Projected tensors reciprocal vectors

We also define dynamical reciprocal basis vectors that are closely related to the dynamical projection tensor defined above. We define a set off vectors... 

In this section, we introduce generalized definitions of sets of reciprocal basis vectors, and of corresponding projection tensors, which include the dynamical reciprocal vectors and the dynamical projection tensor introduced in Section VI as special cases. These definitions play an essential role in the analysis of the constrained Langevin equation given in Section IX. 

We now show, conversely, that for each projection tensor P j, there exists a unique set of corresponding reciprocal basis vectors that are related to P j, by Eq. (2.195). To show this, we show that the set of arbitrary numbers required to uniquely define such a projection tensor at a point on the constraint surface is linearly related to the set of fK arbitrary numbers required to uniquely specify a system of reciprocal vectors. A total of (3A) coefficients are required to specify a tensor P v- Equation (2.193) yields a set of 3NK scalar equations that require vanishing values of both the hard-hard components, which are given by the quantities n P = 0, and of the fK mixed hard-soft ... 

We now consider several possible ways of defining a system of reciprocal vectors and a corresponding projection tensor. 

By repeating the reasoning applied in Section VI to the dynamical reciprocal vectors, we may confirm that any vectors so defined will satisfy Eqs. (2.186)-(2.189). It will hereafter be assumed that (except for pathological choices of S v) they also satisfy completness relation (2.190). A few choices for the tensors S v and T yield useful reciprocal vectors and projection tensors, for which we introduce special notation ... 

Dynamical reciprocal vectors and m) , which were introduced in Section VI, are defined by taking T = H in Eq. (2.208). The corresponding projection tensor is the dynamical projection tensor P v-... 

The pseudoforce associated with the dynamical projection tensor may be calculated by using dynamical reciprocal vectors to evaluate Eq. (2.205). In the simple case of a coordinate-independent mobility as in a free-draining model or a model with an equilibrium preaveraged mobility, we may use Eq. (A. 17) to express as a derivative... 

Note that the soft reciprocal vectors b are expanded in a basis of tangent vectors, and so are manifestly parallel to the constraint surface (as indicated by the use of a tilde), while the hard reciprocal vectors ihi are expanded in normal vectors, and so lie entirely normal to the constraint surface (as indicated by the use of a caret). These basis vectors may be used to construct a geometric projection tensor... 

The inertial and geometrical projection tensors, and associated reciprocal vectors, are identical for models with equal masses for all beads, in which the mass tensor is proportional to the identity. 

These are the reciprocal basis vectors corresponding to the generalized projection tensor Z v identified in Eq. (2.298). With this definition,... 

Derivatives of the determinants S and f of the generic projected tensors Sah and Tij defined in Eqs. (2.20) and (2.24) may be expressed compactly in terms of the reciprocal vectors that are generated by applying Eqs. (2.207) and (2.208) to the corresponding Cartesian tensors S v and respectively. Using Eq. (A.14) to differentiate In 5 with respect to a soft variable gives... 

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