Gibbs oscillations

By taking Equation 2.49 rather than Equation 2.42 as a finite Fourier transform, the Gibbs oscillation due to the finite interval in a numerical calculation of the Fourier transform is reduced. 

It is possible to estimate the error of the approximate time solution //f) defined by Equation 2.42 in comparison with the accurate one/(f) in Equation 2.41, which cannot be evaluated by numerical integration. When/(w) changes suddenly, noticeable oscillation called "Gibbs oscillation" appears in//f). This is the error caused by the finite Fourier transform. A countermeasure to this is to take an average for the time region [f - a, f + a] in the following form ... 

The coefficients a calculated from equation (5) for both the LEPS and Porter-Karplus surfaces are shown in Fig. 1 along with their uncertainties computed from equation (6). For the LEPS coefficients the noise is just equal to the size of the points. In both cases the magnitude of the coefficients falls off rapidly with n. It is apparent from Fig. 1 that the number of coefficients above the noise, M, is approximately 9-11 for the LEPS surface and 14-20 for the Porter-Karplus surface. To minimize the Gibbs oscillations the final coefficient is always chosen, such that a is a local minimum. Thus,... 

This function, and its approximate Fourier representations for TV = 10, 20, are shown in Figure 9.1. The Fourier series representations exhibit artificial Gibbs oscillations that are not foimd in the true square pulse function. 

Figure 9.1 Approximate Fourier representations of a square pulse showing Gibbs oscillations for (a) N= 10and(b)iV=20.

Here we first calculate the commutator [x(/),x(0)] averaged over an equilibrium Gibbs state ensemble with fixed orientations e so as to obtain the GF of a harmonic oscillator ... 

At equilibrium the Gibbs energy of the system is minimal for harmonic oscillators this is the same as requesting that the potential energy is zero. By differentiation we find the equilibrium values of the solvent coordinates ... 

Vq is the frequency of the small oscillation, and AG and AS are, respectively, the difference in Gibbs free energy and entropy of the adatom at the saddle point and the equilibrium adsorption site. Ed is the activation energy of surface diffusion, or the barrier height of the atomic jumps. 

Johonnott [314] explained stratification with alternating molecular forces of attraction and repulsion when a black film thins. Later, Keuskamp and Lyklema [352] pointed out that this phenomenon is caused by oscillation in the Gibbs free energy of the film when its thickness is changed. Two theories have emerged to account for these oscillations in the free... 

Table 7 shows the electron contribution to the energy of activation determined from a GAUSSlAN-03 and the Gibbs energy of transient state calculated with the aid of a MOLTRAN program from the zero oscillation frequencies. Also, Table 7 shows the rate constan we determined for the symmetrical and asymmetrical reaction pathways and the experimental value borrowed from [33]. The latter value was obtained by the method of cryokinetic calorimetry (direct measurements on the rate constants of the reaction of TFE with pure ozone were performed at 90-150 K). The measurements showed that that the rate constant may be described as follows ... 

As mentioned above, hrst consider the diamagnetic, transparent, insulator already mentioned. When such a sample is exposed to an electric held, it responds. The electrons around each atom move in such a way as to reduce the potenhal energy (or minimize the Gibbs free energy if the system is at a constant temperature and pressure) of the system. Electrons have a low mass compared to the nucleus which they surround and can move in response to the electric held. If the held is oscillating, the electrons will oscillate as well with the same frequency. The electric held, E, induces in the material a polarizahon ... 

The solution obtained is plotted at t = 0. We observe oscillations about initial condition 1. This is called Gibb s phenomenon. Theoretically it will take N = oo for this profile to become u = 1 at t = 0. 

In order to visualize the energy barrier between reactants and products, it is assumed that each system can be represented as a classical harmonic oscillator along the reaction coordinate. This is illustrated in fig. 7.15. The left-hand parabola gives the Gibbs energy of the reactants and the right-hand parabola, that of... 

Distribution of fibers Aspect ratio Filler Frequency of oscillation Coefficient of expansion of filler Shear modulus Gibbs free energy Heat of reaction, change Hildebrand unit, solubility parameter value High-density (linear) polyethylene Heat deflection temperature... 

In recent years, several theoretical and experimental attempts have been performed to develop methods based on oscillations of supported drops or bubbles. For example, Tian et al. used quadrupole shape oscillations in order to estimate the equilibrium surface tension, Gibbs elasticity, and surface dilational viscosity [203]. Pratt and Thoraval [204] used a pulsed drop rheometer for measurements of the interfacial tension relaxation process of some oil soluble surfactants. The pulsed drop rheometer is based on an instantaneous expansion of a pendant water drop formed at the tip of a capillary in oil. After perturbation an interfacial relaxation sets in. The interfacial pressure decay is followed as a function of time. The oscillating bubble system uses oscillations of a bubble formed at the tip of a capillary. The amplitudes of the bubble area and pressure oscillations are measured to determine the dilational elasticity while the frequency dependence of the phase shift yields the exchange of matter mechanism at the bubble surface [205,206]. 

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