Idealised models for closed system relaxation
A class of model non-linear systems which describe the energy exchange of molecules with internal degrees of freedom is examined. The models, which arise purely from a consideration of the combinatorics of energy exchange, may also be formulated as models of 'collective games' and 'economic systems'. Their transport properties are determined by a Boltzmann-like equation which is first studied in a more general formalism giving useful insight into this type of problem. The equation's properties and it's approximations (viz. the linearised equation and a non-linear Fokker-Planck equation ) are derived from first principles and studied in detail. Four particular inter-related models are then considered with both discrete and continuous state variables. Two of these (one discrete and one continuous ) fall under the heading of 'molecules with diffuse scattering', and have a definite analogy with the case of Maxwell molecules in the classical Boltzmann equation, whilst the other two may be termed 'persistent scattering molecules' and in allowing persistence of state can be said to be more realistic. For each of these models an exact similarity solution of the non-linear Boltzmann equation is found in closed form, and in the case of diffuse scattering molecules this is compared with the solution of the corresponding linearised equation. These similarity solutions are only valid for a particular class of initial distributions, which although apparently close to equilibrium still differ appreciably from the linearised solution. An exact solution of these models for an arbitrary initial distribution can only be found in terms of the moments which can be calculated sequentially, and this is discussed for each model. This is the first known case of a soluble Boltzmann equation with persistent rather than diffuse scattering, moreover only in the former case is a Fokker-Planck approximation conceivable.