Phase space shell

Classical statistical mechanics is concerned with the probability distribution of phase points. In a classical microcanonical ensemble the phase space density is constant. Loosely speaking, aU phase points with the same energy are equally likely. In consequence the number of states of the classical system in a given energy range E to E + dE is proportional to the volume of the phase space shell defined by this energy range. 

Orthant sampling [10] works in the classical phase space of the molecular Hamiltonian H(P, Q). For a microcanonical ensemble, each phase-space point in the volume element dP dQ has equal probability [17]. The classical density of states is then proportional to the surface integral of the phase-space shell with H P, Q) = and is given by [17]... 

The point q = p = 0 (or P = Q = 0) is a fixed point of the dynamics in the reactive mode. In the full-dimensional dynamics, it corresponds to all trajectories in which only the motion in the bath modes is excited. These trajectories are characterized by the property that they remain confined to the neighborhood of the saddle point for all time. They correspond to a bound state in the continuum, and thus to the transition state in the sense of Ref. 20. Because it is described by the two independent conditions q = 0 and p = 0, the set of all initial conditions that give rise to trajectories in the transition state forms a manifold of dimension 2/V — 2 in the full 2/V-dimensional phase space. It is called the central manifold of the saddle point. The central manifold is subdivided into level sets of the Hamiltonian in Eq. (5), each of which has dimension 2N — 1. These energy shells are normally hyperbolic invariant manifolds (NHIM) of the dynamical system [88]. Following Ref. 34, we use the term NHIM to refer to these objects. In the special case of the two-dimensional system, every NHIM has dimension one. It reduces to a periodic orbit and reproduces the well-known PODS [20-22]. 

For Newtonian dynamics and a canonical distributions of initial conditions one can reject or accept the new path before even generating the trajectory. This can be done because Newtonian dynamics conserves the energy and the canonical phase-space distribution is a function of the energy only. Therefore, the ratio plz ]/p z at time 0 is equal to the ratio p[.tj,n ]/p z ° at the shooting time and the new trajectory needs to be calculated only if accepted. For a microcanonical distribution of initial conditions all phase-space points on the energy shell have the same weight and therefore all new pathways are accepted. The same is true for Langevin dynamics with a canonical distribution of initial conditions. 

Fig. 18 Phase space of PI-fc-PS-fc-PEO in vicinity of ODT. Filled and open circles-. ordered and disordered states, respectively, within experimental temperature range 100 < T/° C< 225. Outlined areas compositions with two- and three-domain lamellae (identified by sketches) shaded regions three network phases, core-shell double gyroid (Q230), orthorhombic (O70), and alternating gyroid (Q214). Overlap of latter two phase boundaries indicates high- and low-temperature occurrence, respectively, of each phase. Dashed line condition tfin = 0peo associated with symmetric PI-fc-PS-fc-PEO molecules. From [75]. Copyright 2004 American Chemical Society...

In statistical mechanics of equilibrium one assigns equal probabilities to equal volume elements of the energy shell in phase space. 510 This assignment is determined... 

D. J. Tannor To understand the role of dissipation in quantum mechanics, it is useful to consider the density operator in the Wigner phase-space representation. Energy relaxation in a harmonic oscillator looks as shown in Fig. 1, whereas phase relaxation looks as shown in Fig. 2 that is, in pure dephasing the density spreads out over the energy shell (i.e., spreads in angle) while not changing its radial distribution... 

Hence, for small perturbations the system is nearly integrable Most classical trajectories are restricted to two-dimensional phase-space structures that are often called KAM tori. Since two classical trajectories cannot cross each other, a toms such as shown in Fig. 1 is in fact an impenetrable phase-space stmcture, dividing the three-dimensional energy shell into disjoint regions. With stronger perturbations, more KAM tori are expected to be destroyed and therefore more trajectories become chaotic. 

While in some analogous works, it was possible to devise surfaces of section or even full representations of phase space this is hardly thinkable here. Let us recall that an on-shell (or constant energy H = E = 0.001 atomic units) Poincare section would be of dimension = D(phasespace) — 1 — 1=6. Instead we... 

