The Feller equation

To make evaluations more definite, we use optical and microwave experimental data, as well as calculations of molecular dynamics of certain simple liquids which usually fit the experiment. Rotation is everywhere considered as classical, and the objects are two-atomic and spherical molecules, as well as hard ellipsoids. 

In impact theory the result of a collision is described by the probability /(/, /)dJ of finding angular momentum J after the collision, if it was equal to / before. The probability is normalized to 1, i.e. / /(/, /)d/=l. The equilibrium Boltzmann distribution over J is 

The above information is enough to define J(t) as a random process, 

Here 0(a) is the density of distribution over a after collision and tc is the average collision time. The popular Keilson-Storer model, presented in Eq. (1.6) and Fig. 1.2, uses the single numerical parameter y to characterize the strength of inelastic collisions. It will be discussed in Section 1.3. 

The proposed specification of the kernel for m- and J-diffusion models is mathematically closed, physically clear and of quite general character. In particular, it takes into consideration that any collisions may be of arbitrary strength. The conventional m-diffusion model considers only strong collisions (0(a) = 1 /(27c)), while J-diffusion considers either strong (y = 0) or weak (y = 1) collisions. Of course, the particular type of kernel used in (1.6) restricts the problem somewhat, but it does allow us to consider kernels with arbitrary y 1. 

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In the case of weak collisions, the moment changes in small steps AJ (1 — y)J < J, and the process is considered as diffusion in J-space. Formally, this means that the function /(z) of width [(1 — y2)d]i is narrow relative to P(J,J, x). At t To the latter may be expanded at the point J up to terms of second-order with respect to (/ — /). Then at the limit y -> 1, to — 0 with tj finite, the Feller equations turn into a Fokker-Planck equation... 

In order to determine the actual strength of collisions, it is desirable to have a general solution of the Feller equation that holds for any y. Consider the problem of the eigenvalues of the integral operator L, which appears in the Feller equation... 

In the case of weak collisions the change in J is so slight that one may proceed from an integral description of the process to a differential one, just as in Eq. (1.23). However, the kernel of the integral equation (3.26) specified in Eq. (3.28) is different from that in the Feller equation. Thus, the standard procedure described in [20] is more complicated and gives different results (see Appendix 3). The final form of the equation obtained in the limit y — 1, to —> 0 with... 

An algorithm for performing a constant-pressure molecular dynamics simulation that resolves some unphysical observations in the extended system (Andersen s) method and Berendsen s methods was developed by Feller et al. [29]. This approach replaces the deterministic equations of motion with the piston degree of freedom added to the Langevin equations of motion. This eliminates the unphysical fluctuation of the volume associated with the piston mass. In addition, Klein and coworkers [30] present an advanced constant-pressure method to overcome an unphysical dependence of the choice of lattice in generated trajectories. 

The space-fractional equation (3.91) can be derived from the Kolmogorov-Feller equation (3.74) by using the assumption that the random jump Z has a Levy-stable PDF Wg,(z), symmetric with respect to zero, with power-law tails as z -> oo. There is no general explicit form for u (z), but the characteristic function of iv z), the structure function, is... 

In this section we show how to obtain subdiffusive transport by using the idea of inverse subordination [278, 371]. Assume that the density 0 obeys the Kolmogorov-Feller equation... 

UX = const, w(z) does not depend on x, and v(x) = 0, we obtain the Kolmogorov-Feller equation (3.74), for which the underlying microscopic random movement is a compound Poisson process. 

By using the same cmicepts, a very large niun-ber of other problems may be solved. Such an example the probability density function of a random variable may be obtained with the same technique here used for representing cross-correlations in terms of FSMs. It follows that Fokker-Planck equation, Kolmogorov-Feller equation, Einstein-Smoluchowski equation, and path integral solution (Cottone et al. 2008) may be solved in terms of FSM. Moreover, wavelet transform and classical or fractional differential equations may be easily solved by using fractional calculus and Mellin transform in complex domain. 

Let us illustrate, in short, the results found by the authors of Ref. 71. Let us study the dynamic approach to Levy diffusion using the Levy walk perspective. Let us use the condition p > 2, which is compatible with the existence of the stationary correlation function (t). As we have shown earlier, the Levy scaling can be derived using the arguments behind Eq. (118). This equation implies that the number of events is proportional to time. This is not quite correct. The exact formula was found by Feller [72], and it reads... 

Thus, we observe that when the memory kernel in the fractional Langevin equation is random, the solution consists of the product of two random quantities giving rise to a multifractal process. This is Feller s subordination process. We apply this approach to the SRV time series data discussed in Section II and observe, for the statistics of the multiplicative exponent given by Levy statistics, the singularity spectrum as a function of the positive moments... 

Equation (502) is interpreted as the pdf of a process that occurs at integer times n (the operational time). Feller calls the random variable N(t) the randomized operational time. The operational times need not necessarily be discrete and need not be distributed according to the Poisson law, which has been only used for illustration purposes. 

For equation (5) to be defined, at least three points are required. We now introduce the notation Feller(/mn) for the result of equation (5) from CCSD(T)/cc-pV/Z, cc-pVmZ, and cc-pVnZ calculations, and Feller(/mn)4-core for the same plus an additivity correction for core-correlation. The use of aug -cc-pVnZ instead of regular cc-pVnZ basis sets can then be denoted using the aug -Feller(/mn) and aug -Feller(/mn)4-core notations. 

Although anisotropic pressure coupling is the simplest approach computationally, especially when the weak coupling scheme has already been implemented in the code used, it may not be entirely correct from a. statistical mechanical point of view. Zhang and -Feller et al. identified a series of ensembles that can be obtained with simulations of liquid-liquid interfaces and derived equations of motion using an extended system Hamiltonian that includes a piston for pressure coupling. ... 

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