Differential recurrence relation

Calculation of Mean First Passage Times from Differential Recurrence Relations... 

A quite different approach from all other presented in this review has been recently proposed by Coffey [41]. This approach allows both the MFPT and the integral relaxation time to be exactly calculated irrespective of the number of degrees of freedom from the differential recurrence relations generated by the Floquet representation of the FPE. 

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). 

The corresponding Fourier coefficients satisfy the differential recurrence relations... 

The use of the differential recurrence relations to calculate the mean first passage time is based on the observation that if in Eq. (5.48) one ignores the term sY(x, s) (which is tantamount to assuming that the process is quasi-stationary, i.e., all characteristic frequencies associated with it are very small), then one has... 

The derivative of any Si can be computed by means of the differential recurrence relation [23]... 

One finds these complex coefficients by solving numerically the infinite set of differential recurrence relations obtained by substitution of expansion (4.56) into Eq. (4.27) ... 

In the case of isotropic magnetic particles, that is, TJ = -p(e H), both linear and cubic dynamic susceptibilities may be obtained analytically. To show this, we first transform Eq. (4.90) into an infinite set of differential recurrence relations ... 

Substituting Eq. (4.260) into Eq. (4.90), and then integrating over e, one arrives at the pentadiagonal set of differential recurrence relations for the mean... 

We calculate %(a>) by converting the solution of the fractional diffusion Eq. (55) into the calculation of successive convergents of a differential-recurrence relation just as normal diffusion [8,62]. By expanding the distribution function W(< ), f) in Fourier series... 

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... 

Substituting Eq. (267) into Eq. (265), taking the inner product, and utilizing the orthogonal properties and known recurrence relations [51] for the associated Legendre functions Pf cosi ) and the Hermite polynomials H (z) then yields the infinite hierarchy of differential recurrence relations for the clnm(t) governing the orientational relaxation of the system, namely,... 

Multiplying Eq. (33) by (q0o) (0o denotes a sharp initial value) and averaging over a Maxwell Boltzmann distribution of 0o and 0o, we automatically have a differential recurrence relation fory i, (t), namely,... 

The result given here may also be extended to rotation in space. As far as the cage motion is concerned, the complex susceptibility will still be governed by the Rocard equation because the equations of motion factorize. However, the solution for the dipole correlation function is much more complicated because of the difficulty of handling differential recurrence relations pertaining to rotation in space in the presence of a potential. 

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