State space curve

Fig. 13a and b. Intensity contour maps around the 5.9-nm and 5.1-nm actin layer lines (indicated by arrows) a resting state b contracting state. Z is the reciprocal-space axial coordinate from the equator. M5 to M9 are myosin meridional reflections indexed to the fifth to ninth orders of a 42.9-nm repeat, (c) intensity profiles (in arbitrary units) of the 5.9- and 5.1-nm actin reflections. Dashed curves, resting state solid curves, contracting state. Intensity distributions were measured by scanning the intensity data perpendicular to the layer lines at intervals of 0.4 mm. The area of the peak above the background was adopted as an integrated intensity and plotted as a function of the reciprocal coordinate (R) from the meridian... 

Figure 1.2 illustrates the difference between the transitions involved in van der Waals dimer bands which Welsh and associates hoped to find, and the collision-induced absorption spectra that were discovered instead. Intermolecular interaction is known to be repulsive at near range and attractive at more distant range. As a consequence, a potential well exists which for most molecular pairs is substantial enough to support bound states. Such a bound state is indicated in Fig. 1.2 (solid curve b). When infrared radiation of a suitable frequency is present, the dimer may undergo various transitions from the initial state (solid curve) to a final state which may have a rather similar interaction potential (dashed curve b ) and dimer level spacings. Such transitions (marked bound-bound) often involve a change of the rotovibrational state(s) EVj of one or both molecule(s),... 

DAE case. Even at a = 20, the steady-state multiplicity curves (Regenass and Aris, 1965) are not very close to those of a single component (a = < ). The region of multiplicity seems to increase as a increases, that is, in parameter space the single component case is the most likely to result in multiplicity. 

To examine gle at elevated pressures we return to Figure 4.10. At T, which is below the critical temperature of both A and B, the bubble- and dew-point curves both start at the low-pressure end at the vapor pressure of B, which is the less volatile component. At higher pressure, the two curves diverge and finally both end at the vapor pressure of A, the more volatile component. At T2, which is between the critical temperatures of A and B, the bubble- and dew-point curves at the low-pressure end stiU start at the vapor pressure of B, but at the higher-pressure side they meet at the mixture critical state. The critical points of mixtures of varying composition form the mixture critical loci, a space curve that connects the critical states of A and B. 

In physics such oscillatory objects are denoted as self-sustained oscillators. Mathematically, such an oscillator is described by an autonomous (i.e., without an explicit time dependence) nonlinear dynamical system. It differs both from linear oscillators (which, if a damping is present, can oscillate only due to external forcing) and from nonlinear energy conserving systems, whose dynamics essentially depends on initial state. Dynamics of oscillators is typically described in the phase (state) space. Periodic oscillations, like those of the clock, correspond to a closed attractive curve in the phase space, called the limit cycle. The limit cycle is a simple attractor, in contrast to a strange (chaotic) attractor. The latter is a geometrical image of chaotic self-sustained oscillations. 

Fig. 11.1 Two-dimensional bifurcation diagram calculated by continuation from the Citri-Epstein mechanism for the chlorite-iodide system. Plot of 2D space of constraints is chosen to be the ratio of input concentrations, [CIO ]o/[I ]o> versus the logarithm of the reciprocal resideuce time, logfco-Notation SSI, SS2, region of steady states with high [1 ] and low [I ], respectively Osc, region of periodic oscillations Exc, region of excitahihty snl, sn2, curves of saddle-node bifurcations of steady states hp, curve of Hopf bifiucatious sup, ciuve of saddle-node bifurcations of periodic orbits swt, swallow tail (a small area of tristability) C, cusp point TB, Takens-Bogdanov point (terminus of hp on snl). (From [5].)...

Spiegelmann and Malrieu have proposed improved versions of this procedure,where the model space is spanned by several multi-configurational zeroth-order descriptions of the various diabatic eigenfunctions. This proposal, which fits very well the architecture of the CIPSI algorithm, has received applications on Arf excited states, NaCl curve crossing, HeNe and the Cs (7p) -i- HjfX Sg ) CsH(X Zg ) + H reactive collision. ... 

Figure 15.4 Phase portraits of the chlorite-iodide system with pH = 1.56, [r]o = 1.8 X 10 M, [CIOJlo = 5.0 x lO" M. Points with Roman numerals are steady states, closed curves are limit cycles. Arrows indicate how the system evolves in concentration space. Row A. shows evolution of the system without piemixing at low stirring rate (< 550 rpm) as flow rate increases Row B shows intermediate stirring rate without premixing Row C show high stirring rate or premixing. (Reprinted with permis.sion from Luo, Y. Epstein, I. R. Stirring and Premixing Eflects in the Oscillatory Chlorite Iodide Reaction," J. Chem. Phys. 85, 5733 5740. it) 1991 American Institute of Physics.)...

The low-frequency period error remains nearly unaffected unaffected by the extended state-space formulation of the damping as illustrated by the fully drawn curve in Fig. 2b. In fact, the asymptotic behavior is given by (20) also in this case. 

This section is devoted to a reconsideration of system (3.2.2), to obtain the frequency change as a power series in e. A slightly careful examination of Method I reveals that a major obstacle to systematic perturbation expansions lies in the fact that the surfaces of constant phase are generally curved in state space. Since the definition of the surfaces of constant phase is entirely at our disposal. 

If a fluid system contains a single substance in a single phase, its equilibrium state can be specified by the values of three variables such as T, V, and . We can define a three-dimensional space in which T, V, and n are plotted on the three axes. We call such mathematical space a state space. Each equilibrium state is represented by a state point located in the state space. A reversible process proceeds infinitely slowly, so that the system has sufficient time to come to equilibrium during any part of the process. The system passes through a succession of equilibrium states and the state point fiaces out a curve in the equilibrium state space, such as that shown in Figure 2.3 for a fixed value of n. 

Because the integrand in this integral depends on T and V we must specify how T depends on V in order to carry out the integration. This dependence corresponds to a curve in state space, and the letter c under the integral stands for this curve. This kind of integral is called a line integral. 

Equation (2.4-21) can be used for a reversible adiabatic compression as well as for an expansion. It is an example of an important fact that holds for any system, not just an ideal gas For a reversible adiabatic process in a simple system the final temperature is a function of the final volume for a given initial state. All of the possible final state points for reversible adiabatic processes starting at a given initial state lie on a single curve in the state space, called a reversible adiabat. This fact will be important in our discussion of the second law of thermodynamics in Chapter 3. 

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