Phase space sphere

To be more precise, let us assume, as Boltzman first did in 1872 [boltz72], that we have N perfectly elastic billiard balls, or hard-spheres, inside a volume V, and that a complete statistical description of our system (be it a gas or fluid) at, or near, its equilibrium state is contained in the one-particle phase-space distribution function f x,v,t) ... 

Integrating ovtir all of phase space must give us back the total number of hard-spheres in our system J f x, v, t) (Px (Pv = N. 

In general, the distribution function changes in time because of the underlying motion of the hard-spheres. Consider first the nonphysical case where there are no collisions. Phase-space conservation, or Louiville s Theorem [bal75], assures us that... 

The classical spin motion that follows from the above considerations can be viewed as a dynamics on the sphere S2 driven by the Hamiltonian dynamics in phase space. To see this one first transforms the (redundant) fourdimensional representation of the matrix degrees of freedom, corresponding... 

This equation describes the Thomas precession of a classical spin (Thomas, 1927), which is driven by the underlying Hamiltonian dynamics in phase space. In combination the two types of dynamics, Hamiltonian on phase space and driven precession on the sphere, yield the following picture There is a combined phase space It3 x It3 x S2 with two dynamical systems,... 

Figure 3 depicts the spectmm of Lyapunov exponents in a hard-sphere system. The area below the positive Lyapunov exponent gives the value of the Kolmogorov-Sinai entropy per unit time. The positive Lyapunov exponents show that the typical trajectories are dynamically unstable. There are as many phase-space directions in which a perturbation can amplify as there are positive Lyapunov exponents. All these unstable directions are mapped onto corresponding stable directions by the time-reversal symmetry. However, the unstable phase-space directions are physically distinct from the stable ones. Therefore, systems with positive Lyapunov exponents are especially propitious for the spontaneous breaking of the time-reversal symmetry, as shown below. 

Consider a statistically homogeneous dispersion of N particles contained in a volume V. Although generalization to the case of dissimiliar particles is not difficult, the particles are assumed for simplicity to be identical spheres (radii a). For any given experimental realization, the configuration of the latter particulate system at time t is completely determined by the values of the 3N coordinates xN representing the particle locator points (chosen to lie at the sphere centers so as to preclude individual particle orientational issues) of the N spheres. Note that the dynamical variables (i.e., momenta) need not be included in the phase space since inertia has been systematically neglected. 

As already remarked, the idea underlying the Thomas-Fermi (TF) statistical theory is to treat the electrons around a point r in the electron cloud as though they were a completely degenerate electron gas. Then the lowest states in momentum space are all doubly occupied by electrons with opposed spins, out to the Fermi sphere radius corresponding to a maximum or Fermi momentum pt(r) at this position r. Therefore if we consider a volume dr of configuration space around r, the volume of occupied phase space is simply the product dr 47ipf(r)/3. However, we know that two electrons can occupy each cell of phase space of volume h3 and hence we may write for the number of electrons per unit volume at r,... 

Let us now lift this disk D2 into phase space. To do so, one must go back to the sphere equation, Eq. (37). There are several ways of depicting a 3-sphere one is particularly appropriate here [24]. The sphere is dynamically composed of two identical harmonic oscillators without explicit coupling, but whose total energy is a constant, hs3 > 0. Let us thus transform the Hamiltonian (37) in action angle variables, where N,Iy are the actions of the two oscillators and Q, 0, are the two associated angles. Since... 

Here again, we take a classical trajectory as a reference path in a reaction tube that passes across the transition region between two basins a and b with the flow direction b —> a. Set the time origin t = 0 at just the moment of transition. At a given time t, we take a sphere of a radius rt in 30-dimensional phase space, the center of which proceeds along the reference trajectory. Pick random points in this sphere, and let them run backward in time. Some of them will go back to the basin b if the sphere still lies inside the same tube, and the others will move to some other basins if this sphere is already out of the tube. Should the latter happen, a similar procedure is to be redone with a smaller radius r,. Repeating... 

