Third-order linear system

Equations (2.9), (2.10) and (2.11) are linear differential equations with constant coefficients. Note that the order of the differential equation is the order of the highest derivative. Systems described by such equations are called linear systems of the same order as the differential equation. For example, equation (2.9) describes a first-order linear system, equation (2.10) a second-order linear system and equation (2.11) a third-order linear system. 

The cost, however, is that some of our terminology i s nontraditional. For example, the forced harmonic oscillator would traditionally be regarded as a second-order linear equation, whereas we will regard it as a third-order nonlinear system, since (3) is nonlinear, thanks to the cosine term. As we ll see later in the book, forced oscillators have many of the properties associated with nonlinear systems, and so there are genuine conceptual advantages to our choice of language. 

The above theory is usually called the generalized linear response theory because the linear optical absorption initiates from the nonstationary states prepared by the pumping process [85-87]. This method is valid when pumping pulse and probing pulse do not overlap. When they overlap, third-order or X 3 (co) should be used. In other words, Eq. (6.4) should be solved perturbatively to the third-order approximation. From Eqs. (6.19)-(6.22) we can see that in the time-resolved spectra described by x"( ), the dynamics information of the system is contained in p(Af), which can be obtained by solving the reduced Liouville equations. Application of Eq. (6.19) to stimulated emission monitoring vibrational relaxation is given in Appendix III. 

For multi-molecular assemblies one has to consider whether the total interaction energy can be written as the sum of pairwise interactions. The first-order electrostatic interaction is exactly pairwise additive, the dispersion only up to second order (in third order a generally small three-body Axilrod-Teller term appears [73]) while the induction is not at all pairwise it is non-linearly additive due to the interference of electric fields from different sources. Moreover, for polar systems the inducing fields are strong enough to change the molecular wave functions significantly. 

To add to the confusion noted for conventions of polarizabilities, both cgs and recommended SI units for linear and non-linear optical polarizabilities coexist in the literature. We strongly advocate the use of SI units. The SI unit of the electric dipole moment is Cm (Cohen and Giacomo, 1987). Thus, consistent SI units of an nth-order polarizability are defined as C m(mV )" = C m " V ", cf. (34)-(37). Conversions from the SI to the esu system for the dipole moment, the first-, second-, and third-order polarizability, are given in (38)-(41). 

A full quantum mechanical description of the third order non-linearity requires forty-eight terms, which are more complex than those describing second order effects, see for example Boyd (2003). The simple model of molecular non-linearity considered above (Fig. 3.22) indicates that the delocalised 7r-electron systems of conjugated polymers will have large third order non-linearity. Other factors governing the third order non-linearity of conjugated polymers will be discussed in Chapter 9, Section 9.4.2. 

For each set of constant input and output concentration constraints a system of linear chemical reactions has a unique steady state. For a network of nonlinear biochemical reactions, however, there could be several steady states compatible with a given set of constraints. The number and character of these steady states are determined by the structure of the network including the extent of nonlinearity, the number and connectivity of the individual chemical reactions and the values of the reaction rate constants and the concentrations of the reactants. The higher the order of a chemical reaction, the more steady states may be compatible with a given set of chemical constraints. The simple trimolecular reaction system of Schlogl [13] illustrates how a third-order chemical reaction can have two stable steady states compatible with a single set of chemical constraints ... 

The linearity of dependencies In Fig. 9 can be predicted assuming miscibility In a blend composed of polymers with similar molecular weights. For such systems the general, third order mixing rule for relaxation spectrum (55, 56) ... 

This set of function values does not correspond to a converged solution either so we must compute the Jacobian matrix a second time and solve the linear system (Eq. C.2.5) in order to obtain a third estimate of the solution... 

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