Response Experiments for Reaction-Diffusion Systems

In chemical kinetics a marked species can be a molecule containing a radioactive isotope and we neglect the kinetic isotope effect. In fluid mechanics a marked species can be a colored fluid for which the hydrodynamic properties (density, viscosity, diffusion coefficients) are the same as the ones of the main fluid. In population genetics a marked species can be an individual carrying a neutral mutation, and for which the main functions describing the vital statistics (natality and mortality functions, diffusion coefficients) are the same as in the case of a nonmutant individual. In the following we denote by (r, t) and p (r,t), u = 1,2. the concentrations of the not marked and marked species, respectively, and by p (r, r) = Pj r,t)+p r,t),u = 1,2. the total concentrations of the species. 

At the beginning of the experiment the system contains only not marked species. The system need not be but may be in a stationary state. The experiment consists in 

We assume the existence of generalized neutrality conditions [8,11], in the form of scaling laws, which connect the kinetic and transport laws for the whole system to the corresponding laws for the marked and not marked species, respectively  

Equations (12.132) express the fact that the marked and not marked species contribute equally to the transport process. We assume that the output fluxes, J [p (r, t)], are expressed by kinetic laws, and thus they also obey a scaling condition similar to eq. (12.132)  

It is possible to show that the susceptibility function has a physical interpretation, similar to the one existing for homogeneous systems. It is related to the probability density 

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