For a detailed description of the money exchange models, see the paper A. A.
Dragulescu and V. M. Yakovenko, "Statistical mechanics of money",
The European Physical Journal B, v. **17**, pp. 723-729
(2000), pdf.

The computer animation videos presented below were produced by Justin Chen, an undergraduate student at Caltech, as a summer project in 2007 under guidance of Victor Yakovenko.

Initially, all agents are given the same amount of money. After simulation starts, certain amounts of money D*m*
are repeatedly transferred from one randomly selected agent to another. If
the selected agent does not have enough money to pay D*m*
, the transaction does not take place, and simulation continues with another
pair of agents. The transfers of money are supposed to represent payments
from one agent to another for certain products and services. However, we
do not keep track of the goods offered in exchange for money and only keep track
of money balances of all agents. The random character of money transfers
is supposed to reflect the wide
variety of products and connections in modern economy.

The histogram on the upper plot shows time evolution of the money distribution between the agents. As time goes on, the initial d-function distribution of money broadens. The vertical scale is adjusted with time, so that the histogram fits into the screen. Eventually, the money distribution stabilizes at the exponential shape, shown by the reference line in red, when the system reaches statistical equilibrium.

The bottom plot shows time evolution of the entropy of the money distribution. The entropy increases in time from the initial value 0 to the maximal value achieved when the system reaches statistical equilibrium.

Models with different rules for the transferred amount D*m*
were considered.

**Model 1:**video file**animation-1.avi**, 1.4 MB

In each transaction, D*m*is selected to be a random fraction of $1000, which is the average amount of money per agent in the system. The simulation is performed with 5000 agents. The histogram shown in the animation is obtained by averaging the histograms of 10 runs of simulations to produce a smoother distribution.**Model 2:**video file**animation-2.avi**, 2.6 MB

In this model, D*m=*$1 has the same value for all transactions. To speed up convergence, each agent is initially given the smaller amount $10. The simulations are performed with 500 agents and repeated 1000 times. The histogram shown in the animation is obtained by adding the histograms of all runs. Because D*m*is small, the money distribution evolves in a diffusive manner. The initial distribution first broadens into a symmetric, Gaussian curve. Then, probability starts to accumulate around*m*=0, which acts as the impenetrable boundary, because money balances of agents cannot go below zero. As a result, the probability distribution acquires a skewed (asymmetric) exponential shape.

We observe that a narrow initial distribution, where all agents have the same amount of money, is unstable and evolves in time into a broad and skewed distribution, where many agents have low money balances and few agents have high money balances. Eventually, the distribution of money reaches statistical equilibrium at the exponential shape (the Boltzmann-Gibbs distribution), in agreement with general principles of statistical physics and the principle of maximal entropy.

For more information on this subject, see the papers at
http://www2.physics.umd.edu/~yakovenk/econophysics/. When the rules for
money transfers have different symmetries, e.g. D*m*
is proportional to the money balance of an agent, other distributions may be
obtained.

*Last updated
August 22, 2007*

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