increases or decreases $m_p$ by [@hughesbook] $$|\langle \Delta m_p\rangle| \simeq \sqrt{(2/\pi)\mathcal{N}}  \simeq \sqrt{4\Delta \tilde{t}\over \pi} m_p.
\label{eq:Dmpmp}$$ Identifying Eq. (\[eq:DApAp\]) with (\[eq:Dmpmp\]) under the relation $m_p = M \, A_p$, we find that $c \simeq \sqrt{4\Delta \tilde{t}\over \pi}$ yielding $\Delta \tilde{t}\simeq 0.01$. We use the rescaled time $$\tau = {\tilde{t}\over \Delta\tilde{t}}$$ such that the increase of $\tau$ by $1$ corresponds to one year in reality.

Fitting the simulated second moment to data
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![Comparison of simulation results and empirical data. (a) The share distribution at different rescaled times $\tau$ obtained by $P_\tau(A) = M\, p_\tau(m=A\, M)$ from the simulations of the urn model with $b=2$, the number of sites $N=508$, and the particle density $\rho=50$. The dashed line represents the power law decay. (b) The second moment $\langle A^2\rangle$ versus the rescaled time $\tau$ for different $\rho$’s. The shaded region indicates the empirical-data range $2.2\times 10^{-5} \leq \langle A^2\rangle \leq 4.0\times 10^{-5}$. Inset: The rescaled time entering and leaving the empirical-data range $\tau_i$ and $\tau_f$ are given for each value of $\rho$. (c) The share distribution $P(A)$ in year 1998 and from model simulations at $\tau=\tau_f$ for each $\rho$. The Boltzmann distribution $P(A)=N e^{-N\, A}$ with $N=508$ is also shown. 