### Following the Money around the Eurozone

Take a fistful of euro coins and examine the obverse sides; you may be surprised at the wide variety of designs. The eurozone is a monetary union of 19 member states of the European Union that have adopted the euro as their primary currency. In addition to these countries, Andorra, Monaco, San Marino and Vatican City use euro coins so, from 2015, there have been 23 countries, each with its own national coin designs. For the €1 and €2 coins, there are 23 distinct national patterns; for the smaller denominations, there are many more. Thus, there is a wide variety of designs in circulation.

The euro was introduced in January 2002. The common sides of the €1 and €2 coins depict the denomination, the currency, a map of Europe and twelve stars representing the Union. The obverse, or national side, is distinct for each country. For Ireland, all the coins feature the harp — a national symbol — the country name, Éire, and the year of minting. Some countries use several designs for different denominations. It is generally easy to recognise the country of origin of the coins.

The euro currency is managed by the European Central Bank. As of December 2021, there were approximately 140 billion coins in circulation around the eurozone.

This dispersion process provided an opportunity to track the paths of coins between nations. Approximately 10% of the coins in each country are exported every year, with a similar volume of “foreign” coins flowing inwards.

If the dispersion process were perfect, the proportion of different coins at equilibrium would be similar everywhere in the eorozone, with the percentage of each variety being equal to its proportion relative to the totality of coins.

Currents of Currency

An applied mathematician, faced with the problem of understanding the distribution of euro coins, might begin by treating the entire collection as a fluid covering the eurozone. The fluid flow would represent the movement of coins from one place to another. Assuming that the total number of coins does not change, a continuity equation can be formulated, ensuring that inflows and outflows are in balance.

The dispersion process could be modelled using a diffusion equation similar to that first formulated by Joseph Fourier, a partial differential equation like $\displaystyle \frac{\partial\theta}{\partial t} = \nu \nabla^2 \theta$

which governs the flow of heat and many other processes. Initially, the fluid over each nation could be given a distinct colour. Over time, the colours would mix as the fluid parcels diffused into each other, ultimately becoming a murky brown colour.

However, the diffusion model has a serious shortcoming: diffusion processes modelled in this way are local, while the mixing of coins involves long-range changes. A sun-seeking Finn may carry coins from Helsinki to the Canary Islands in a few hours. So, the model might work for cross-border exchanges, it is unlikely to be useful for continental-scale mixing.

More general transport processes must be included. For fluids, the Navier-Stokes equations $\displaystyle \frac{\partial \mathbf{V}}{\partial t} + \mathbf{V\cdot}\boldsymbol{\nabla}\mathbf{V} + \frac{1}{\rho}\boldsymbol{\nabla}p = \nu\boldsymbol{\nabla}^2\mathbf{V} \,,$ $\displaystyle \frac{\partial \rho}{\partial t} + \boldsymbol{\nabla\cdot}\rho\mathbf{V} = 0$

simulate an enormous range of flows. Some variation of this might be formulated to simulate coin migration. It is not too difficult to define quantities corresponding to fluid velocity ${\mathbf{V}}$ and density ${\rho}$.  If we assume the fluid is incompressible, conservation of coinage implies a continuity equation ${\boldsymbol{\nabla\cdot}\mathbf{V} = 0}$. But what corresponds to pressure ${p}$?

Perhaps the pressure represents fluid depth, as for the shallow water equations. But we would not expect the pressure gradient to drive currency outwards from a maximum; indeed, quite the opposite might apply! The diffusion term ${\nu\boldsymbol{\nabla}^2\mathbf{V}}$ represents local spread, but rapid long-distance exchanges must also be considered, and the means of doing this are not obvious. A new equation is required.

Numerical Modelling

If we wish to simulate the movement of all national varieties of coins through all the regions, a mathematical model with hundreds of differential or difference equations may be used. Numerous parameters are needed, and these must be determined by studying the observed patterns [Seitz, et al., 2009]. Many factors may be modelled, including the mobility of travellers, preference for different destinations, cross-border commuting, and the activities of hoarders. Temporary fluctuations like ski holidays and major sporting events can be observed. Another factor is the outflow of coins from the eurozone, where collectors may keep rather than spend them.

Seitz et al (2009) also model coin migration using a Markov Chain model. Their transition matrix represents the probability that a German coin will drift abroad in a fixed period, the probability that a foreign coin will enter Germany, and so on. This model approach yields much the same results as the model based on difference equations.

Both predictions and surveys suggest that the distribution of coins is close to equilibrium. However, economic and social disruptions can change this, and any expansion or contraction of the eurozone needs to be allowed for.

Entropy

Entropy is a useful quantity for measuring the distribution of coins. Entropy is a measure of the disorder of the distribution. At the initial introduction of the currency, each national design was found only in one country. Thus, in Ireland, the probability was 1 that a random coin was Irish and 0 that it was from another country.

Suppose there are in total ${N}$ one-euro coins, with ${n}$ Irish coins. Initially, all the Irish coins are located in Ireland. Thus, the probability is ${p(\mbox{Irish}) = 1}$ that a coin randomly selected in Ireland is Irish. The probability that it is not is ${p(\mbox{Not Irish}) = 0}$. More generally, we can define the entropy as $\displaystyle S = - [ p(N_\mathrm{I})\log p(N_\mathrm{I}) + p(N_\mathrm{J})\log p(N_\mathrm{J}) ] \,.$

where ${N_\mathrm{I}}$ is the proportion of Irish coins and ${N_\mathrm{J}}$ the proportion of non-Irish coins in Ireland.

The entropy is a measure of the disorder of a system. At the initial time, the entropy is zero. As the coins become mixed through travel and other processes, this quantity increases. Ultimately, if the number of coins in Ireland remains fixed and transfers of coins continue, the fraction of Irish coins in the country tends to the ratio of Irish coins to the total number of coins. This is the equilibrium state of complete mixing and maximum entropy. ${\bullet}$ Seitz, Franz, Stoyan, Dietrich and Tödter, Karl-Heinz, 2009: Coin migration within the euro area. Deutsche Bundesbank, Frankfurt, Discussion Paper Series 1 No. 27/2009. ${\bullet}$ Deutsche Bundesbank, 2019: The mixing of euro coins in Germany. Deutsche Bundesbank Monthly Report, December 2019, No. 73.