Many particle interaction in the kinetic exchange models

Another idea that may be used to improve kinetic exchange models is the introduction of many-particle interactions. In the ideal gasses, these interactions do not occur, but in the social systems, they might have interesting consequences.

In this post, let us modify the elementary kinetic exchange model by assuming that only \(N\)-particle interactions occur. Let the combined wealth (energy) of all agents (particles) \(W(t,I)=\sum_{i\in I}w_i(t)\), be redistributed uniformly among themselves:

\begin{equation} w_i(t+1) = \frac{\varepsilon_i}{E(I)} W(t, I) , \quad\forall i \in I . \end{equation}

In the above \(I\) is a set of agents (particles) selected for the interaction. Random number \(\varepsilon_i\) is divided by \(E(I)=\sum_{i\in I}\varepsilon_i\) factor, to use the conservation of wealth (energy).

It should be evident that the wealth (energy) distribution will approach uniform distribution when \(N\) grows and approaches the number of agents (particles) in the system. In the applet below you can see that in case \(N=2\) we recover the exponential distribution. As \(N\) grows larger and closer to the total number of agents (particles) in the system, probabilities to observe agents with lower wealth (particles with lower energies) flatten out.

Note 2025: Earlier iteration of this post also featured another model, which is many particle interaction model, but has little common with kinetic exchange models. That model is now featured in a dedicated post.