Diffusion equation rotational

A small step rotational diffusion model has been used to describe molecular rotations (MR) of rigid molecules in the presence of a potential of mean torque.118 120,151 t0 calculate the orientation correlation functions, the rotational diffusion equation must be solved to give the conditional probability for the molecule in a certain orientation at time t given that it has a different orientation at t = 0, and the equilibrium probability for finding the molecule with a certain orientation. These orientation correlation functions were found as a sum of decaying exponentials.120 In the notation of Tarroni and Zannoni,123 the spectral denisities (m = 0, 1, 2) for a deuteron fixed on a reorienting symmetric top molecule are ... 

P(Qol, t) is the conditional probability of the orientation being at time t, provided it was Qq a t time zero. The symbol — F is the rotational diffusion operator. In the simplest possible case, F then takes the form of the Laplace operator, acting on the Euler angles ( ml) specifying the orientation of the molecule-fixed frame with respect to the laboratory frame, multiplied with a rotational diffusion coefficient. Dr. Equation (44) then becomes identical to the isotropic rotational diffusion equation. The rotational diffusion coefficient is simply related to the rotational correlation time introduced earlier, by tr = 1I6Dr. 

The Debye theory [220] in which a sphere of volume V and radius a rotates in a liquid of coefficient of viscosity t has already been mentioned. There is angular momentum transfer across the sphere—liquid interface that is, the liquid sticks to the sphere so that the velocity of the sphere and liquid are identical at the sphere s surface. Solution of the rotational diffusion equation... 

This particular solution of the rotational diffusion equation can be interpreted as the transition probability that is, the probability density for a rod to have orientation u at time t, given that it had an orientation Uo initially. 

The underlying fractional rotational diffusion equation is Eq. (52) with... 

The formal solution of the fractional rotational diffusion equation is obtained just as that of Eq. (135) from the Sturm-Liouville representation [7,57]... 

These and many other examples suggest the development of a model wherein the presence of individual sites connected by pathways or reaction channels to a reaction center (the consequence of having a surface broken up into domains or differentiated into clusters) is built into the formulation of the problem from the very outset. That is, rather than assuming that all sites are accessible to a diffusing coreactant so that a rotational diffusion equation can be written down... 

Letting c(u, t)d2u be the fraction of rods2 with orientation u in the solid angle, d2u ( = sin Odddfy at time t, and substituting Eq. (7.3.2) into Eq. (7.3.1) we obtain the rotational diffusion equation (or Debye equation)... 

Those who are familiar with elementary quantum mechanics should recognize that the differential operator in Eq. (7.3.3), that is, the angular part of the Laplacian operator, is — 12 where I is the dimensionless orbital angular momentum operator of quantum mechanics (see Dicke and Witke, 1960). Thus the rotational diffusion equation can be written as... 

This simplifies the solution of the rotational diffusion equation considerably. As is well known, the spherical harmonics Yim 0, [Pg.120]

These formulas apply regardless of whether the rotations can be described by a rotational diffusion equation. In the event that the rotational diffusion equation applies, Eq. (7.B. 16) reduces to the results found in Sections 7.3. and 7.5. If the molecules freely rotate, Eqs. (7.B. 16) reduce to the results of Section 7.6. 

CONCEPTS More about the effect of collisions on distribution functions microscopic theory of dielectric loss The Debye theory can define a distribution function which obeys a rotational diffusion equation. Debye [22, 23] has based his theory of dispersion on Einstein s theory of Brownian motion. He supposed that rotation of a molecule because of an applied field is constantly interrupted by collisions with neighbors, and the effect of these collisions can be described by a resistive couple proportional to the angular velocity of the molecule. This description is well adapted to liquids, but not to gases. 

The rotational part of the motion is described by the rotational diffusion equation for >p(0,[Pg.263]

In some systems, ip(0,(p t) does not depend on probability density i/>(u t) evolves for a rod with u(0) parallel to the polar axis, the distribution is a function of 0 and t only. Another example is a rodlike molecule that has a permanent dipole moment along the axis in an electric field. The natural choice of the polar axis is the direction of the electric field. When ip depends on 6 and t only, the rotational diffusion equation is simplified to... 

The full solution of the rotational diffusion equation including a general single particle potential of D , symmetry has been investigated [36], and it is found that the dipole correlation function, which can be related to the permittivity as a function of frequency, is a sum over many exponential terms each characterised by a different relaxation time. Extending Eq. (20) for an anisotropic fluid gives ... 

As described in depth in Ref. 4, the starting point for this treatment is the rotational diffusion equation of a rigid body, which can be elegantly solved by exploiting its correspondence with the time-dependent Schrodinger equation. The solution is an infinite series of spherical harmonics. Recalling that the reorientational correlation functions are expressible as spherical harmonics, the orthonormality condition leads to the... 

In general, short intense laser pulses are required to create appreciable molecular alignment in liquids. To quantitatively describe the pnlsed laser-induced effect, a time-dependent approach is needed. In this regime /(0) obeys a Debye rotational diffusion equation ... 

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