The above integral can be performed analytically [7] (Ref. [7] p. 28) as the phase space maps onto a 3N-dimensional sphere with radius, E, to yield... 

As polysaccharides tend to functionalize environmental phase space, for specification of polysaccharide dimensions, geometry and dynamics must be distinguished, although the transition is diffuse Whereas the dimensions of polysaccharide molecules in terms of sphere equivalent radii of mean excluded volume for macroscopic periods are up to maybe lOOnm only, dynamics of coherent supermolecular structures provide sphere equivalent radii that are more than one magnitude larger and enter the micrometer range. However, these structures are hidden if mass fractions are taken for illustration. Their identification typically needs sophisticated detection and specific scaling, for instance, photon correlation spectroscopy and representation of detected populations with respect to the square of coherent (occupied) volumes (Fig. 6C). 

Solution Repellers are incompatible with volume contraction because they are sources of volume, in the following sense. Suppose we encase a repeller with a closed surface of initial conditions nearby in phase space. (Specifically, pick a small sphere around a fixed point, or a thin tube around a closed orbit.) A short time later, the surface will have expanded as the corresponding trajectories are driven away. Thus the volume inside the surface would increase. This contradicts the fact that all volumes contract. ... 

For second-order point processes such as hard-sphere collisions, the total number of particles remains unchanged. However, the number of particles with a specific phase-space vector will always increase or decrease. 

However, the effect of a small perturbation in action-action-angle type flows is quite different. The two-parameter family of invariant cycles coalesce into invariant tori that are connected by resonant sheets defined by the u(h,l2) = 0 condition. The consequence of this is that contrary to action-angle-angle flows in this case a trajectory can cover the whole phase space and no transport barriers exist. Thus, in this type of flows global uniform mixing can be achieved for arbitrarily small perturbations. This type of resonance induced dispersion has been demonstrated numerically in a low-Reynolds number Couette flow between two rotating spheres by Cartwright et al. 

Figure 9.13 Phase space trajectories and polyad phase spheres for polyad N = 3 of H2O. H2O is an example of a molecule with two identical coupled anharmonic bond stretch oscillators. The local mode (Iz, ip) phase space trajectories in part (a) contain the same information as the normal mode (lz,ip) trajectories in part (b). The corresponding polyad phase spheres in parts (c) and (d) are identical except for a rotation by tt/2 about the y axis (from Xiao and Kellman, 1989).

Figure 9.13 displays the phase space structure of local and Hnormal for the N = vs + va = vr + vl = 3 [I = (N + l)/2 = 2]f polyad of H2O. Just as HloCAL and H qRMAL provide identical quality representations of the observed spectrum, so too do Wlocal and Tinormal- The phase space structures displayed in parts (a), LOCAL, and (b), NORMAL, of Fig. 9.13 are equivalent. The appearance of qualitatively different structures in the LOCAL and NORMAL representations is largely due to the mapping of the information onto an Iz, ip (or Iz, tp) planar rectangle rather than a polyad phase sphere. As shown in parts (c) and (d) of Fig. 9.13, the structures from parts (a) and (b) differ only by a rotation of the phase sphere by 7t/2 about the y axis. 

The fixed points on the phase space diagrams or phase spheres in Fig. 9.13 are labeled A, B, Ca, and C. Each corresponds to a periodic orbit that is said to organize the surrounding region of phase space that is filled with topologically similar quasiperiodic trajectories. 

