Asymptotic state

Many, possibly all, rules appear to generate asymptotic states which are block-related to configurations evolving according to one of only a small subset of the set of all rules, members of which are left invariant under all block transformations. That is, the infinite time behavior appears to be determined by evolution towards fixed point rule behavior, and the statistical properties of all CA rules can then, in principle, be determined directly from the appropriate block transformations necessary to reach a particular fixed point rule. 

Although it is certainly not immediately obvious from the rule itself, it turns out that, just as is the case for generalized threshold rules, the only possible asymptotic states of finite symmetric muib -threshold rules are either fixed points or cycles of period two [golesQO]. Unlike their binary brethren, however, multi-threshold rules possess some intriguing additional properties. 

If the turbulent flame is ever proven to have asymptotically a constant flame brush thickness and constant speed in constant, i.e., nondecaying, turbulence, then the aforementioned turbulent flame speed and the flame brush thickness (5 give a well-defined sufficient characterization of the flame in its asymptotic behavior. However, it is not proven up to now that the studied experimental devices have been large enough to ensure that this asymptotic state can be reached. Besides, the correct definitions for the turbulent flame speed or flame brush thickness, as given above, are far from... 

A theoretical framework for considering how the behavior of dynamical systems change as some parameter of the system is altered. Poincare first apphed the term bifurcation for the splitting of asymptotic states of a dynamical system. A bifurcation is a period-doubling, -quadrupling, etc., that precede the onset of chaos and represent the sudden appearance of a qualitatively different behavior as some parameter is varied. Bifurcations come in four basic varieties flip bifurcations, fold bifurcations, pitchfork bifurcations, and transcritical bifurcations. In principle, bifurcation theory allows one to understand qualitative changes of a system change to, or from, an equilibrium, periodic, or chaotic state. 

The situation is not so simple with HLSP functions. They do not have the antisymmetry characteristic mentioned above, and the asymptotic state requires a sum of three of them as shown in Table 11.14. 

Table 16.9. The leading Rumer tableaux for the asymptotic state offormaldehyde, H2 + CO. In this case the standard tableaux functions are the same for these terms.

The concept of correlation enters via the inclusion of asymptotic base states. Translated into the particle model, one type of asymptotic state is given by a doubly occupancy of an anti-bonding state the fragments show a repulsive... 

The systems considered here are isothermal and at mechanical equilibrium but open to exchanges of matter. Hydrodynamic motion such as convection are not considered. Inside the volume V of Fig. 8, N chemical species may react and diffuse. The exchanges of matter with the environment are controlled through the boundary conditions maintained on the surface S. It should be emphasized that the consideration of a bounded medium is essential. In an unbounded medium, chemical reactions and diffusion are not coupled in the same way and the convergence in time toward a well-defined and asymptotic state is generally not ensured. Conversely, some regimes that exist in an unbounded medium can only be transient in bounded systems. We approximate diffusion by Fick s law, although this simplification is not essential. As a result, the concentration of chemicals Xt (i = 1,2,..., r with r sN) will obey equations of the form... 

These questions are irrelevant for scattering which may be described in terms of asymptotic states corresponding to t -> + oo. 

Again the situation is much simpler when only asymptotic states containing stable particles are considered. Then unstable particles enter neither into the completeness relation nor into the unitary relations of the theory.5 However, in the intermediate states unstable particles may appear. They manifest themselves as poles exactly as in Eq, (16). We may then describe such poles by various approximate formulas of the Breit-Wigner type. But again this approach is severely limited. By definition we have to exclude the production or destruction processes involving unstable particles. It is even not easily seen how this can be done in a consistent manner. 

Of course, one may try an even more phenomenological approach in which unstable particles are introduced as well in the asymptotic states. In his recent book Barut3 goes so far as to write There is one S-matrix for the whole universe we shall look at individual elements and submatrices of this one continuously infinite matrix. For a human being that seems quite an assertion as our average lifetime does not permit us to take an asymptotic point of view in respect to the whole history of the universe. 

Many problems appear to be ripe for a more quantitative discussion. What is the error involved in the introduction of unstable states as asymptotic states in the frame of the 5-matrix theory 16 What is the role of dissipation in mass symmetry breaking What is the consequence of the new definition of physical states for conservation theorems and invariance properties We hope to report soon about these problems. We would like, however, to conclude this report with some general remarks about the relation between field description and particles. The full dynamical description, as given by the density matrix, involves both p0 and the correlations pv. However, the particle description is expressed in terms of p (see Eq. (50)). Now p has only as many elements as p0. Therefore the... 

In order to work with the on-shell (physical) scattering T matrix, we must consider the kinetic equation in the space of the asymptotic states which is the direct sum of the channel subspaces In this space we have the following completeness relation... 

The initial and final asymptotic states are always expanded in the time independent basis associated with the molecular hamiltonian the scattering matrix is unitary. Note again that the basis contains all possible resonance and compound states. If there is no interaction, the scattering matrix is the unit matrix 1. Formally, one can write this matrix as S= 1+iT where T is an operator describing the non-zero scattering events including chemical reactions. Thus, for a system prepared in the initial state Op, the probability amplitude to get the system in the... 

Control of bimolecular collisions is achieved by constructing an initial state E, q, am ) composed of a superposition of N energetically degenerate asymptotic states E, q, m 0) ... 

The Fence description is completed with the asymptotic states of the EM field k lB), where the emission source is located at the origin of the k-vector that also identifies a direction k/ k in reciprocal space. Thus, conventionally, IT jl-kMa) signals an EM field in a direction pointing to the I-frame sustaining IT ) whereas ITqi+k 1 ) stands for one photon emission from the I-frame. These base states supplement a real-space (Fence) description. 

Two (or more) noninteracting quantum states each associated with a corresponding I-frame so that in laboratory space they are located at will in an experiment these are asymptotic states. 

Quantum states for systems type (1) and (2) are not commensurate. I-frames belong to laboratory space, and consequently, asymptotic states evolve in real space separately, whereas the one-I-frame states evolve in Hilbert space following the I-frame motion, the internal quantum states are not changed unless real-space interaction sources are allowed for. 

The quantum states for the one-I-frame system involve a nonseparable materiality these states should not include I-frame-related asymptotic states. 

Asymptotic states are simple products labeled with box (nk) and internal U-km) quantum numbers. For example, nn,knl) > nm,Xmi.), etc. 

All sorts of asymptotic states together with the one-I-frame system quantum states cover all possibilities meaning is that all thinkable processes can be described as changes of quantum states using as base states simple products, and I-frame base states provided the total number of material elements are conserved. Basically, a sort of abstract quantum chemistry emerges if base states can be related to those characterizing chemical species. 

Consider an entangled base state m,Kn) and an asymptotic state n,Xm) ffm, mb>, the quantum state is the linear superposition ... 

The box-Hilbert space is now rigged with the asymptotic states that can be probed at the laboratory as if they were space separate quantum objects. 

---