Rescaling of time

The cumulative functions of the diffusive modes can here also be constructed by using Eq. (60) with trajectories integrated with a numerical algorithm based on the rescaling of time at the singular collisions. The initial position is taken on a small circle around a scattering center at an angle 0 with respect to the... 

We have seen that an appropriate rescaling of time and renormalizing the stochastic process leads to the Brownian motion. Here we define the homogeneous-in-time... 

Let us prove that the same holds true for the trajectories of the non-autonomous system (10.5.10). First, let us consider more carefully the rescaling of time given by (10.5.13). The meaning of this formula is that the old time t, which parametrizes the trajectories of (10.5.12), is a function of R (p) and of the new time r, which parametrizes the trajectories of (10.5.14), and it is defined by... 

Lemma 12.3. For /i > 0, there exists a smooth transformation of the angular variable (/ and a rescaling of time which reduce the system to the... 

The first term is momentum at nth site p), the second and third terms are nonlinear potential formulation and the last term contains an interaction constant (k). From the above Eq. (14), after deducing the elassieal equation of motion and rescaling of time, we get ... 

Both the determination of the effective number of scatterers and the associated rescaling of variances are still in progress within BUSTER. The value of n at the moment is fixed by the user at input preparation time for charge density studies, variances are also kept fixed and set equal to the observational c2. An approximate optimal n can be determined empirically by means of several test runs on synthetic data, monitoring the rms deviation of the final density from the reference model density (see below). This is of course only feasible when using synthetic data, for which the perfect answer is known. We plan to overcome this limitation in the future by means of cross-validation methods. 

Adolf and Martin [15] postulated, since the near critical gels are self-similar, that a change in the extent of cure results in a mere change in scale, but the functional form of the relaxation modulus remains the same. They accounted for this change in scale by redefinition of time and by a suitable redefinition of the equilibrium modulus. The data were rescaled as G /Ge(p) and G"/Ge(p) over (oimax(p). The result is a set of master curves, one for the sol (Fig. 23a) and one for the gel (Fig. 23 b). 

Let us reiterate that whenever a similarity solution to (3.2.5), (3.2.4c) exists, the physical space coordinate of any given value of concentration between 0 and Cq propagates as const s/t. The shape of the solution (the concentration profile) evolves accordingly in terms of the physical space variable x, whereas it is preserved unchanged (either after a rescaling with some function of time only as in (3.2.9) or without it as in (3.2.13b)) in terms of the similarity variable. 

The fraction rj/rja= d is thus the rescaling of the classical Kramers rate according to the parameters classifying the multiple trapping system with broadly distributed waiting times. Similarly, in the underdamped case, one finds the fractional Kramers rate... 

We use the bosonization technique [17] to calculate the ratchet current. After an appropriate rescaling of the time variable, the system can be described by the action [13]... 

For a particle evolving in a thermal bath, we focused our interest on the particle displacement, a dynamic variable which does not equilibrate with the bath, even at large times. As far as this variable is concerned, the equilibrium FDT does not hold. We showed how one can instead write a modified FDT relating the displacement response and correlation functions, provided that one introduces an effective temperature, associated with this dynamical variable. Except in the classical limit, the effective temperature is not simply proportional to the bath temperature, so that the FDT violation cannot be reduced to a simple rescaling of the latter. In the classical limit and at large times, the fluctuation-dissipation ratio T/Teff, which is equal to 1 /2 for standard Brownian motion, is a self-similar function of the ratio of the observation time to the waiting time when the diffusion is anomalous. 

As far as quantitative comparison with the experimental data is concerned, we decided to compare directly the experimentally determined orientation with the theoretical prediction. This requires a rescaling of the theoretical orientation in order to match it with the experimental data at times of the order of xa- A satisfactory agreement for the average orientation is observed as shown in Figure 14. 

Temperature is defined in molecular dynamics simulations in the microcanonical ensemble (constant N, V, T) by making use of the fact that the average kinetic energy K per degree of freedom is kT. Consequently, one initially assigns velocities to the molecules that add up to this value and then solves the equations of motion for a period of time sufficient to reach equilibrium while rescaling the velocities to hold the system at the desired temperature. After equilibrium has been achieved, the velocity scaling is turned off and the system is allowed to evolve in... 

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