Action variables

Wirkungs-losigkeit, /. inactivity, ineffectiveness, inefficiency, -moglichkeit, /. possible effect, -quant, -quantum, n. quantum of action, effective quantum, -querschnitt, m. effective cross section, -sphkre, /. sphere of action, -variabel, n. action variable, -ver mSgen, n. power of action, working power. 

It is now shown how the abrupt changes in the eigenvalue distribution around the central critical point relate to changes in the classical mechanics, bearing in mind that the analog of quantization in classical mechanics is a transformation of the Hamiltonian from a representation in the variables pR, p, R, 0) to one in angle-action variables (/, /e, Qr, 0) such that the transformed Hamiltonian depends only on the actions 1r, /e) [37]. Hamilton s equations diR/dt = (0///00 j), etc.) then show that the actions are constants of the motion, which are related to the quantum numbers by the Bohr correspondence principle [23]. In the present case,... 

Instead of the reactive-mode energy ), it is convenient to study the associated action variable... 

Just as K, the Hamiltonian // depends on APX and AQ only through the action variable 7, which is a constant of the motion. 

One can therefore transform away a resonance condition by going over to a new set of n action variables, n— 1 of which are conserved. Having eliminated a primary resonance [i.e., one having a large coefficient Vm in Eq. (3.29)], one can eliminate the next one in turn, etc. Of course, with each additional term in Eq. (3.29) that is included in the Hamiltonian, a larger range of actions become accessible to the dynamics. The books cited in Note 2 will all provide more details on this point. 

The point made in Eq. (3.31), namely, that the coupled, old, n action variables can be transformed to new, uncoupled, n - 1 conserved action variables is one to which we shall repeatedly return, in the quantum-algebraic context, in Chapters 4—6. Of course, we shall first discuss H0, which has n good quantum numbers, and which we shall call a Hamiltonian with a dynamical symmetry. At the next order of refinement we shall introduce coupling terms that will break the full symmetry but that will still retain some symmetry so that new, good, but fewer quantum numbers can still be exactly defined. In particular, we shall see that this can be done in a very systematic and sequential fashion, thereby establishing a hierarchy of sets of good quantum numbers, each successive set having fewer members. 

Under these conditions, the distribution of the action variables (e.g., the momenta) [the vacuum, pg] tends irreversibly toward the thermodynamic equilibrium after a sufficiently long time. Under the same conditions, the correlations are determined by the vacuum (technically, they become functionals of the vacuum distribution pg) (see Appendix). 

They would become the stars of Prigoginian statistical mechanics. Their importance lies in the fact that, whenever it is possible to determine these variables by a canonical transformation of the initial phase space variables, one obtains a description with the following properties. The action variables / ( = 1,2,..., N, where N is the number of degrees of freedom of the system) are invariants of motion, whereas the angles a increase linearly in time, with frequencies generally action-dependent. The integration of the equations... 

So far we have considered quantum systems. Similar considerations can be applied to classical systems and, in particular, to the classical Friedrichs model [13, 19]. In the A representation we have flucmations. We can define transformed action variables like in the quanmm case. 

In Ref. 13, we have proved that the A transformation constructed is invertible for the classical model discussed in the previous section. Here, using the same system discussed in the previous section, we demonstrate the invertiblity of our transformation by a numerical calculation of the time evolution of the action variable J (f) for an initial condition where all the field actions are zero [20]. Due to radiation damping, J t) follows an approximately exponential decay. However, there are deviations from exponential in the exact evolution both at short and long time scales as compared with the relaxation time scale. In Fig. 1, we present numerical results. 

In integrable systems, the periodic orbits are not isolated but form continuous families, which are associated with so-called resonant tori. In action-angle variables, the Hamiltonian depends only on the action variables, similar to the Dunham expansion, ... 

The periodic orbits (2.31) are referred to as bulk periodic orbits in the sense that all the F actions are nonvanishing. Therefore, all the F degrees of freedom are excited in this periodic motion. On the other hand, there exist edge periodic orbits in the subsystems in which one or several action variables vanish (see Fig. 1). These subsystems have a lower number of excited degrees of freedom, but their periodic orbits also contribute to the trace formula. However, they have smaller amplitudes, related to the amplitude of the bulk periodic orbits as... 

