## Tuesday, July 23, 2013

### A diffusion analogy for the quantity theory of money

One of the applications of the information transfer framework to describe physical processes is for diffusion. In that context, we look at the equation $$(\Delta u)^{1/\kappa} \sim \Delta q$$ Which becomes $$(\Delta x)^{1/\kappa} \sim \Delta t$$ $$(\Delta x)^{2} \sim (\Delta t)^{2 \kappa}$$ The authors of the linked paper take $\kappa = 1/2$ and recover Fick's law. In the model of supply and demand, we have $$(Q^s)^{2} \sim (Q^d)^{2 \kappa}$$ The quantity theory of money ($\kappa = 1/2$) corresponds to standard diffusion. The results in this blog have $\kappa > 1/2$, which would correspond to "anomalous diffusion" or "superdiffusion". See, e.g. here. Superdiffusion has a tendency to exhibit Levy flights which are observed in markets. However, the analogy is not terribly intuitive. "Time" can go forward (GDP growth) or backward (recession).