Phase space dynamics

2 Phase Space Dynamics Phase space dynamics describes a methodology for defining the optimal set of external forces for transferring a charged particle beam between two points in space. 

The phase space of interest refers to the particle s position in space and its momentum. As momentum can be described by classical statistical mechanics, equations of motion can be expressed in terms of the particle s position in space and its momentum, hence the term Phase Space Dynamics. Considering that an ion beam is composed of a large number of charged particles, it then follows that the optical properties of the beam can be described as a collection of such parameters. 

The parameters include the particle s position (x, y, and z) with respect to the axis of motion and the particle s momentum py, and p ) in each of the three spatial dimensions. The boundaries of the phase space, which can otherwise be considered the walls of a balloon containing a fixed amount of water, are defined by the hypervolume bounded by the maximum displacement and momentum of the ions within the charged particle beam at any point in space. 

Optimizing the transmission of such a charged particle beam is then simply a case of optimizing the phase space shape at that point in space (this is referred to as the beam s emittance) with the various obstacles the beam may experience (this is referred to as the beam s acceptance). This matching procedure is otherwise referred to as emittance-acceptance matching. 

---

A situation that arises from the intramolecular dynamics of A and completely distinct from apparent non-RRKM behaviour is intrinsic non-RRKM behaviour [9], By this, it is meant that A has a non-random P(t) even if the internal vibrational states of A are prepared randomly. This situation arises when transitions between individual molecular vibrational/rotational states are slower than transitions leading to products. As a result, the vibrational states do not have equal dissociation probabilities. In tenns of classical phase space dynamics, slow transitions between the states occur when the reactant phase space is metrically decomposable [13,14] on the timescale of the imimolecular reaction and there is at least one bottleneck [9] in the molecular phase space other than the one defining the transition state. An intrinsic non-RRKM molecule decays non-exponentially with a time-dependent unimolecular rate constant or exponentially with a rate constant different from that of RRKM theory. 

Consider a quantity of some liquid, say, a drop of water, that is composed of N individual molecules. To describe the geometry of this system if we assume the molecules are rigid, each molecule must be described by six numbers three to give its position and three to describe its rotational orientation. This 6N-dimensional space is called phase space. Dynamical calculations must additionally maintain a list of velocities. 

In the analysis of the periodic orbits, it is convenient to first study the edge periodic orbits of relevant 1F, 2F, or 3F subsystems rather than to start with the bulk periodic orbits for which all the F = 4 degrees of freedom are active. The amplitudes of the edge orbits are significantly smaller than those of the bulk periodic orbits, as explained in Section II. Nevertheless, they help to localize the bulk periodic orbits because the edge periodic orbits form a skeleton for the bulk phase-space dynamics in integrable systems. 

Generally speaking, the degrees of freedom in many-body systems, such as Ar7, are too many to analyze the phase-space dynamics, and only limited methods originally developed to investigate chaotic systems with a few degrees of freedom can be applicable for the analysis. Seko et al. calculated the phase volume—that is, the configuration entropy—of Ar7 and proposed a new concept of the temperature in micro-clusters based on this phase volume [17], A phase-space analysis seems to be prospective even for many-body systems, such as Ar7. However, most of the currently available methods concern statistical properties. The methods and quantities that are directly related to the dynamics are expected for a detailed analysis. 

The Wigner-Weyl formulation of quantum mechanics provides an appropriate quantum analogue to classical phase space dynamics and consists of a particular representation of quantum mechanics. 

Armed as we are now with the KAM theorem, the Center Manifold theorem, and the Stable Manifold theorem, we can begin to visualize the phase space of reaction dynamics. Returning to our original system (see Uncoupled Reaction Dynamics in Two Degrees of Freedom ), we now realize that the periodic orbit that sews together the half-tori to make up the separatrix is a hyperbolic periodic orbit, and it is not a fixed point of reflection. From our previous visualization of uncoupled phase-space dynamics, we know that the separatrix is completely nontwisted. In the terminology of Poliak and Pechukas, the hyperbolic periodic orbit is a repulsive PODS. ... 

M. J. Davis,. Chem. Phys., 86, 3978 (1987). Phase Space Dynamics of Bimolecular Reactions and the Breakdown of Transition State Theory. 

II. Classical Phase Space Dynamics for vdW Systems with Three Degrees of Freedom... 

Finally, we would like to elaborate the proposed protocol of the high-friction map, eqn (13.17). Its construction is based purely on the thermodynamic consideration, eqn (13.15), validated by the central limit theorem. Therefore it may offer a general rule to obtain the Smoluchowski limit to any phase-space dynamics under study. The protocol proposed in this chapter is based on the fact that the map is universal at formal level and is therefore obtainable with thermodynamic consideration. It means the Smoluchowski dynamics can be taken care of by the related Fokker-Planck equation, upon the universal map being carried out. It is worth pointing out that the resultant diffusion operator in eqn (13.18) clearly originates from only the Hamiltonian part of the... 

Phase Space Dynamics and Energy Levels, Classical/ Quantum Correspondence, and RRKM Theory... 

There is a good correspondence between the above classical description of the phase space dynamics and quantum dynamics. Classically there is an increase in the fraction of trajectories which are chaotic with increase in energy, while quantum mechanically all the states are assignable at low energy, then... 

Quadrupole performance slowly improved but in the 1970s new applications to organic analysis and particularly the implementation of the combination of GC and MS placed increasing emphasis on better understanding of how the instruments worked in order to overcome their limitations. Computer simulations of performance became important. Combined with detailed experimental analysis, these led to a much improved knowledge of real-world quadru-poles with fringing fields and field imperfections. An important advance came with the application of phase space dynamics for calculating quadrupole performance. The new advances were incorporated in the classic textbook of the field written by Dawson and various collaborators and published in 1976. This book was re-issued by the American Institute of Physics in 1995 as a paper-back classic . 

The finite diameter of the field (r ) means that ions are only accepted for transmission when they enter the field with a small initial transverse displacement from the axis and small transverse velocities. The combination of displacement and transverse velocities that are possible defines the acceptance of the instrument. The acceptance and so the overall sensitivity becomes smaller as the resolution is increased. The acceptance is calculated using phase space dynamics. 

---