Phase space separation

Local Trajectory Instability That is, the local (in phase space) exponential sensitivity of trajectories to changes in initial conditions. This property is the primary characteristic of a C system. Specifically, consider, in an N degree of freedom system, two trajectories with initial coordinates and momenta q0, p0 and qo, Po- For convenience we denote the column vector [q(f), p(t)] associated with p0, q0 as x(t) and the trajectory [q ( ), p (t)] associated with Po, q 0 as x (t). The phase space separation d(t) between trajectories is given by... 

Hamiltonian) trajectory in the phase space of the model from which infonnation about the equilibrium dyuamics cau readily be extracted. The application to uou-equilibrium pheuomeua (e.g., the kinetics of phase separation) is, in principle, straightforward. 

We want to examine the relative behaviors of the two neighboring phase space trajectories, x (t) and x(t), starting from the initial conditions x (0) and x(0) = x (0) -f-Jx(0), respectively. The time evolution of their separation, x(i), may be approximated by linearizing the equations about the reference trajectory, x (t) ... 

Rosenstock (55) pointed out that the initial formulation of the theory failed to consider the effect of angular momentum on the decomposition of the complex. The products of reaction must surmount a potential barrier in order to separate, which is exactly analogous to the potential barrier to complex formation. Such considerations are implicit in the phase space theory of Light and co-workers (34, 36, 37). These restrictions limit the population of a given output channel of the reaction com-... 

Usually we are only interested in mutual intensity suitably normalised to account for the magnitude of the helds, which is called the complex degree of coherence 712 (r). This quantity is complex valued with a magnitude between 0 and 1, and describes the degree of likeness of two e. m. waves at positions ri and C2 in space separated by a time difference r. A value of 0 represents complete decorrelation ( incoherence ) and a value of 1 represents complete eorrelation ( perfect coherence ) while the complex argument represents a difference in optical phase of the helds. Special cases are the complex degree of self coherence 7n(r) where a held is compared with itself at the same position but different times, and the complex coherence factor pi2 = 712(0) which refers to the case where a held is correlated at two posihons at the same time. 

In the case of a space separation of the laser beams (i.e. if the atomic velocity is perpendicular to the direction of the laser beams), the interferometer is in a Mach-Zehnder configuration. Then, the interferometer is also sensitive to rotations, as in the Sagnac geometry (Sagnac, 1913) for light interferometers. For a Sagnac loop enclosing area A, a rotation Q, produces a phase shift ... 

VOF or level-set models are used for stratified flows where the phases are separated and one objective is to calculate the location of the interface. In these models, the momentum equations are solved for the separated phases and only at the interface are additional models used. Additional variables, such as the volume fraction of each phase, are used to identify the phases. The simplest model uses a weight average of the viscosity and density in the computational cells that are shared between the phases. Very fine resolution is, however, required for systems when surface tension is important, since an accurate estimation of the curvature of the interface is required to calculate the normal force arising from the surface tension. Usually, VOF models simulate the surface position accurately, but the space resolution is not sufficient to simulate mass transfer in liquids. 

To achieve the desired separation of the reactive degree of freedom from the bath modes, we use time-dependent normal form theory [40,99]. As a first step, the phase space is extended through the addition of two auxiliary variables a canonical coordinate x, which takes the same value as time t, and its conjugate momentum PT. The dynamics on the extended phase space is described by the Hamiltonian... 

A two-dimensional cartoon is helpful in understanding overlap relationships between the important phase spaces /, and /. As illustrated in Fig. 6.1, there are four possible ways that /, and / can be related (a) / can form a subset of /, (b) /, and r may almost coincide, i.e., have complete overlap (c) T( and / may have partial overlap and (d) / and r may have no overlap. For simplicity, in Fig. 6.1 we have only considered the case in which / is continuous in space but the same principle applies to more-complicated situations, in which regions of / are separated in space. 

As the initial and final states are set by the problem under study, their important phase space relationship could be any one of the cases illustrated in Fig. 6.1. For cases Fig. 6.1c, d, it is impossible to construct a funnel path from 0 to 1 directly. To satisfy the funnel requirement, similar to the MFEP calculation, a staged NEW calculation can be performed. For example, in the case Fig. 6.1c, one can first construct an intermediate in the common region of / ,[ and /), then perform two separate NEW calculations following the paths 0 —> M and 1 —> M, respectively. This NEW-overlap sampling (NEW-OS) technique will be discussed in detail in Sect. 6.6. 

The accretion history of a parent galaxy is constructed using a semi-analytical code. The full phase-space evolution during each accretion event is then followed separately with numerical simulations [2]. Star-formation and chemical evolution models are implemented within each satellite. The star formation prescription matches the number and luminosity of present-day galaxies in the Local Group, whereas the chemical evolution model takes into account the metal enrichment of successive stellar populations as well as feedback processes. Below we present results of a sample of four such simulated galaxy halos, denoted as Halos HI, H2, H3 and H4. 

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