Linear time dependence

Complex systems can often be represented by linear time-dependent differential equations. These can conveniently be converted to algebraic form using Laplace transformation and have found use in the analysis of dynamic systems (e.g., Coughanowr and Koppel, 1965, Stephanopolous, 1984 and Luyben, 1990). 

Case n transport is an interesting special case of sorption because the linear time dependence of the relaxation process means that a constant swelling rate could be observed [119,121,128,129,132-135], Mechanistically, case II transport... 

Figure 2 Probability that the quantum of vibrational excitation is on the same molecule at time t that it was on at time zero for a cluster of 20 Br2 molecules. Time is inpicoseconds. The two solid curves are results from two different runs, with the difference rejecting the statistical uncertainly in the calculations. The dotted straight line is drawn to indicate the region of linear time dependence.

The results of the calculations shown in Fig. 2.32 represent a complete quantitative solution of the problem, because they show the decrease in the induction period in non-isothermal curing when there is a temperature increase due to heat dissipation in the flow of the reactive mass. The case where = 0 is of particular interest. It is related to the experimental observation that shear stress is almost constant in the range t < t. In this situation the temperature dependence of the viscosity of the reactive mass can be neglected because of low values of the apparent activation energy of viscous flow E, and Eq. (2.73) leads to a linear time dependence of temperature ... 

Given that Eq. 6.1 (with D 2) applies to reaction-controlled flocculation kinetics, Eq. 6.54 implies that MM(t) [or MN(t)] must also exhibit an exponential growth with time. Therefore, by contrast with transport-controlled flocculation kinetics, a uniform value of the rate constant kmn cannot be introduced into the von Smoluchowski rate law, as in Eq. 6.17, to derive a mathematical model of the number density p,(t). Equations 6.22 and 6.24 indicate clearly that a uniform kinil leads to a linear time dependence in the... 

A number of issues preliminary to questions of control and process selectivity are afeo discussed. In particular we ask What determines the final outcome of a photodissociation process Although in quantum mechanics the fate of a system can only be known in a probabilistic sense, the linear time dependence of the Schrodinger Equation does guarantee that the probability of future events is completely determined by the probability of past events. (That is, quantum mechanics is a determi-, nistic theory of distributions of various observables). Hence by identifying attributes AMjne quantum state at earlier times we learn what is required to alter, that is, control,. hs stem dynamics in the future. 

Murabayashi coworkers (Nakajima et al., 2001, 2002) investigated the PL properties of rutile and anatase powders at room temperature in air in the absence or presence of reactants such as methanol or ethanol, with the goal of understanding the mechanisms of gas-phase photocatalytic reactions. In experiments with rutile, they observed that the PL intensity in the presence of methanol or ethanol increased linearly with the square root of UV-irradiation time, as shown in Figure 15. They also found a linear time dependence of the integrated amount of photodesorbed O2 (Nakajima et al., 2002). The time dependence of the PL intensity and the effect of O2 photodesorption were explained (as mentioned above) in terms of band bending of the powder. These results, however, were not observed for anatase powders. The authors explained that such a difference in PL behavior was related to the difference in photocatalytic activities of these alcohols. 

Dimensional stability is one of the most important properties of solid materials, but few materials are perfect in this respect. Creep is the time-dependent relative deformation under a constant force (tension, shear or compression). Hence, creep is a function of time and stress. For small stresses the strain is linear, which means that the strain increases linearly with the applied stress. For higher stresses creep becomes non-linear. In Fig. 13.44 typical creep behaviour of a glassy amorphous polymer is shown for low stresses creep seems to be linear. As long as creep is linear, time-dependence and stress-dependence are separable this is not possible at higher stresses. The two possibilities are expressed as (Haward, 1973)... 

Equation (20) indicates that < )y assumes a linear time dependence. Furthermore, using Eqs. (16) and (19), one finds that over one oscillation period, the change in... 

If you step back and think about it, the mechanical and rheological properties of many solids and liquids can be modeled fairly well by just two simple laws, Hooke s law and Newton s law. Both of these are what we call linear models, the stress is proportional to the strain or rate of strain. If we examine viscoelastic properties like creep, the variation of strain with time appears decidedly non-linear (see Figure 13-75). Nevertheless, it is possible to model this non-linear time dependence by the assumption of a linear relationship between stress and strain. By this we mean that if, for example, we measure the strain as a function of time in a creep experiment, then for a given time period (say 1 hour) the strain measured when the applied stress is 2o would be twice the strain measured when the stress was o. 

Figure 5.23 illustrates the applicabilily of this equation to the solid state powder reaction between Si02 and BaCOa giving BaSiOa plus CO2 (gas). In Figure 5.23(a) the linear time dependence of [1 - (1 — is plotted for several temperatures. The slopes equal to 2KJR are plotted as a function of in Figure 5.23(6). Figure 5.23(c) shows the Arrhenius expression,... 

A useful relationship that may be derived from a linear time dependence of sedimentation relates, the speed and time of centrifugation. These two parameters may be varied as long as the product of the force and its duration are constant ... 

Another aspect49 is the initial presence of persistent species in nonzero concentrations [Y]o, and it will be discussed more closely in section IV. In the absence of any additional initiation, the excess [Y]o at first levels the transient radical concentration to an equilibrium value [R]s = A[I]o/[Y]o. This is smaller than that found without the initial excess and lowers both the initial conversion rate and the initially large PDI. Further, it provides a linear time dependence of ln-([M]o/[M]), which is directly proportional to the equilibrium constant. Later in the reaction course, [Y] may exceed [Y]0 because of the self-termination, then [R] is given by eq 18. If there is additional radical generation, the first stages will eventually be replaced by a second stationary state that was described above. Further effects are expected from a decay or an artificial removal of the persistent species that increases the concentration of the transient radicals and the polymerization rate (see section IV). Radical transfer reactions to polymer, monomer, or initiator have not yet been incorporated in the analytical treatments. 

