State time variation

There are many examples in nature where a system is not in equilibrium and is evolving in time towards a thennodynamic equilibrium state. (There are also instances where non-equilibrium and time variation appear to be a persistent feature. These include chaos, oscillations and strange attractors. Such phenomena are not considered here.)... 

Throughout this chapter many of the arguments are based on an assumption of steady state. Before the agricultural and industrial revolutions, the carbon cycle presumably was in a quasi-balanced state. Natural variations still occur in this unperturbed environment the Little Ice Age, 300-400 years ago, may have influenced the carbon cycle. The production rate of varies on time scales of decades and centuries (Stuiver and Quay, 1980,1981), implying that the pre-industrial radiocarbon distribution may not have been in steady state. 

MIXED STATE TIME-DEPENDENT VARIATIONAL PRINCIPLE... 

The critical points of the equivalent classical Hamiltonian occur at stationary state energies of the quantum Hamiltonian H and correspond to stationary states in both the quantum and generalized classical pictures. These points are characterized by the constrained generalized eigenvalue equation obtained by setting the time variation to zero in Eq. (4.17)... 

Time variations of maximum flame temperature for Flames I-IV. The inset shows the steady-state flame response for hydrogen/air mixture of 0 = 7.0. Results demonstrate that Flame 1 is dynamically stable. Flame II is monochromatically oscillatory. Flame III exhibits pulsation with period doubling, and Flame IV is extinguished through pulsation. 

Figure 36. Time variation of the wave packet population on the ground X and excited B states of LiH. The system is excited by a single quadratically chirped pulse with parameters 0(a, = 5.84 X 10 eV fs , = 2.319 eV, and / = 1.00 TWcm . The pulse is centered at t = 0...

Under steady-state conditions, variations with respect to time are eliminated and the steady-state model can now be formulated in terms of the one remaining independent variable, length or distance. In many cases, the model equations now result as simultaneous first-order differential equations, for which solution is straightforward. Simulation examples of this type are the steady-state tubular reactor models TUBE and TUBED, TUBTANK, ANHYD, BENZHYD and NITRO. 

Finally, we attack the problem of the transport coefficients, which, by definition, are calculated in the stationary or quasi-stationary state. The variation of the distribution functions during the time rc is consequently rigorously nil, which allows us to calculate these coefficients from more simple quantities than the generalized Boltzmann operators which we call asymptotic cross-sections or transport operators. 

We now introduce the concept of the control parameter X (see Section III. A). In the present scheme the discrete time sequence Xk Q transition probability Wt(C C) now depends explicitly on time through the value of an external time-dependent parameter X. The parameter Xk may indicate any sort of externally controlled variable that determines the state of the system, for instance, the value of the external magnetic field applied on a magnetic system, the value of the mechanical force applied to the ends of a molecule, the position of a piston containing a gas, or the concentrations of ATP and ADP in a molecular reaction coupled to hydrolysis (see Fig. 3). The time variation of the control parameter, X = - Xk)/At, is... 

Adiabatic passage schemes are particularly suited to control population transfer between states, since the adiabatic following condition assesses the stability of the dynamics to fluctuations in the pulse duration and intensity [3]. The time evolution of the wave function does not depend on the dynamical phase, and is therefore slow in comparison with the vibrational time scale. This fact guarantees that the time variation of the observables during the controlled dynamics will be slow. Adiabatic methods can therefore be of great utility to control dynamic observables that do not commute with the Hamiltonian. We are interested in the control of the bond length of a diatomic molecule [4]. 

For steady-state (no time variation) two-dimensional flows, the notion of a streamfunction has great utility. The stream function is derived so as to satisfy the continuity equation exactly. In cylindrical coordinates, there are two two-dimensional situations that are worthwhile to investigate the r-z plane, called axisymmetric coordinates, and the r-0 plane, called polar coordinates. 

For steady-state analysis (i.e., no time variation) the coupled system is essentially elliptic, with some hyperbolic characteristics. The continuity equation alone is clearly hyperbolic, having only first-order derivatives. That is, it carries information about velocity from an inlet boundary, across a domain, to an outlet boundary. By itself, the continuity equation has no way to communicate information at the at the outlet boundary back into the domain. Based on the second-derivative viscous terms, the momentum equation is elliptic in velocity. However, because it is first order in pressure, there is also a hyperbolic character to the momentum equation. Moreover the convective terms have a hyperbolic character. There are situations, for example in high-speed flow, where the viscous terms diminish or even vanish in importance. As this happens, and the second-derivative terms become insignificant relative to the first-derivative terms, the systems changes characteristics to hyperbolic. 