The most natural approach to the density of states is to find the volume of phase space corresponding to the shell defined by the energy range E < p) < + e, and divide this by the span of the energy range e before taking the limit e —> 0. 

In order to obtain a statistical measure for the ionization process we have calculated for an ensemble of trajectories the fraction of ionized orbits as a function of time. The initial internal energy was chosen to correspond to a completely chaotic phase space of the He -ion if the nuclear mass were infinite. The initial conditions for the internal motion have been selected randomly on the energy shell. In Fig. 11 we have illustrated the fraction of ionized orbits as a function of time up to T = 10 a.u. for a series of different CM energies and for a very strong laboratory field strength of B = 10 a.u.. For an initial CM energy of Ec = 0.053 a.u. which corresponds to an initial CM... 

The 8-dimensional phase space of two 2-dimensional oscillators is reduced by the existence of two conserved actions, Ka and Kb, and by the absence of the conjugate angles, classical mechanical polyad 7feff. The conserved actions appear parametrically in 7feff, thus the phase space accessible at specified values of Ka and Kb is four dimensional. Since energy is conserved, in addition to Ka and Kb, all trajectories lie on the surface of a 3-dimensional energy shell. 

Each trajectory is launched at chosen initial values of Jb and ipb and at fra = 0. Since any point on the 3-dimensional energy shell may be specified by three linearly independent coordinates, selection of initial values J , ip%, and ip°, implies a definite value of J°. Thus trajectories are launched at various [Jfi, ip%, ip° = 0, J°(J , ipl, ip°,) ] initial values until all of the qualitatively distinct regions of phase space are represented on the surface of section by either a family of closed curves (quasiperiodic trajectories) that surround a fixed point (a periodic trajectory that defines the qualitative topological nature of the neighboring quasiperiodic trajectories) or an apparently random group of points (chaos). Often, color is used to distinguish points on the surface of section that belong to different trajectories. 

The bound antinucleons are then pulled down into the (lower) continuum. In this way antimatter clusters may be set free. Of course, a large part of the antimatter will annihilate on ordinary matter present in the course of the expansion. However, it is important that this mechanism for the production of antimatter clusters out of the highly correlated vacuum does not proceed via the phase space. The required coalescence of many particles in phase space suppresses the production of clusters, while it is favored by the direct production out of the highly correlated vacuum. In a certain sense, the highly correlated vacuum is a kind of cluster vacuum (vacuum with cluster structure). The shell structure of the vacuum levels (see Figure 8.21) supports this latter suggestion. Figure 8.23 illustrates this idea. 

As discussed in textbooks (Blatt 1968), the interaction term in eq. (15) does not give rise to any resistivity unless the fermion gas is imbedded in a lattice so that umklapp scattering can occur. Furthermore, since at T Tp only those electrons within an energy shell of width k T around the Fermi level can conduct electric current, the pair of electrons affected by the interaction must be within this energy shell both before and after scattering. Thus, the phase space argument shows that the resistivity must be scaled by (T/Tp), as observed experimentally. Unfortunately, the theory contains too many parameters to allow a reliable determination of the interaction potential from the resistivity data of real materials. 

We introduced the microcanonical analysis in Section 2.7 and found that the density of states g E) already contains all relevant information about the phases of the system. Alternatively, one can also use the phase space volume AG(E) of the energetic shell that represents the macrostate in the microcanonical ensemble in the energetic interval (E,E+ AE) with AE being sufficiently small to satisfy AG E)=g E)AE. In the limit AE —> 0, the total phase space volume up to the energy E can thus be expressed as G E) = dE g(E ). Since g E) is positive for all E, G(E) is a monotonically increasing function and this quantity is suitably related to the microcanonical entropy S(E) of the system. In the definition of Hertz,... 

Following a similar process, we first compute the size of phase space for = 1 confined in three dimensions (Fig. 4.6). This is the volume of a spherical shell with... 

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