By Ilya Prigogine, Stuart A. Rice
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Extra info for Advances in Chemical Physics, Volume 43
37) where one phase is assumed ( P = I), and the number of components is C = n - d. As parameters for this system, choose the reciprocal temperature /3 = 1/ RT, and the dimensionless product p*, j E [ 1, n - d], where is the chemical potential of speciesj . These parameters are the fundamental variables of the entropy representation of thermodynamics (Callen) and are the most profound parameters because the partition functions of the statistical mechanical ensembles all take the same general form only when expressed in these parameters.
N corresponds to the parameter point (h,j). The existence of multiple steady states for this j means that these equations have other solutions, which we call a’(j),a”(j), . . If the solution a’@ corresponds to the steady state (h’,j’),whose concentration at steady state is Xi”= hi-’, the dimensionless concentration at steady state is therefore h / h and this is a’@. 51) using x = a’(i). Hence for j = 1, . . 57) 37 STABILITY OF COMPLEX REACTION NETWORKS If this sytem of equations is solved for j' as a function of j, the solution will be unique only if II, is a simplex because the redundant parameters have been used.
22) implies that M@) has at least p zero eigenvalues. 2 1) and a perturbation in the corresponding eigenspace (space spanned by the corresponding eigenvectors) takes the system into a new steady state. Thus there is a p-dimensional set of steady states near any particular steady state. Suppose the original steady state lies in n(@. t Since C E R:-d the nearby steady states form an ( n - d)-dimensional set. If M@) is n X n, these steady states are the steady states associated with the generalized eigenspace of the n - d vanishing eigenvalues.