Slow manifold

As we shall see below, a useful strategy to identify the slow manifold is to calculate an average of the momentum, p, over r time units during a molecular dynamics propagation... 

In order for the averaged momentum p to point along the slow manifold (i.e., for the components of momentum in the fast manifold to average to zero), one has to choose the averaging time r so as to be several times longer than the period of the fast modes but shorter than those of the slow modes. 

The identification of the slow manifold introduced in the previous section for the MEHMC method turns out to be effective not only for enhanced thermodynamic... 

The objective of the method presented here is to develop a momentum distribution that will bias path dynamics along the slow manifold, permitting the efficient calculation of kinetic properties of infrequent reactions. 

Just like in the MEHMC method described in Sect. 8.7, we can identify the slow manifold from the time average of the momentum, e.g., by choosing a conformational direction es = Po/ Po where po is calculated as in (8.31). 

In cases when the slow manifold is higher than one-dimensional (which is likely to be the case for complex biomolecular conformational changes), the guiding vector es is to be calculated for each initial configuration, or, if there is little variation in the slow direction for certain initial regions, a single es can be applied to some or all of the initial points. 

An example of a smart tabulation method is the intrinsic, low-dimensional manifold (ILDM) approach (Maas and Pope 1992). This method attempts to reduce the number of dimensions that must be tabulated by projecting the composition vectors onto the nonlinear manifold defined by the slowest chemical time scales.162 In combusting systems far from extinction, the number of slow chemical time scales is typically very small (i.e, one to three). Thus the resulting non-linear slow manifold ILDM will be low-dimensional (see Fig. 6.7), and can be accurately tabulated. However, because the ILDM is non-linear, it is usually difficult to find and to parameterize for a detailed kinetic scheme (especially if the number of slow dimensions is greater than three ). In addition, the shape, location in composition space, and dimension of the ILDM will depend on the inlet flow conditions (i.e., temperature, pressure, species concentrations, etc.). Since the time and computational effort required to construct an ILDM is relatively large, the ILDM approach has yet to find widespread use in transported PDF simulations outside combustion. 

Figure 6.7. Sketch of a one-dimensional, non-linear slow manifold. The dashed curves represent trajectories in composition space that rapidly approach the slow manifold. 

Note that the dimensions of the fast and slow manifolds will depend upon the time step. In the limit where At is much larger than all chemical time scales, the slow manifold will be zero-dimensional. Note also that the fast and slow manifolds are defined locally in composition space. Hence, depending on the location of 0q], the dimensions of the slow manifold can vary greatly. In contrast to the ILDM method, wherein the dimension of the slow manifold must be globally constant (and less than two or three ), ISAT is applicable to slow manifolds of any dimension. Naturally this flexibility comes with a cost ISAT does not reduce the number (Ns) of scalars that are needed to describe a reacting flow.168... 

Everything that we have done so far in this example is completely standard. The next step in which we identify the slow fundamental thermodynamic relation in the state space M2 (i.e., we illustrate the point (IV) (see (120))) is new. Having found the slow manifold Msiow in an analysis of the time evolution in Mi, we now find it from a thermodynamic potential. We look for the thermodynamic potential ip(q,p, e, //. q, e, v) so that the manifold Msiow arises as a solution to... 

We shall organize the Chapman-Enskog reduction into four steps Step 1) Initial suggestion M w C Mi for the slow manifold Msiow C Mi is made. It is a manifold that has a one-to-one relation with the space M2. We can regard it as an imbedding of M2 in Mi. Step 2) The vector field v.f.) 1 is projected on M w (i.e., v.f. jf 1 denoting the vector field v. /.) 1 attached to a point of M, is projected on the tangent plane of M ow a at point). The projected vector field is denoted by the symbol... 

M w and its projection (vf.y 1"" becomes smaller (as small as possible) than for the slow manifold M w. Step 4) The slow fundamental thermodynamic relations associated with the slow manifolds M w and M w are identified. 

It is easy to verify that the slow time evolution on M2 corresponding to the choice (135) of the slow manifold (i.e., the time evolution governed by the vector field (vf.jp) is the Euler time evolution (see (95) with... 

The slow time evolution corresponding to the slow manifold is... 

We shall make the fourth step. We shall make it first for the slow manifold M wand then for its deformation M w. 

Let the slow manifold be M w (see 135) and the slow time evolution the Euler equations generated by the vector field (vf, )f (see 136)). We look... 

The fast time evolution, i.e., the time evolution describing the approach the slow manifold, is governed by (82) (i.e., M jw with the dissipation potential E given by (79) and with — J -). 

Now we turn to the slow manifold M w and repeat the analysis that we have just made above for M w. We begin by looking for the thermodynamic potential

[Pg.126]

The reduction techniques which take advantage of this separation in scale are described below. They include the quasi-steady-state approximation (QSSA), the computational singular perturbation method (CSP), the slow manifold approach (intrinsic low-dimensional manifold, ILDM), repro-modelling and lumping in systems with time-scale separation. They are different in their approach but are all based on the assumption that there are certain modes in the equations which work on a much faster scale than others and, therefore, may be decoupled. We first describe the methods used to identify the range of time-scales present in a system of odes. 

Fig. 4.7. A phase plot showing the approach of trajectories to the slow manifold for the...

Since, in the simulation of combustion problems, the calculation of the reaction terms must be carried out many times, it is important that this is done efficiently. For this reason look-up tables are often used in preference to an explicit calculation of each function such as the rates of change of species concentration. Evaluations can then be performed using interpolations from the table rather than by integrating large systems of non-linear equations. The main purpose of the ILDM technique is to produce tables of species and system properties relating to points on the slow manifold for use in the CFD code. The reduction of the dimension of phase space by its restriction to a manifold reduces the size of the tables and, therefore, the burden on computer storage and look-up times. 

Ql is the submatrix of corresponding to the n - n c fastest variables, and so the slow manifold is defined by assuming that motion in the direction of the Schur vectors associated with the fast variables is zero. This means that the system is in local equilibrium with respect to its fastest time-scales. 

---