Statistical model with a standard $\Gamma$ distribution

We study a statistical model consisting of $N$ basic units which interact with each other by exchanging a physical entity, according to a given microscopic random law, depending on a parameter $\lambda$. We focus on the equilibrium or stationary distribution of the entity exchanged and verify through numerical fitting of the simulation data that the final form of the equilibrium distribution is that of a standard Gamma distribution. The model can be interpreted as a simple closed economy in which economic agents trade money and a saving criterion is fixed by the saving propensity $\lambda$. Alternatively, from the nature of the equilibrium distribution, we show that the model can also be interpreted as a perfect gas at an effective temperature $T\left(\lambda \right)$, where particles exchange energy in a space with an effective dimension $D\left(\lambda \right)$.

Figure 1

Equilibrium money distributions for different values of the saving propensity $\lambda$, in the closed economy model defined by Eqs. (2, 4). The continuous curves are the fitting functions, defined in Eq. (6) with the values of $n$ governed by Eq. (5). Note that for the simulation $⟨x⟩=1$.

Figure 2

Same quantities as in Fig. 1, but on a linear-logarithmic scale. Note that for the simulation $⟨x⟩=1$.

Figure 3

The parameters $n$ (top) and $a_n$ (bottom) vs $\lambda$, obtained from numerical data (dots) and the corresponding analytical formulas (continuous curves) given by Eqs. (5, 6), respectively. Note that for the simulation $⟨x⟩=1$.

Figure 4

Variation of the mode with the parameter $\lambda$. Dots represent the numerical data, while the dashed curve was obtained theoretically.