Adiabatic elimination of fast relaxing variables in socio-economic dynamical models (Q1072977)

The paper deals with the application of projection techniques, known as adiabatic elimination procedures (AEP), to the quantitative descriptions of complex socio-economical systems. The need to use reduced descriptions can be traced back to the human brain's limits in dealing with several variables simultaneously. While computer simulations allow one to deal with systems of many interacting variables, it is argued that some sort of projection upon a subset of few relevant variables is still necessary, in order to extract meaningful information out of the simulations. A method is envisaged to deal with complex cultural systems, based on interplay between reduced and detailed descriptions. The emphasis is placed on formal projection techniques which allow one to extract a reduced dynamical model from a more detailed one, in the case where there are two separated time scales and the relevant variables are also the slower ones. The reduced models are therefore meaningful for a long time description of the system. Two such techniques are applied to some models of the man-environment interaction in a tourist settlement, namely the direct AEP in the case of deterministic dynamical systems and the Zwanzig AEP in the case where stochastic models should be preferred. Among the different phenomena described we mention the presence of noise induced transitions. An important question studied in this paper concerns the ''Itô or Stratonovich'' choice in stochastic calculus. The case considered here is one in which the nonlinear Langevin equation with multiplicative noise is obtained by a more detailed standard nonlinear Langevin equation through a crude adiabatic elimination. It is shown that the correct PDE associated to the reduced Langevin equation can be either of the Itô or of the Stratonovich type depending upon the relative time constants of the fast irrelevant variables. Such result is obtained, with a technique due to P. Grigolini, applying the Zwanzig AEP up to fourth order to the Fokker-Planck equation for the detailed system, thus obtaining an evolution equation for the reduced probability density which reduces either to the Itô or to the Stratonovich form if certain conditions on the damping constants of the fast variables are satisfied.