Legendre polynomial averaging

If we now replace the coefficients by their respective Legendre polynomial averages recalling that... 

Here u is a unit vector oriented along the rotational symmetry axis, while in a spherical molecule it is an arbitrary vector rigidly connected to the molecular frame. The scalar product u(t) (0) is cos 0(t) in classical theory, where 6(t) is the angle of u reorientation with respect to its initial position. It can be easily seen that both orientational correlation functions are the average values of the corresponding Legendre polynomials ... 

The anisotropy of the liquid crystal phases also means that the orientational distribution function for the intermolecular vector is of value in characterising the structure of the phase [22]. The distribution is clearly a function of both the angle, made by the intermolecular vector with the director and the separation, r, between the two molecules [23]. However, a simpler way in which to investigate the distribution of the intermolecular vector is via the distance dependent order parameters Pl+(J") defined as the averages of the even Legendre polynomials, PL(cosj r)- As with the molecular orientational order parameters those of low rank namely Pj(r) and P (r), prove to be the most useful for investigating the phase structure [22]. 

At the simplest level the orientational correlation of molecular pairs can be characterised by the averages of the even Legendre polynomials Pl(cos J ij) where is the angle between the symmetry axes of molecules i and j separated by a distance r. This correlation coefficient is denoted by... 

The values of these autocorrelation functions at times t = 0 and t = 00 are related to the two order parameters orientational averages of the second- and fourth-rank Legendre polynomial P2(cos/ ) and P4 (cos p). respectively, relative to the orientation p of the probe axis with respect to the normal to the local bilayer surface or with respect to the liquid crystal direction. The order parameters are defined as... 

The thermodynamical average 5 (a) over the Legendre polynomials P occur in the expressions for the susceptibilities, the specific heat, and the dipolar fields in Section II.B. For uniaxial anisotropy these averages read... 

The order parameter S is the orientational average of the second-order Legendre polynomial P2(a n) (n = the director), and if the orientational distribution function is approximated by the Onsager trial function, it can be related to the degree of orientation parameter ot by... 

To evaluate the averages like those in Eq. (4.78), it is very convenient to pass from cosines ((en)k) to the set of corresponding Legendre polynomials for which a spherical harmonics expansion (addition theorem)... 

The other components, namely, b f and b j, may be constructed straightforwardly using their relations with the given ones [see Eqs. (4.192)]. For a random system, that is, for an assembly of noninteracting particles with a chaotic distribution of the anisotropy axes, the average of any Legendre polynomial is zero, so that b[1> = b, and the linear dynamic susceptibility reduces to... 

For a random system, the averages of Legendre polynomials drop out and = b. With respect to formalism constructed in Section III.A, these expressions yield the asymptotic representations for formulas (4.110) and (4.111) there. 

The orientational order parameter is the second Legendre polynomial of cos9 and, where the chemical shift dispersion is reduced by rotational averaging, is given by... 

Segmental orientation in a material submitted to uniaxial elongation may be conveniently described by the average of the second Legendre polynomial ... 

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