Gaussian diffusion is by no means ubiquitous, despite the appeal of the central limit theorem. Indeed, many systems exhibit deviations from the linear time dependence of Eq. (1). Often, a nonlinear scaling of the form [14—16]... 

In equation (17) 5 is defined as the overlap of two electronic determinantal wave functions S = z,R, P z,R,P ) and the energy is E = Pl[/2Mi + (z,7 ,Pl/7eiecl, ./ ,E)/(z,7 ,PIz,7 ,PX This level of theory can be characterized as fully non-linear time-dependent Hartree-Fock for quantum electrons and classical nuclei. It has been applied to a great variety of problems involving ion-atom [12,14,15,23-25], and ion-molecule reactive collisions... 

Single analyte kinetic determinations. The analytical signal is linearly time dependent and the slope of the recorded curve is proportional to the analyte concentration in the injectate. Using this approach, the injection of blanks is not necessary [66]. 

The overall value in of the economic potential increases significantly with the increased value of the product, but each case shows the same maximum and at the same point in time. The changes in mass of B and D are apparently linear in time over most of the run up to 10,000 time units. Why should this be Linear time dependence indicates a constancy of slope. However, this is a fully transient, that is, time-dependent, system. How can we have a constant slope, that is, a constant rate of change, in the concentrations for a fully transient system To understand this we must reintroduce the concept of the pseudo-steady state. 

The presented TH simulations are essentially simplified ID predictions of the relative humidity by means of calibration of the temperature boundary conditions. The actual geometry, behaviours of the heater, liner and rock, and the deformation of the buffer are not considered. Convection heat transfer and variations of pressure, viscosity of liquid, or liquid density are neglected, and the gas diffusion is of a simple form. The main simulational simplifications are the approximation of the heater adjustment period by a linear time dependence of temperature, and the use of a full saturation boundary condition at the rock surface. 

Crossover studies in healthy volunteers examining pharmacokinetic linearity, time dependency, and intra- and intersubject variability over the anticipated clinical dose range are generally required to ensure that the pharmacokinetic model developed for the drug candidate is suitable and predictive. Exceptions could include drugs with such low variability that definitive data on linearity are obtained from dose tolerance studies, or in cases where Phase 11b studies use crossover designs with extensive pharmacokinetic sampling. 

Integration of the system of equations (9) yields trajectories of classical nuclei dressed with END. This approach can be characterized as being direct, and non-adiabatic or as fully non-linear time-dependent Hartree-Fock (TDHF) theory of quantum electrons and classical nuclei. This simultaneous dynamics of electrons and nuclei driven by their mutual instantaneous forces requires a different approach to the choice of basis sets than that commonly encountered in electronic structure calculations with fixed nuclei. This aspect will be further discussed in connection with applications of END. 

In principle, the rate equations for surface reaction kinetics are linear and describe a linearly time-dependent growth of the corrosion layer. However, during this growth the oxygen activity on the surface increases and gradually approaches the value for equilibrium of gas phase and oxide surface. Because of the dependence on ao with a negative exponent, the rate gradually decreases, and several authors have misinterpreted this kinetics as parabolic kinetics (see Sect. G.2.3.2). 

With a large external resistance one can force the internal recombination. This was achieved in the example seen in Figure 9.13 with an external resistance of 1 MO. The internal recombination followed a second-order rate equation as is shown by the linear time dependence of the reciprocal values. 

W satisfies a system of linear (time-dependent) differential equations involving the matrix f (the Jacobian matrix of vector field) evaluated along the solution z(t). W may be seen as a fundamental matrix solution of the indicated Unear system. 

Chemical reaction network is a typical example of complexity, where the reactants can interact in a variety of ways depending on the nature of interaction (chemical as well as non-chemical). Oscillatory reactions involve a number of steps, including positive and negative feedbacks. The complexity leads to periodic as well as aperiodic oscillations (multi-periodic, bursting/intermittency sequential oscillations separated by a time pause, relaxation and chaotic oscillations). The mechanism is usually determined by non-linear kinetics and computer modelling. Once the reaction mechanism has been postulated, the non-linear time-dependent kinetic equation can be formulated in terms of concentrations of different reactants, which would yield a multi-variable equation. Delay differential equations are sometimes used to characterize oscillatory behaviour as in economics (Chapter 14). 

Using the ground-state density matrix as an input, the CEO procedure - computes vertical transition energies and the relevant transition density matrices (denoted electronic normal modes ( v)mn = g c j Cn v ), which connect the optical response with the underlying electronic motions. Each electronic transition between the ground state and an electronically excited state v) is described by a mode which is represented hy K x K matrix. These modes are computed directly as eigenmodes of the linearized time-dependent Hartree—Fock equations of motion for the density matrix (eq A4) of the molecule driven by the optical field. 

Non-linear, time-dependent characteristics of viscoelastic materials such as polyethylene have been mathematically modelled and the model compared with experimental results. Mechanical properties such as creep and stress relaxation are non-linear because they include time-dependent and irreversible components. The time-dependent component is non-linear when relaxation time is longer than the timeframe of the experiment. This becomes increasingly so at high stress. Low stress will act on faster responding deformation modes and as stress increases slower modes will respond. The slower modes will be non-linear relative to the timescale of the experiment. Some slower modes such as relative translation of molecules are irreversible. Stress relaxation is complementary to creep in that strain is applied creating a stress that may relax according to the relative times of the experiment and molecular processes. 

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