Exponent

As described in a previous section, the Lyapunov exponents are a generalized measure of the growth or decay of perturbations that might be applied to a given dynamical state they are identical to the stability eigenvalues for a steady state and the Floquet exponents for a limit cycle. For aperiodic motion at least one of the Lyapunov exponents will be positive, so it is generally sufficient to calculate just the largest Lyapunov exponent. 

For chaotic behavior, at least one of the Lyapunov exponents would be expected to be positive. In practice, it is difficult to find the entire spectrum of Lyapunov exponents from experimental data because it is not always clear how many dependent variables are involved, and the number of Lyapunov exponents is equal to the number of dependent variables. For models, however, it is possible to compute the complete spectrum of Lyapunov exponents. If these are 

/ is the maximum integer for which Xi + X2 +. . . + X,- S 0, that is, / is die number of positive Lyapunov exponents in the spectrum. 

An example of a calculation of the Lyapunov exponents and dimension, for a simple four-variable model of the peroxidase-oxidase reaction will help to clarify these general definitions. The following material is adapted from the presentation in Ref. 94. As described earlier, the Lyapunov dimension and the correlation dimension, D, serve as upper and lower bounds, respectively, to the fractal dimension of the strange attractor. The simple four-variable model is similar to the Degn—Olsen-Ferram (DOP) model discussed in a previous section but was suggested by L. F. Olsen a few years after the DOP model was introduced. It remains the simplest model the peroxidase-oxidase reaction which is consistent with the most experimental observations about this reaction. The rate equations for this model are  

Computational techniques are centrally important at every stage of investigation of nonlinear dynamical systems. We have reviewed the main theoretical and computational tools used in studying these problems among these are bifurcation and stability analysis, numerical techniques for the solution of ordinary differential equations and partial differential equations, continuation methods, coupled lattice and cellular automata methods for the simulation of spatiotemporal phenomena, geometric representations of phase space attractors, and the numerical analysis of experimental data through the reconstruction of phase portraits, including the calculation of correlation dimensions and Lyapunov exponents from the data. 

Let s regroup the like terms (multiplication is associative, so it is okay to group the terms together in any order we like). 

1 x 1 x 1 = 1, but what about the cm Since there are three cm terms, the exponential expression of cm x cm x cm is cm3. Therefore, the volume of our cube is 1 cm3. Notice that cm connotes length but cm3 connotes volume, a different concept than length. 

There are some special cases of exponents. Any quantity (except zero) raised to the zero power equals one. Any quantity (except zero) to the -1 power is equal to the reciprocal of the quantity. 

Moving a quantity across a division bar changes the sign of the exponent. This is true for numbers as well as variables or units. 

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The pore system is described by the volume fraction of pore space (the fractional porosity) and the shape of the pore space which is represented by m , known as the cementation exponent. The cementation exponent describes the complexity of the pore system i.e. how difficult it is for an electric current to find a path through the reservoir. 

The volume fraction of water (S J and the saturation exponent n can be considered as expressing the increased difficulty experienced by an electrical current passing through a partially oil filled sample. (Note is only a special case of C, when a reservoir... 

Only even values of Wi -t- m2 -t- m3 are used for the FCC lattice. The numerical values of these lattice sums are dependent on the exponents used for U(r), and Eq. VII-11 may be written... 

The exponents n and m in Eq. VII-1 are expected to be 1 and 3, respectively, under some conditions. Assuming spheres and letting Vr = V/Vq, derive the rate law dVrjdt =f(Vr). Expound on the peculiar nature of this rate law. 

As discussed elsewhere in diis encyclopaedia, the critical exponents are related by the following expressions ... 

The individual values of the exponents are detennined by the symmetry of the Hamiltonian and the dimensionality of the system. 

This is the well known equal areas mle derived by Maxwell [3], who enthusiastically publicized van der Waal s equation (see figure A2.3.3. The critical exponents for van der Waals equation are typical mean-field exponents a 0, p = 1/2, y = 1 and 8 = 3. This follows from the assumption, connnon to van der Waals equation and other mean-field theories, that the critical point is an analytic point about which the free energy and other themiodynamic properties can be expanded in a Taylor series. 

The equation of state detemiined by Z N, V, T ) is not known in the sense that it cannot be written down as a simple expression. However, the critical parameters depend on e and a, and a test of the law of corresponding states is to use the reduced variables T, and as the scaled variables in the equation of state. Figure A2.3.5 bl illustrates this for the liquid-gas coexistence curves of several substances. As first shown by Guggenlieim [19], the curvature near the critical pomt is consistent with a critical exponent (3 closer to 1/3 rather than the 1/2 predicted by van der Waals equation. This provides additional evidence that the law of corresponding states obeyed is not the fomi associated with van der Waals equation. Figure A2.3.5 (b) shows tliat PIpkT is approximately the same fiinction of the reduced variables and... 

The divergence m the correlation length is characterized by the critical exponent v defined by... 

Table A2.3.4 simnnarizes the values of these critical exponents m two and tliree dimensions and the predictions of mean field theory.

There are 2 temis in the sum since each site has two configurations with spin eitlier up or down. Since the number of sites N is fmite, the PF is analytic and the critical exponents are classical, unless the themiodynamic limit N oo) is considered. This allows for the possibility of non-classical exponents and ensures that the results for different ensembles are equivalent. The characteristic themiodynamic equation for the variables N, H and T is... 

This equation is analogous to the compressibility equation for fluids and diverges with the same exponent y as the critical temperaUire is approached from above ... 

The correlation length C = T -T diverges with the exponent v. Assuming that when T>T the site... 

We now turn to a mean-field description of these models, which in the language of the binary alloy is the Bragg-Williams approximation and is equivalent to the Ciirie-Weiss approxunation for the Ising model. Botli these approximations are closely related to the van der Waals description of a one-component fluid, and lead to the same classical critical exponents a = 0, (3 = 1/2, 8 = 3 and y = 1. 

An essential feature of mean-field theories is that the free energy is an analytical fiinction at the critical point. Landau [100] used this assumption, and the up-down symmetry of magnetic systems at zero field, to analyse their phase behaviour and detennine the mean-field critical exponents. It also suggests a way in which mean-field theory might be modified to confonn with experiment near the critical point, leading to a scaling law, first proposed by Widom [101], which has been experimentally verified. 

To detennine the critical exponents y and S, a magnetic interaction temi -hm is added to the free energy and... 

This implies that the critical exponent y = 1, whether the critical temperature is approached from above or below, but the amplitudes are different by a factor of 2, as seen in our earlier discussion of mean-field theory. The critical exponents are the classical values a = 0, p = 1/2, 5 = 3 and y = 1. 

The assumption that the free energy is analytic at the critical point leads to classical exponents. Deviations from this require tiiat this assumption be abandoned. In mean-field theory. 

More generally, for other lattiees and dimensions, nmnerioal analysis of the high-temperature expansion provides infonnation on tire eritieal exponents and temperature. The high-temperature expansion of the suseeptibility may be written in powers of = p J as... 

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