Viewed in the time domain, the replacement of M(a>) by M washes out the details of the time variation within Q space. For this approximation to be useful, all strongly coupled states should be included in the P space and the Q space should not include any states that couple strongly to the P space (weak coupling assumption). We now find that the population dynamics of the m levels within the P space is governed by the equations of motion... 

If we now imagine that AE varies in time, but slowly, its only effect is to cause a time variation of the energy of the n state. We assume that the spatial wavefunction is unaffected by AE and that no transitions occur. This approximation is the adiabatic approximation of Autler and Townes.11 Now let us consider the time variation of AE of particular interest to us, Emw cos cot. If the assumptions stated above are valid, we can use the energy of Eq. (15.6) as the unperturbed Hamiltonian H0 in the Schroedinger equation. Explicitly,... 

Fig. 22 shows the results of photometry of plates similar to that illustrated in Fig. 21. The relative intensities of suitable transitions were determined from the asymptotic limit at long time delays when the system attains equilibrium. (These resemble, but are not identical to, the relative/ values because of the usual instrumental effects which depend on line width.) The time variation of the relative concentrations is shown in Fig. 23 the upper four levels attain Boltzmann equilibrium amongst themselves after 100 /isec, to form a coupled (by collision) system overpopulated with respect to the 5DA state. The equilibration of the upper four levels causes the initial rise (Fig. 22) in the population of Fe(a5D3). Thus relaxation amongst the sub-levels is formally similar to vibrational relaxation in most polyatomic molecules, in which excitation to the first vibrational level is the rate determining step. In both cases, this result is due to the translational overlap term, for example, in the simple form of equation (14) of Section 3.

Let us perform this study like that carried out for the adsorption mechanism. We will analyze time variations in the solutions of the unsteady-state model (2)-(3). Typical phase patterns are represented in Fig. 13. The heavy closed lines are two isochrones [in this case they are geometric sites of the... 

The simplest example of how the adsorption system non-ideality affects the number of its elementary stages has been discussed in Ref. [132] (see also Ref. [89]). The stages (a = 1) A + Z <-> ZA, and (a = 2) B + ZA- Z+C, in the absence of interaction involves one stationary state. The kinetic equation describing time variations of the surface coverage with the adspecies A is written down as follows ... 

Similarly, in the glassy state the variation of relaxation time with inverse temperature is... 

A time variation in inductance arises because the permeability diminishes after the establishment of a fresh magnetic state this effect is known as disaccommodation and is illustrated in Fig. 9.47. 

Fig. 5.6. Time variations of the wavepacket populations in the X1 X 1 and /111 states of NaK. The system is excited by a quadratically chirped pulse with parameters = 3.13 x 102eV/fs2, plo = 1.76eV and / = 0.20TW/cm2. The pulse is...

On the (assumed) much longer time scale over which SeOj and Mn2+ begin to appear in the aqueous-solution phase from the decomposition of = Mn" - 0Se020H, Eqs. 4.52c-4.52e can be solved under an appropriate imposed condition regarding the time variation of [=MnM - 0SeO2OH] based on the surface oxidation-reduction kinetics. For example, under steady-state conditions that yield constant concentrations of the adsorbed and dissolved selenite species, Eqs. 4.52a and 4.52b lead to a constant concentration of adsorbed selenate and therefore a constant rate of selenate detachment from the mineral surface (Eq. 4.52c). If the reasonable assumption is also made that the proton reaction with =MnH - OH equilibrates rapidly, then... 

We have investigated the transitions among the types of oscillations which occur with the Belousov-Zhabotinskii reaction in a CSTR. There is a sequence of well-defined, reproducible oscillatory states with variations of the residence time [5]. Similar transitions can also occur with variation of some other parameter such as temperature or feed concentration. Most of the oscillations are periodic but chaotic behavior has been observed in three reproducible bands. The chaos is an irregular mixture of the periodic oscillations which bound it e.g., between periodic two peak oscillations and periodic three peak oscillations, chaotic behavior can occur which is an irregular mixture of two and three peaks. More recently Roux, Turner et. al. 

Figure 24. Autothermal reaction control with direct (regenerative) heat exchange for an irreversible reaction [14], A) Basic arrangement B) Local concentration and temperature profiles prior to flow reversal in steady state C) Variation of outlet temperature with time in steady state.

---