Money Equilibrium? Maximizing Entropy w/ Given Denominations

[alxm]

Just a musing I had.. how far from thermodynamic equilibrium is the change in my pocket?

Obviously having any given amount in the smallest denomination only, maximizes a degree of freedom in the sense that I can form the largest number of sums with it, meaning higher entropy. But on the other hand, I have lower entropy in the sense that I then only have one kind of coin.

So the question is.. for a given distribution of denominations, e.g. {1, 5, 10, 25, 50, 100} and a given amount of money (e.g. 100 cents), which distribution of coins will maximize the entropy?

Any combinatorics fans want to take a crack at this?

 

[davee123]

I guess the question is how you measure entropy given the two possibilities. My instinct answer is that with 5 pennies, 2 nickels, 1 dime, 1 quarter, and 1 half dollar, you can create ALL integer values between 0-100, and have the widest variety of coins possible that still allows you to create all values. But that's far from a scientific answer.

I think the interesting question would be somewhere along the lines of:

Given T coins totaling 100 cents in value, N coins are chosen randomly where 1 <= N <= T. The value of the N coins is V. What distribution of coins should be chosen to maximize the probable number of distinct values for V?

In other words, if you had 100 pennies, then V = N for all values of N. But with a different distribution, V is variable. And obviously, with 95 pennies and 1 nickel, V you have potential for V = N or V = N+4. But also, N+4 is relatively unlikely when choosing only a few coins.

So, how would you measure that? I'm pretty sure I could write a program to simulate this action, and get a good guess at the best distribution. But how do you measure the randomness of a given sample set of V for a given value N? Do you take the standard deviation? The number of distinct values? Some combination of the two? I honestly don't recall how you assign a specific number for a value of entropy...

DaveE

 

[Architeuthis Dux]

I find this question intriguing and it raises some interesting points about the concept of equilibrium and entropy in a monetary system.

First, let's define what is meant by equilibrium in this context. In thermodynamics, equilibrium refers to a state where there is no net flow of energy or matter between different parts of a system. In the case of a monetary system, we can think of equilibrium as a state where there is no net flow of money between different denominations. This would mean that the distribution of coins in circulation is stable and does not change over time.

Now, let's consider the concept of entropy. In thermodynamics, entropy is a measure of the disorder or randomness in a system. In the context of a monetary system, we can think of entropy as a measure of the diversity or variety of coins in circulation. A higher entropy would mean a greater variety of coins in circulation, while a lower entropy would mean a more limited variety of coins.

So, how does this relate to the distribution of denominations and the amount of money in circulation? As the original post mentions, having a larger number of smaller denominations would maximize the number of possible sums that can be formed, leading to a higher entropy. However, this would also mean a lower entropy in terms of the diversity of coins, as there would only be one type of coin in circulation.

On the other hand, having a larger number of larger denominations would lead to a lower number of possible sums, resulting in a lower entropy in that sense. However, it would also mean a higher entropy in terms of the diversity of coins, as there would be a greater variety of coins in circulation.

So, to answer the question of which distribution of coins would maximize entropy, it would depend on what aspect of entropy we are considering. If we are looking at the diversity of coins, then a distribution with a greater variety of denominations would likely maximize entropy. However, if we are looking at the number of possible sums that can be formed, then a distribution with a larger number of smaller denominations would likely maximize entropy.

In terms of the distance from thermodynamic equilibrium, it's difficult to say without knowing the specific distribution of denominations and the amount of money in circulation. However, it's safe to say that any distribution that leads to a stable and balanced circulation of money would be closer to equilibrium than one that is constantly changing and fluctuating.

In conclusion, the concept of equilibrium and entropy can provide an