The mass action variable X can be rendered into an experimentally more meaningful form. To do this, let X be a reference value for given values of the concentration and temperature T. Then, by a linear Taylor expansion of the dimensionless free energy f3g around the reference temperature T, we have... 

According to Equation (3), the mass action variable X/X depends strongly, that is, exponentially, on the temperature. Depending on whether the assembly is endo or exothermic, that is, whether ht /0,... 

Figure 3 Fraction of material in the polymerized state, f, as a function of the mass action variable X relative to its value X at the half-way point f =1/2. Indicated are predictions for the isodesmic and the self-catalyzed nucleated polymerization models. Activation constant of the nucleated polymerization Ka = 10-4.

Figure 8 Dimensionless relaxation rate l/pf, as a function of the dimensionless mass action variable X according the kinetic Landau model discussed in the main text, with I, the nucleation rate. Shown is the prediction in the limit where the nucleation reaction is rate limiting. Inset experimental results from measurements on actin (Attri et al., 1991). Notice the zero growth atX = l, the critical polymerization point.)...

Either type of periodic motion can be described by the variable J, designed to replace o, as the constant (transformed) momentum. This action variable is defined as... 

The Kepler model was ceased upon by Sommerfeld to account for the quantized orbits and energies of the Bohr atomic model. By replacing the continuous range of classical action variables, restricting them to discrete values of... 

The classical counterpart of v is the action variable /, which equals fpdq, where the integral is over one vibrational cycle of the vibration. In old quantum theory (or, later, the WKB theory), / is related to v by / = (u + l 2)h. Thus, it occurred to me that the above lowering of the energy barrier for the motion along the reaction coordinates could be rewritten as l(v— v ) and so the results of Wall imply that the classical vibrational action I was constant along the reaction coordinate in this system. 

It is assumed that the tendency of a molecular mixture to interact can be analyzed as a function of the chemical (quantum) potential energy field and some action variable that reflects mass ratios or amounts of substance. Spontaneous chemical change occurs as the chemical potential of a system decreases, i.e. while Ap < 0, and ceases when Ap = 0, at equilibrium. The quantity here denoted by Ap, also known as the affinity, a of the system, is the sum over all molecules, reactants and products... 

Figure 1. (a) Angle-action variables (4)i,, /2) the invariant torus for a two-oscillator... 

Suppose that Hamiltonian 7/(p, q) is expressed in a region around a saddle point of interest as an expansion in a small parameter e, so that the zero-order Hamiltonian Hq is regular in that region specifically, it is written as a sum of harmonic-oscillator Hamiltonians. Such a zero-order system is a function of action variables J of Hq only, and it does not depend on the conjugate angle variables 0. The higher-order terms of the Hamiltonian are expressed as sums of... 

Once the Hamiltonian has been transformed into normal form, the quantization of the nonreactive DOFs is straightforward. While complications can occur, the present example is free of the worst of these. The vibrational modes normal to the reactive coordinate are not in resonance. Consequently, the quantization is straightforward and accomplished by quantizing the classical action variables... 

This means that the unperturbed motion of jc may depend on the action variables of the vibrational modes. Thus, we do not need a separable Hamiltonian as the unperturbed part. The second term 77q describes vibrational motions, which are integrable and, in general, nonlinear. 

As for the dynamics of JC under the unperturbed Hamiltonian Hq x,I), we assume that the reaction coordinate jc has a saddle X I) = (Q I),P I)). Its location, in general, depends on the action variables 7. Suppose that the saddle X I) has a separatrix orbit JCo(t,7) connecting it with itself. See Fig. 9 for a schematic picture of the phase space jc = (q,p) under the unperturbed Hamiltonian Hq x,I). Here, we show the saddle X and the separatrix orbit on the two-dimensional phase space jc = (q,p). 

Fig. 10 for an example where the potential explicitly depends on / as V q, I). Reaction processes involving enzymes can correspond to such cases. These reactions take place within a specific range of temperature. This temperature dependence results from the dependence of the saddle on I, where the height of the saddle decreases for a specific range of the action values of the vibrational modes. Then, the reaction proceeds at those temperatures where the action variables are excited specifically to these values. 

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