The central region of the local mode representation of the phase space trajectories is called the resonance region. In this resonance region of phase space the trajectories ( 3 and 4) are not free to explore the full 0 < ip < n range and are threfore classified as normal mode trajectories. Points A and B are fixed points which he at the maximum and minimum E extremes, Ea(I) and Eb(I), of the resonance region for a particular value of I. Point A at Iz = 0 (vr = vi) and ip = 7t/2 (out-of-phase motion of the R and L oscillators) is stable and corresponds to a pure antisymmetric stretch. Point B at Iz = 0 (vr = vl) and ip = 0 and 7r (in-phase motion) is unstable (because it lies on the separatrix) and corresponds to a pure symmetric stretch. Quasiperiodic trajectories that circulate about a stable fixed point resemble the fixed point periodic trajectory. At E > Ea(I) no trajectories of any type can exist. At E < Er(I) the B-like trajectories vanish and are replaced by trajectories that circulate about the Ca, Cb fixed points and are therefore C-like. The Ca and Cb lines (Iz = / = 2, 0 < ip < 7r) are actually the north and south poles on the local mode polyad phase sphere (Fig. 9.13(c)). The stable fixed points he near the poles and trajectories la and 2a circulate about the fixed point near Ca and trajectories lb... 

The normal mode representation of the phase space trajectories contains the same information as the local mode representation. However, the resonance region on the normal mode phase space map contains the local mode trajectories (la, lb 2a, 2b) and the stable fixed points Ca and C t,. The trajectories contained within the resonance zone are not free to explore the entire 0 < tp < n range whereas the trajectories outside the resonance zone do explore the 0 < ip < n range and are therefore classified as normal mode trajectories. The fixed point B (Iz = I = +2) is unstable, because it lies on a separatrix, and is located at the north pole of the normal mode polyad phase sphere. The stable fixed point A (7Z = — I = —2) is located at the south pole. 

Phase space trajectories and polyad phase spheres for polyad N —... 

Fredrickson and Helfand [58] used a selfconsistent Hartree approximation in Eq. (199) to study the Gaussian fluctuations around the solution, Eq, (200). While for ordinary second order transition the local function <4>2(r) > — << >>2 stays small even at Tc, this local fluctuation diverges here as T - Tc The reason is that normally the phase space for critical fluctuations is only the vicinity of a point in reciprocal space (the surroundings of q = 0 for a ferromagnet, the surroundings of a few discrete points qB at the Brillouin zone boundary for antiferromagnets, etc.), while here it is the vicinity of a sphere ( q = q ). Fluctuations lead here to a divergence of the mean square displacement of 4> similar as it happens due to phonons in one-dimensional crystals. 

Figure 6.1 The three-dimensional translational phase space represented as the volume of the sphere with the momentum as the radius. The surface of the sphere is related to the...

Locally, the 2d — l)-dimensional energy surface E(E) has the structure of X R, i.e., fhe Cartesian product of a 2d — 2)-dimensional sphere and a line, in fhe 2rf-dimensional phase space. The energy surface S(E) is splif locally info two components, "reactants" and "products," by a (2d — 2)-dimensional "dividing surface" fhaf is diffeomorphic to and which we therefore denofe by S (E). The dividing surface that we construct has the following properties ... 

First, consider the case of trapped motion within a single isomer. The phase space of 2 is (always) an ellipse, which has the same topology as a onedimensional sphere (which a mathematician would name S ). However, the phase space of is also elliptical and has the same topology (S ). The topology of the two-dimensional phase-space surface on which the dynamics lies is the Cartesian product of these two, which is a two-dimensional torus, or a phase-space doughnut (T = SI X The toroidal geometry is shown in... 

Next considering the case of reactive motion at the same energy E, we realize that the situation at hand is not very different. The phase space of qi is still elliptical. The phase space of is not elliptical, but it is a simple closed curve, and it still therefore has the same topology as a one-dimensional sphere (every point on a closed curve can be uniquely mapped onto a sphere). Thus, the phase space of reactive motion consists of foliated tori that span both sides of the potential barrier. These reactive tori will be skinny when sliced along the ( 2 Pz) compared to the trapped tori, because they have less energy in the vibrational coordinate and more in the reaction coordinate. In Figure 8 these are labeled Qab j... 

---