Mean-field theory of an asset exchange model with economic growth and wealth distribution

We develop a mean-field theory of the growth, exchange, and distribution (GED) model introduced by Liu et al. [K. K. L. al., preceding paper, Phys. Rev. E 104, 014150 (2021)] that accurately describes phase transition in limit number agents $N$ approaches infinity. The GED is generalization yard-sale which additional wealth added economic growth nonuniformly distributed to according their way determined parameter $\ensuremath{\lambda}$. shown numerically have at $\ensuremath{\lambda}=1$ be characterized critical exponents slowing down. Our treatment correctly predicts existence transition, down, values introduces an energy whose probability satisfies Boltzmann for $\ensuremath{\lambda}<1$, implying system thermodynamic equilibrium $N\ensuremath{\rightarrow}\ensuremath{\infty}$. show obtained varying $\ensuremath{\lambda}$ fixed value do not satisfy usual scaling laws, but if combination parameters, we refer as Ginzburg parameter, much greater than one held constant. discuss possible implications our results understanding systems subtle nature with both additive multiplicative noise.