Agent-Based Wealth Model: Proving Inequality Consider the following agent-based model:

*

*There are $N$ agents

*Every agent starts with $1

*At each time interval (i.e. at each step), every agent gives \$1 to a randomly chosen agent.

I want to find how unequal the wealth distribution becomes over a long period of time.
After running a simulation for a large number of agents, I find that the wealth distribution becomes over a long period of time approaches (what I am "by eye" guessing to be) a Boltzmann distribution.
I am curious as to why this happens from a derivation standpoint. I have tried to find sample derivations online, but only find kinetic-model related Boltzmann distribution derivations. Can anyone point me to any resources or share a derivation that explain this result?
 A: Not a full answer (but too long for a comment).
Consider the set $E$ of all vectors in $\mathbb Z_{\ge 0}^N$ which sum to $N$. Your game can be seen as a homogeneous Markov chain with state space $E$.
It is irreducible (exercise) and therefore, since $E$ is finite, it is positively recurrent (see 1). The chain is also aperiodic since there is a probability $>0$ that you go from one state to itself (everyone who has money gives themselves money, this is defined to happen with probability $>0$).
Therefore, by the Markov chain convergence Theorem (cf. [2; Satz 17.52 and Satz 18.13]), starting from any probability distribution over $E$, the distribution over $E$ of state $n$ of the Markov chain (corresponding to iterating your game $n$ times) will converge, in the sense of the total variation distance, to the unique distribution on $E$ which is invariant with respect to one iteration of the Markov chain.
Therefore, you should compute this irreducible measure on $E$ and then check that the „typical vector in $E$“ according to this distribution corresponds approximately to your conjectured wealth distribution. (Not an easy task in my opinion.)
Footnote: If the agents cannot give money to themselves, the chain is not always irreducible: If you have $N=2$  then the two agents keep swapping one dollar so you can never leave your initial state in that case.
References
[2]: Achim Klenke, Wahrscheinlichkeitstheorie, 4. Auflage, 2020.
