The Inequality Process as a wealth maximizing process

The One Parameter Inequality Process (OPIP) long predates the Saved Wealth Model (SWM) to which it is isomorphic up to the different choice of stochastic driver of wealth exchange. Both are stochastic interacting particle systems intended to model wealth and income distribution. The OPIP and other versions of the Inequality Process explain many aspects of wealth and income distribution but have gone undiscussed in econophysics. The OPIP is a jump process with a discrete 0,1 uniform random variate driving the exchange of wealth between two particles, while the SWM, as an extension of the stochastic version of the ideal gas model, is driven by a continuous uniform random variate with support at [0.0,1.0]. The OPIP's stationary distribution is a Lévy stable distribution attracted to the Pareto pdf near the (hot) upper bound of the OPIP's parameter, ω, and attracted to the normal (Gaussian) pdf toward the (cool) lower bound of ω. A gamma pdf model approximating the OPIP's stationary distribution is heuristically derived from the solution of the OPIP. The approximation works for ω<.5, better as ω→0. The gamma pdf model has parameters in terms of ω. The Inequality Process with Distributed Omega (IPDO) is a generalization of the OPIP. In the IPDO each particle can have a unique value of its parameter, i.e., particle i has ωi. The meta-model of the Inequality Process implies that smaller ω is associated with higher skill level among workers. This hypothesis is confirmed in a test of the IPDO. Particle wealth gain or loss in the OPIP and IPDO is more clearly asymmetric than in the SWM (λ≠0). Time-reversal asymmetry follows from asymmetry of gain and loss. While the IPDO scatters wealth, it also transfers wealth from particles with larger ω to those with smaller ω, particles that according to the IPDO's meta-model are more productive of wealth, nourishing wealth production. The smaller the harmonic mean of the ωi's in the IPDO population of particles, the more wealth is concentrated in particles with smaller ω, the less noise and the more ω signal there is in particle wealth, and the deeper the time horizon of the process. The IPDO wealth concentration mechanism is simpler than Maxwell's Demon.