Gaussian phase

In the case of emulsions, the oil phase is confined in a spherical geometry (droplets), and the random motion of the oil molecules is restricted to the droplet boundary. In this case, the signal decay function S/S0, assuming a Gaussian phase... 

I. Horenko, C. Salzmann, B. Schmidt, and C. Schutte. Quantum-classical li-ouville approach to molecular dynamics Surface hopping gaussian phase-space packets. J. Chem. Phys., 117(24) 11075-11088, 2002. 

J. Ma, D. Hsu, J.E. Straub, Approximate solution of the classical Liouville equation using Gaussian phase packet dynamics application to enhanced equilibrium averaging and global optimization, J. Chem. Phys. 99 (1993), 4024. 

The properties of food products such as margarine or low-fat spreads are extremely dependent on the droplet size distribution. Water and fat diffusion in cheese have been studied in this context (Callaghan et al., 1983) and the size distribution of water droplets in margarine products has been measured (Van den Enden et al., 1990). The Gaussian phase distribution (GPD) approximation has been used to determine the size distribution of water droplets in margarine and low-fat spreads (Balinov et al., 1994). NMR diffusometry studies have been compared in some cases with results from image analysis (Fourel et al., 1995). However, up to now, most studies assume that the emulsion droplets are more or less spherical. 

Figure 1 Results of a simulation of the diffusion of water molecules inside an emulsion droplet of radius R, given as the echo amplitude vs. the duration S of the field gradient pulse. The ratio DA/R is 1. The dotted line is the prediction of the Gaussian phase approximation [Eq. (4)], whereas the solid line is the prediction of the short gradient pulse [Eq. (3)]. (Adapted from Ref. 17.)...

Thus, we have at our disposal two equations wifli whieh to interpret PFG data from emulsions in terms of droplet radii, neither of whieh are exact for all values of experimental and system parameters. As the conditions of flie SGP regime are technically demand ing to achieve, Eq. (4) (or limiting forms of it) have been used in most cases to determine the droplet radii. A key question is then under what conditions Eq. (4) is valid. That it reduces to the exact result in the limit of R —K is easy to show and also obvious fi om the fact that we are then approaching flic case of free diffusion, in which the Gaussian phase approximation becomes exact. 

Density functional theory from statistical mechanics is a means to describe the thermodynamics of the solid phase with information about the fluid [17-19]. In density functional theory, one makes an ansatz about the structure of the solid, usually describing the particle positions by Gaussian distributions around their lattice sites. The free... 

Consider simulating a system m the canonical ensemble, close to a first-order phase transition. In one phase, is essentially a Gaussian centred around a value j, while in the other phase tlie peak is around Ejj. 

Far from the transition, one or other of these will apply. Close to the phase transition we will see contributions from both Gaiissians, and a double-peaked distribution. The weight of each Gaussian changes as the temperature is varied. Thus, a smooth... 

Even expression ( B3.4.31), altiiough numerically preferable, is not the end of the story as it does not fiilly account for the fact diat nearby classical trajectories (those with similar initial conditions) should be averaged over. One simple methodology for that averaging has been tln-ough the division of phase space into parts, each of which is covered by a set of Gaussians [159, 160]. This is done by recasting the initial wavefunction as... 

To remedy this diflSculty, several approaches have been developed. In some metliods, the phase of the wavefunction is specified after hopping [178]. In other approaches, one expands the nuclear wavefunction in temis of a limited number of basis-set fiinctions and works out the quantum dynamical probability for jumping. For example, the quantum dynamical basis fiinctions could be a set of Gaussian wavepackets which move forward in time [147]. This approach is very powerfLil for short and intemiediate time processes, where the number of required Gaussians is not too large. 

The random-bond heteropolymer is described by a Hamiltonian similar to (C2.5.A3) except that the short-range two-body tenn v.j is taken to be random with a Gaussian distribution. In this case a tliree-body tenn with a positive value of cu is needed to describe the collapsed phase. The Hamiltonian is... 

In the collapse phase the monomer density p = N/R is constant (for large N). Thus, the only confonnation dependent tenn in (C2.5.A1) comes from the random two-body tenn. Because this tenn is a linear combination of Gaussian variables we expect that its distribution is also Gaussian and, hence, can be specified by the two moments. Let us calculate the correlation i,) / between the energies and E2 of two confonnations rj ]and ry jof the chain in the collapsed state. The mean square of E is... 

By substituting these expressions into Eq. (55), one can see after some algebra that ln,g(x, t) can be identified with lnx (t) + P t) shown in Section III.C.4. Moreover, In (f) = 0. It can be verified, numerically or algebraically, that the log-modulus and phase of In X-(t) obey the reciprocal relations (9) and (10). In more realistic cases (i.e., with several Gaussians), Eq. (56-58) do not hold. It still may be due that the analytical properties of the wavepacket remain valid and so do relations (9) and (10). If so, then these can be thought of as providing numerical checks on the accuracy of approximate wavepackets. 

Finally, Gaussian wavepacket methods are described in which the nuclear wavepacket is described by one or more Gaussian functions. Again the equations of motion to be solved have the fomi of classical trajectories in phase space. Now, however, each trajectory has a quantum character due to its spread in coordinate space. 

The fundamental method [22,24] represents a multidimensional nuclear wavepacket by a multivariate Gaussian with time-dependent width niaUix, A center position vector, R, momentum vector, p and phase, y,... 

To deal with the problem of using a superposition of functions, Heller also tried using Gaussian wave packets with a fixed width as a time-dependent basis set for the representation of the evolving nuclear wave function [23]. Each frozen Gaussian function evolves under classical equations of motion, and the phase is provided by the classical action along the path... 

The Hemian-Kluk method has been developed further [153-155], and used in a number of applications [156-159]. Despite the formal accuracy of the approach, it has difficulties, especially if chaotic regions of phase space are present. It also needs many trajectories to converge, and the initial integration is time consuming for large systems. Despite these problems, the frozen Gaussian approximation is the basis of the spawning method that has been applied to... 

While it is not essential to the method, frozen Gaussians have been used in all applications to date, that is, the width is kept fixed in the equation for the phase evolution. The widths of the Gaussian functions are then a further parameter to be chosen, although it appears that the method is relatively insensitive to the choice. One possibility is to use the width taken from the harmonic approximation to the ground-state potential surface [221]. 

As usual there is the question of the initial conditions. In general, more than one frozen Gaussian function will be required in the initial set. In keeping with the frozen Gaussian approximation, these basis functions can be chosen by selecting the Gaussian momenta and positions from a Wigner, or other appropriate phase space, distribution. The initial expansion coefficients are then defined by the equation... 

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