Matrix continued fractions

We also remark that Eq. (5.44) may be decomposed into separate sets of equations for the odd and even ap(t) which are decoupled from each other. Essentially similar differential recurrence relations for a variety of relaxation problems may be derived as described in Refs. 4, 36, and 73-76, where the frequency response and correlation times were determined exactly using scalar or matrix continued fraction methods. Our purpose now is to demonstrate how such differential recurrence relations may be used to calculate mean first passage times by referring to the particular case of Eq. (5.44). 

These recurrence relations are solved using the matrix-continued fraction method [35,107,108]. Figure 4.15 shows how the decay of SNR takes place as a result of gradual shoaling of one of the minima under growth of the bias field Ho-To characterize the latter, in Eqs. (4.240) and (4.241) we have defined the dimensionless parameter = ilIo/T with respect to the bias field. Under that choice, the ratio... 

Risken, Vollmer, and Mdrsch studied the Kramers equation, that is, the Fokker-Planck equation (1.9), by expanding the distribution function p(x, o /) in Hermitian polynomials (velocity part) and in another complete set satisfying boundary conditions (position part). The Laplace transform of the initial value problem was obtained in terms of continued fractions. An inverse friction expansion of the matrix continued fraction was then used to show that the first Hermitian expansion coefficient may be determined by a generalized Smoluchowski equation. This provides results correcting the standard Smoluchowski equation with terms of increasing power in 1/y. They evaluated explicit expressions up to order y . ... 

As we have already mentioned, it is difficult to evaluate dielectric parameters from Eq. (201) because a knowledge of all the eigenvalues X[ and corresponding amplitudes c7k is required. A more simple (from the computational point of view) solution can be given in terms of matrix continued fractions. The general transient response solution of Eq. (172) for t > 0, one can be sought in the form... 

We now present the solution of Eqs. (204) and (205) in terms of matrix continued fractions. The advantage of posing the problem in this way is that exact formulae in terms of such continued fractions may be written for the Laplace transform of the aftereffect function, the relaxation time, and the complex susceptibility. The starting point of the calculation is Eqs. (204) and (205) written as the matrix differential recurrence relation... 

The three-term matrix recurrence relation, Eq. (208), may now be solved for the Fourier-Laplace transform C (oo) in terms of matrix continued fractions to yield [8]... 

Equations (210) and (211) constitute the exact solution of our problem formulated in terms of matrix continued fractions. Having determined the Laplace transform Y(co) and noting that CY(lco) = f (i )/fj (0), one may calculate the susceptibility xT( ) from Eq. (201). 

Figure 16. xj (co) and xj ( ) versus cox evaluated from the exact matrix continued fraction solution (solid lines) for a = 0.5, t v = 10 and various values of and compared with those calculated from the approximate Eq. (215) (filled circles) and with the low-(dotted lines) and high-frequency (dashed lines) asymptotes Eqs. (184) and (185), respectively. 

Figure 18. Xj ( ) and xK ) versus cor evaluated from the exact matrix continued fraction...

As we have already mentioned, we choose as an example of an internal field potential a double-well potential (N = 2) that will allow us to treat overbarrier relaxation (for N = 1 corresponding to a uniform electric field this process does not exist). In order to solve Eq. (319), we shall use matrix continued fractions [8,26]. This is accomplished as follows. We introduce the column vectors... 

By invoking the general method for solving the matrix recurrence Eq. (322) [8], we have the exact solution for the spectrum Ci (co) in terms of a matrix continued fraction, namely,... 

As an example the results of numerical calculations for linear (lz = 0) molecules based on the above matrix continued fraction solution have been compared with that of Ref. 67 presented in terms of ordinary continued fractions. The numerical calculations show that both matrix and ordinary continued fraction solutions yield the same results. 

Exact Solution for the Complex Susceptibility Using Matrix Continued Fractions Approximate Expressions for the Complex Susceptibility Numerical Results and Comparison with Experimental Data Conclusions... 

The first objective of this review is to describe a method of solution of the Langevin equations of motion of the itinerant oscillator model for rotation about a fixed axis in the massive cage limit, discarding the small oscillation approximation in the context of dielectric relaxation of polar molecules, this solution may be obtained using a matrix continued fraction method. The second... 

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