TY - JOUR

T1 - Rigorous Proof of the Boltzmann–Gibbs Distribution of Money on Connected Graphs

AU - Lanchier, Nicolas

N1 - Funding Information:
The author would like to thank an anonymous referee for pointing out some relevant literature on money dynamics. Research supported in part by NSA Grant MPS-14-040958.
Publisher Copyright:
© 2017, Springer Science+Business Media New York.

PY - 2017/4/1

Y1 - 2017/4/1

N2 - Models in econophysics, i.e., the emerging field of statistical physics that applies the main concepts of traditional physics to economics, typically consist of large systems of economic agents who are characterized by the amount of money they have. In the simplest model, at each time step, one agent gives one dollar to another agent, with both agents being chosen independently and uniformly at random from the system. Numerical simulations of this model suggest that, at least when the number of agents and the average amount of money per agent are large, the distribution of money converges to an exponential distribution reminiscent of the Boltzmann–Gibbs distribution of energy in physics. The main objective of this paper is to give a rigorous proof of this result and show that the convergence to the exponential distribution holds more generally when the economic agents are located on the vertices of a connected graph and interact locally with their neighbors rather than globally with all the other agents. We also study a closely related model where, at each time step, agents buy with a probability proportional to the amount of money they have, and prove that in this case the limiting distribution of money is Poissonian.

AB - Models in econophysics, i.e., the emerging field of statistical physics that applies the main concepts of traditional physics to economics, typically consist of large systems of economic agents who are characterized by the amount of money they have. In the simplest model, at each time step, one agent gives one dollar to another agent, with both agents being chosen independently and uniformly at random from the system. Numerical simulations of this model suggest that, at least when the number of agents and the average amount of money per agent are large, the distribution of money converges to an exponential distribution reminiscent of the Boltzmann–Gibbs distribution of energy in physics. The main objective of this paper is to give a rigorous proof of this result and show that the convergence to the exponential distribution holds more generally when the economic agents are located on the vertices of a connected graph and interact locally with their neighbors rather than globally with all the other agents. We also study a closely related model where, at each time step, agents buy with a probability proportional to the amount of money they have, and prove that in this case the limiting distribution of money is Poissonian.

KW - Boltzmann–Gibbs distribution

KW - Econophysics

KW - Interacting particle systems

UR - http://www.scopus.com/inward/record.url?scp=85012305504&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85012305504&partnerID=8YFLogxK

U2 - 10.1007/s10955-017-1744-8

DO - 10.1007/s10955-017-1744-8

M3 - Article

AN - SCOPUS:85012305504

VL - 167

SP - 160

EP - 172

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1

ER -