Time evolution of a system of two valued interacting elements : a microscopic interpretation of birth and death equations

We consider here one of the simplest possible systems with N interacting particles. It has the following features : (i) the state variable of each particle takes the values σ_i(= ?1 ; (ii) the interaction is chosen in such a way to preserve the symmetry of the distribution function p(σ₁, σ₂, tdot], σ _N ; t) with respect to the σ _i and (iii) the evolution of the system is defined in a stochastic way by the transition probabilities of each particle as depending on the state of all other particles. The master equation of this Markov process is shown to be the equation of a general birth and death process in one dimension. More precisely, the birth and death process is : linear if the particles are independent ; quadratic if there is a binary interaction ; or cubic if there is a third-order interaction. We develop the reduced distribution equations hierarchy (which is the analogue of the BBGKY hierarchy) and we study under what conditions this hierarchy closes. Then we show that for specific systems there is a conserved quantity (in the mean) and we discuss for what kind of intercation there is respectively an H-theorem and a postulate of equal a priori probabilities at equilibrium. It appears in particular that this postulate should not be true in the strong form in which it is usually stated.