Evolution equation, steady-state solution

Steady-state solution of the evolution equation. The steady-state solution, of the evolution eqnation at a polarization point Eg, is obtained by pntting d/dt = 0, which leads to ... 

Auchmuty, J., Nicolis, G. Bifurcation analysis of nonlinear reaction-diffusion equations—I. Evolution equations and the steady state solutions. BuU. Math. Biol. 37(4), 323-365 (1975). http //dx.doi.org/10.1007/BF02459519... 

In order to consider the linear stability of thin film wavy motions governed by the B-M problem, we must first determine the steady-state solutions of the evolution equations presented. Then we can perturb about the steady-states using the normal modes. Assuming that the perturbing quantities are small, we linearize the evolution equations about the steady-state solutions to obtain an eigenvalue system for the considered B-M problem. 

Auchmuty, J. F. G., Nicolis, G. (1975) Bifurcation analysis of nonlinear reaction-diffusion equations I. Evolution equations and the steady state solutions. Bull. Math. Biol. 37, 323 Auchmuty, J. F. G., Nicolis, G. (1976) Bifurcation analysis of nonlinear reaction-diffusion equations III. Chemical oscillations. Bull. Math. Biol. 38, 325 Bogoliubov, N. N., Mitropolskii, I. A. (1961) Asymptotic Methods in the Theory of Nonlinear Oscillations (Gordon and Breach, New York) [English transl.]... 

The Maxwellian velocity d istribution function (2.152) represents a steady-state solution of the Boltzmann equation describing the microscopic evolution of a... 

For some values of the rate constants, these approximate forms give a good prediction of the intermediate concentrations, at long enough times. Equations (1.31) and (1.32) do not match the initial conditions, i.e. ass tends to (k0/ku)p0 and bss to (k0/k2)p0 rather than to zero as t - 0. Thus there must always be some initial evolution of a and b even if the system eventually settles close to the pseudo-steady states. Equations (1.31) and (1.32) can in fact be seen as a long time limit of the exact solutions (1.29) and (1.30), provided fcu is greater than k0 and k2 is greater than ku the terms exp( — kat) and exp( — k2t) then tend to zero more quickly than exp( — k0t) which becomes the dominant term. 

Usually, experiments are performed with steady-state photolysis or radiolysis of the solution and the yield of scavenger products determined optically or by ESR methods. There is no direct interest in the actual time evolution of the density or recombination (survival) probability. Consequently, the creation of ion-pairs may be pictured as occurring at a constant rate, say 1 s 1, from time t0 = 0 to infinity. The steady-state ion-pair density distribution, which arises when dp/dt = 0, is the balance between continuous creation of ion pairs at a rate Is-1, recombination and scavenging. Removing the instantaneous creation of an ion-pair at time t = t0 (i.e. removing the 6(f — f0) in the source term), means that ion-pairs were continuously formed from time t = — 00 to t. At long times, f > — oo the density distribution is independent of t and, of course, t0. Let pss(r cs r0) = /i p(r, t cs t0, 0)d 0 be the steady-state ion-pair density distribution for ion pairs continuously formed at r0, and note d/dt J" f pd 0 = 0. The diffusion equation (169) becomes... 

The multi-mode model for a tubular reactor, even in its simplest form (steady state, Pet 1), is an index-infinity differential algebraic system. The local equation of the multi-mode model, which captures the reaction-diffusion phenomena at the local scale, is algebraic in nature, and produces multiple solutions in the presence of autocatalysis, which, in turn, generates multiplicity in the solution of the global evolution equation. We illustrate this feature of the multi-mode models by considering the example of an adiabatic (a = 0) tubular reactor under steady-state operation. We consider the simple case of a non-isothermal first order reaction... 

This system of equations shows, through even orders, that polarized light irradiation creates anisotropy and photo-orientation by photoisomerization. A solution to the time evolution of the cis and trans expansion parameters cannot be found without approximations this is when physics comes into play. Approximate numerical simulations are possible. 1 will show that for detailed and precise comparison of experimental data with the photo-orientation theory, it is not necessary to have a solution for the dynamics, even in the most general case where there is not enough room for approximations, i.e., that of push-pull azo dyes, such as DRl, because of the strong overlap of the linear absorption spectra of the cis and trans isomers of such chromophores. Rigorous analytical expressions of the steady-state behavior and the early time evolution provide the necessary tool for a full characterization of photo-orientation by photoisomerization. 

Equation was derived without approximations. It is noteworthy that these solutions do not couple tensorial components of different orders and that they confirm that rotational diffusion and cis—>trans thermal isomerization are isotropic processes that do not favor any spatial direction. In Section 3.4, I discuss, through the example of azobenzene, how Equation 3.11 can be used to study reorientation processes during cis—>trans thermal isomerization after the end of irradiation. The next subsection gives analytical expressions at the early-time evolution and steady-state of photo-orientation, for the full quantification of coupled photo-orientation and photoisomerization in A<- B photoisomerizable systems where B is unknown. 

The isotropic and anisotropic distributions obtained from the solution of the steady-state kinetic equation, Eq. (36), related to the undisturbed field are used as initial values for both distributions in the time-dependent treatment of the electron response to the respective field disturbance. Figure 16 illustrates for neon the evolution of the isotropic distribution up to the establishment of the steady state in the undisturbed field for the field pulses of Fig. 15. If the field substantially... 

El < Eo) non-dissociative states (dissociative states are rapidly depopulated by the fast intramolecular dissociation process). As is well known, the time evolution of the populations [A(i)] is given by a series of exponentially decaying terms which ctHTiespond to an initial rovibrational relaxation, a subsequent incubation period with overlap of vibrational rriaxation of upper levels and dissociation, and the final dissociation period with steady-state of all populations [A(i)]. Explicit solutions of the master equation for the dissociation of diatomic molecules have been extensively reviewed by H. O. Pritchard in Volume 1 of this series. Such... 

With this steady-state value of cp for f, but using both roots of the quadratic equation, the solution of eq. (7.37) for an initial value of gives the evolution... 

Because of fast vacancy diffusion (atoms require vacancies to have an opportunity to jump, while vacancies in the atoms company are always self-sufficient ), the relations Eh = Dy ) > Di hold. Hence, the evolution of the combination proceeds much faster than that of and so the solution of Equation 2.34 for i = 2 is of steady-state type ... 

Therefore 71 is a Liapounov functional for the evolution equation (8.1). Steady states of the system, therefore, if exist, are solutions of the equation ... 

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