Space region

Consider a sphere-sphere collision. Let the incoming and outgoing distributions be given by /i, /2 and /i, /2, respectively. Also, assume that the pre- and postcollision velocity-space regions occupied by hard-spheres 1 and 2 are given by cPv, cPv and cPv2, cPv2, respectively. 

A condition to have, together with a fully ordered modification, more or less disordered modifications corresponding to a same unit cell is a substantial equality of steric hindrances in the space regions, where the statistical diadochy is achieved, as will be shown in the following. 

The scaling prescription (59) embodies the assumption that the external force is so weak that it does not drive the TS trajectory out of the phase-space region in which the normal form expansion is valid. In the autonomous version of geometric TST, one generally assumes that this region is sufficiently large to make the normal form expansion a useful tool for the computation of the geometric objects. Once this assumption has been made, the additional condition imposed by Eq. (59) is only a weak constraint. 

Fig. 6.1. A cartoon depiction of the phase space T and important phase space regions and their relationship. The important phase space regions of systems 0 (To ) and 1 (T) ) are abstractly represented by shaded and open oval shapes, respectively. These regions can be related in four ways (a) subset, (b) coincidence, (c) partial overlap, and (d) no overlap. Also sketched is the important phase space region of the intermediate M in a two-stage calculation (see section on Multiple-Stage Design ). The appropriate staging strategy differs according to the different overlap relationships between F0 and 77...

Fig. 6.3. To ensure the accuracy of a nonequilibrium work free energy calculation, the switching paths should go down the funnel. The important phase space regions for the intermediate states along the ideal funnel paths are illustrated in this plot, for the case where r0 and / are partially overlapped. Two funnel paths need to be constructed to transfer the systems from both 0 and 1 to a common intermediate M where rm is inside the r0 and J overlap region. The construction of such paths is discussed in Sect. 6.6...

It would be valuable if one could proceed with a reliable free energy calculation without having to be too concerned about the important phase space and entropy of the systems of interest, and to analyze the perturbation distribution functions. The OS technique [35, 43, 44, 54] has been developed for this purpose. Since this is developed from Bennett s acceptance ratio method, this will also be reviewed in this section. That is, we focus on the situation in which the two systems of interest (or intermediates in between) have partial overlap in their important phase space regions. The partial overlap relationship should represent the situation found in a wide range of real problems. 

A container is said to be under vacuum when the inside pressure is lower than the outer one, usually the atmospheric pressure. If no container exists, vacuum is a space region at a pressure lower than the atmospheric pressure. 

The real crystalline forms are generally intermediate between the limit-ordered and limit-disordered models, the amount of disorder being dependent on the condition of crystallization and thermal and mechanical treatments of the samples. A condition to have more or less disordered modifications, corresponding to the same unit cell, is the substantial equality of steric hindrances in the space regions where a statistical substitution is achieved (Figure 2.29b). 

The orbits are dense in a state space region i.e. the orbits fills the phase space zone of the strange attractor fl. 

The phase-space region Af/ corresponding to the lattice vector / is partitioned into cells A. The probability that the system is found in the cell A at time t is given by... 

The diffusive random walk of the Helfand moment is mled by a diffusion equation. If the phase-space region is defined by requiring Ga(t) < x/2, the escape rate can be computed as the leading eigenvalue of the diffusion equation with these absorbing boundary conditions for the Helfand moment [37, 39] ... 

As shown by TalkneP there is a direct connection between the Rayleigh quotient method and the reactive flux method. Two conditions must be met. The first is that phase space regions of products must be absorbing. In different terms, the trial function must decay to zero in the products region. The second condition is that the reduced barrier height pyl" 1. As already mentioned above, differences between the two methods will be of the order e P. ... 

The results surveyed in the preceding two sections provide a first clue to the origin of chirality chiral patterns can emerge spontaneously in an initially uniform and isotropic medium, through a mechanism of bifurcations far from thermodynamic equilibrium (see Figs. 4 and 5). On the other hand, because of the invariance properties of the reaction-diffusion equations (1) in such a medium, chiral solutions will always appear by pairs of opposite handedness. As explained in Sections III.B and III.C this implies that in a macroscopic system symmetry will be restored in the statistical sense. We are left therefore with an open question, namely, the selection of forms of preferred chirality, encompassing a macroscopic space region and maintained over a macroscopic time interval. 

Traditionally, physics emphasizes the local properties. Indeed, many of its branches are based on partial differential equations, as happens, for instance, with continuum mechanics, field theory, or electromagnetism. In these cases, the corresponding basic equations are constructed by viewing the world locally, since these equations consist in relations between space (and time) derivatives of the coordinates. In consonance, most experiments make measurements in small, simply connected space regions and refer therefore also to local properties. (There are some exceptions the Aharonov-Bohm effect is an interesting example.)... 

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