Metric null eigenvectors

Agreement of (10.7) and (10.22) then requires that the full metric matrix have exactly p independent null eigenvectors satisfying M(c+2)1L = 0 (he., et = 0). We now wish to show that these null eigenvectors can be taken as the vectors of (10.17), namely,... 

Generalized scaling behavior has been hypothesized to underlie certain observed critical-state limits (to be described in Chapter 11). We wish to show here how apparent generalized scaling behavior such as (S 10.4-3) may be inferred from the existence of null eigenvectors (r ) of the metric matrix [cf. (10.24)],... 

As discussed in Section 10.3, equilibrium phase homogeneity is associated with existence of a null eigenvector t] of the full (c + 2)-dimensional metric matrix... 

The phase-boundary conditions are closely associated with the singularity conditions for the full (c + 2)-dimensional metric M(c+2) of a c-component system. In each case, a metric singularity is associated with a null eigenvector of M(c+2),... 

Of course, any linear combination of null eigenvectors also satisfies (12.54a, b), so we are able to choose individual vectors of this null manifold with considerable freedom. We know in general that a / -dimensional manifold of linearly independent null eigenvectors must exist, in order that the rank of the metric matrix,... 

The physical meaning of a null metric eigenvector (10.24) is that the thermodynamic state Ms has no response in the specified direction, i.e., that an overall change dY of the form... 

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