Geometrical Derivation of Equilibrium Distributions in Some Stochastic Systems

Abstract

2Stochastic Control multi-agent economic system with an open or closed economy, respectively. The Maxwellian (Gaussian) distribution is derived in Section 3 from geometrical arguments over the volume or the surface of an N-sphere [7,9]. In this case, the motivation will be a multi-particle gas system in contact with a heat reservoir (non-isolated or open system) or with a fixed energy (isolated or closed system), respectively. And finally, in Section 4, the general equilibrium distribution for a set of many identical interacting agents obeying a global additive constraint is also derived [7]. This distribution will be related with the Gamma-like distributions found in several multi-agent economic models. Other two geometrical collateral results, namely the formula for the volume of high-dimensional symmetrical bodies and an alternative image of the canonical ensemble, are proposed in Section 5. And last Section embodies the conclusions.