critical points, special care must be taken, because the inequality L will almost certainly not be satisfied also, cridcal slowing down will be observed. In these circumstances a quantitative investigation of finite size effects and correlation times, with some consideration of the appropriate scaling laws, must be undertaken. Examples of this will be seen later one of the most encouraging developments of recent years has been the establishment of reliable and systematic methods of studying critical phenomena by simulation. 

Inequality (6.67) is the softest criterion of perturbation theory. Its physical meaning is straightforward the reorientation angle (2.30) should be small. Otherwise, a complete circle may be accomplished during the correlation time of angular momentum and the rotation may be considered to be quasi-free. Diffusional theory should not be extended to this situation. When it was nevertheless done [268], the results turned out to be qualitatively incorrect orientational relaxation time 19,2 remained finite for xj —> 00. In reality t0j2 tends to infinity in this limit [27, 269]. 

Alternatively, the white-noise processes W(f) could be replaced by colored-noise processes. Since the latter have finite auto-correlation times, the resulting Lagrangian correlation functions for U and would be nonexponential. However, it would generally not be possible to describe the Lagrangian PDF by a Fokker-Planck equation. Thus, in order to simplify the comparison with Eulerian PDF methods, we will use white-noise processes throughout this section. 

Relaxors where there is no macroscopic symmetry breaking and where, in view of site and charge disorder, there is an extremely broad distribution of correlation times. The longest correlation time diverges at the freezing transition whereas other correlation times are still finite [e.g., Pb (Mgi/3Nb2/3) O3]. 

The temperature-dependent coupling spectrum is the Fourier transform of the bath response function in Eq. (4.202), and it usually has a certain width proportional to the inverse of the correlation time. The time-dependent modulation spectrum is the finite-time Fourier transform of the modulation function, eft). 

The H NMRD profiles of Mn(OH2)g+ in water solution show two dispersions (Fig. 5.43). The first (at ca. 0.05 MHz, at 298 K) is attributed to the contact relaxation and the second (at ca. 7 MHz, at 298 K) to the dipolar relaxation. From the best fit procedure, the electron relaxation time, given by rso = 3.5 x 10 9 and r = 5.3 x 10 12 s, is consistent with the position of the first dispersion, the rotational correlation time xr = 3.2 x 10 11 s is consistent with the position of the second dispersion and is in accordance with the value expected for hexaaquametal(II) complexes, the water proton-metal center distance is 2.7 A and the constant of contact interaction is 0.65 MHz (see Table 5.6). The impressive increase of / 2 at high fields is due to the field dependence of the electron relaxation time and to the presence of a non-dispersive zs term in the equation for contact relaxation (see Section 3.7.2). If it were not for the finite residence time, xm, of the water molecules in the coordination sphere, the increase in Ri could continue as long as the electron relaxation time increases. 

Garrido and Sancho studied a multiplicative stochastic process with finite correlation time r. By using an ordered cumulant technique they evaluated corrections up to order t. ... 

Since Eq. (49) takes into account only the term of order Dt, the term of order in Eq. (51) is meaningless and the term linear in t in vanishes exactly. For T = 0, our result equals the well-known Smoluchowski rate. The main conclusion we can draw is that the activation rates for non-Markovian processes like Eq. (44) decrease as t increases the exact result of ref. 44 can thus be extended to the case of Gaussian random forces of finite correlation time as well. However, if we take Eq. (50) seriously, we obtain an Arrhenius factor, exp(A /Z)), of T(x) which does not exhibit a dependence on T. This is in contrast to the result found for telegr hic noises, where the Arrhenius factor increases with increasing autocorrelation time r (see ref. 44). The result of a numerical simulation for J(x) based on the bi-... 

Ito [51] obtains different expressions for the Kramers-Moyal coefficients in which the spurious drift term is absent. However use of Ito coefficients involves new rules for calculus and so Stratonovich s method will be used here since it is also in agreement with the original method of Brown [8] and is the correct definition to use in the case of a physical noise which always has a finite correlation time [58] (see B.2). 

The exponential asymptotic decay also holds for temporally random (i.e. with finite correlation time) spatially smooth flows. In this case the corresponding eigenmode is a stationary random function (Fig. 2.21) and the decay process can be characterized by a set of exponents 7n, associated to the moments of the concentration field, defined by... 

From the simulation data, the correlation time is found by integrating the time correlation function as shown in Eq. (6). While the correlation functions may be easily computed from the trajectories, statistical errors due to finite trajectory length limit the useful data to short delay times r [16,17]. The variance of a Gaus-... 

V is the volume of the system. To include finite correlation time the noise was described by the following Ornstein-Uhlenbeck process ... 

The correlation time p characterizes three-bond jumps, and the correlation time B characterizes other processes. Bendler and Yaris have also reconsidered the diamond lattice model, and replaced the discrete jump kinetic formulation by a continuum with adjustable cut-offs in the frequency spectrum. The high cut-off arises from the finite size of the smallest displaceable unit, and the low cut-off from the fact that chain displacements will be damped out as they travel down the chain. Librational motions, hitherto neglected, have been considered by Howarth, with success in interpreting relaxation data on proteins. It is possible that this factor should also be taken into account for synthetic polymers. 

Figures shows the calculated and measured dynamic shear modulus of polystyrene, for values of x and B, chosen so as to ensure agreement with the modulus on the plateau and the length of the plateau. Good agreement is achieved at high and low frequencies. The form of the real part of the modulus confirms the assumption about the finite correlation time r in the memory functions (45), but discrepancies in the region of the plateau (especially seen on the G" ijS) plot) witness that other correlation times in the memory function less than r can exist. In fact, to describe the dependencies for monodis-perse polybutadiene and polystyrene empirically, a few relaxation times were introduced [98, 99]. However, the differences in the region of the plateau can also be attributed to the inevitable polydispersity of the samples.

Unis, SANS yields S(Q, 0), which corresponds to the average of instantaneous stractures, whereas NSE spectroscopy yields additional information on the temporal variation of the stmc-tures that contribute to the scattering intensity. The motions observed are thermal flucmations, their correlation time depends on a balance of friction, and eventually present forces restoring equilibrium configurations. If no restoring forces act, the observed Brownian morion is diflusion without bounds, if microscopic confinement by external potentials or topological effects is present, the diffusion is bounded and the correlation function halts at a finite level. 

Figure C 1.5.10. Nonnalized fluorescence intensity correlation function for a single terrylene molecule in p-terjDhenyl at 2 K. The solid line is tire tlieoretical curve. Regions of deviation from tire long-time value of unity due to photon antibunching (the finite lifetime of tire excited singlet state), Rabi oscillations (absorjDtion-stimulated emission cycles driven by tire laser field) and photon bunching (dark periods caused by intersystem crossing to tire triplet state) are indicated. Reproduced witli pennission from Plakhotnik et al [66], adapted from [118].

The last approximation is for finite At. When the equations of motions are solved exactly, the model provides the correct answer (cr = 0). When the time step is sufficiently large we argue below that equation (10) is still reasonable. The essential assumption is for the intermediate range of time steps for which the errors may maintain correlation. We do not consider instabilities of the numerical solution which are easy to detect, and in which the errors are clearly correlated even for large separation in time. Calculation of the correlation of the errors (as defined in equation (9)) can further test the assumption of no correlation of Q t)Q t )). 

We assume that the sequential errors are not correlated in time, we can write the probability of sampling a sequence of errors as the product of the individual probabilities. We further use the finite time approximation for the delta function and have ... 

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