Projected tensors

Alternatively, the right-hand side can be written in terms of a projection tensor and a convolution (Pope 2000). The form given here is used in the pseudo-spectral method. 

The equivalence of Eqs. (2.133) and (2.136) for is a special case of a more general theorem relating inverses of projected tensors, which is stated and proved in the Appendix, Section B. Both Eqs. (2.133) and (2.136) yield tensors that satisfy Eq. (2.135), and that thus have vanishing hard components. The equivalence of the soft components of these tensors may be confirmed by substituting expansion (2.136) into the RHS of Eq. (2.132), expanding on the... 

The tensor P is a dynamical projection tensor, which operates on the... 

More generally, contracting the transposed projection tensor with a force to its right (or the projection tensor with a force to its left) produces a constrained force given by the sum of the original force and the constraint force induced by it. Such constrained forces may have nonzero hard components, but, on contraction with induce velocities that do not. 

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. 

A generalized projection tensor is defined to be any tensor for which... 

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 ... 

A useful class of alternative expressions for may be derived by inserting an arbitrary projection tensor into the divergence of on the RHS of (2.183). We note that... 

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 product of any 3N vector with this geometric projection tensor isolates the soft component of that vector. The geometrical projection tensor is a symmetric tensor, like the Euclidean identity and unlike the dynamical projection tensor. To reflect this fact, its bead indices are written directly above and below one another, with no offset to indicate whether the implicit Cartesian index associated with each bead index acts to the right or left. 

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. 

Because appears contracted with in the equation of motion, the hard components of have no dynamical effect, and are arbitrary. The values of the soft components of F depend on the form chosen for the generalized projection tensor, and reduce to the metric pseudoforce found by Fixman and Hinch in the case of geometric projection. 

The quantity Pp is a projection tensor, which reduces to the identity 8p in the case of an invertible mobility matrix, and which is always idempotent, since... 

It is straightforward to show that v fits the definition of a generalized projection tensor given in Section VIII, by using Eq. (2.295) to show that P v is idempotent, and using expansion (2.136) for and the relationship... 

Note that the form of the projection tensor P depends on the form chosen for the hard components of Z v Specifically, values of the mixed soft-hard components of P, , which are not specified by the definition of a generalized projection tensor given in Section VIII, are determined in this context by the values chosen for the mixed components of Z v, which specify correlations between hard and soft components of the random forces that are not specified by Eq. (2.295) for Z v... 

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

---