## Abstract:

Econophysics is a field of interdisciplinary research bringing into play the theories and the methods developed by the community of the physicists in order to model and answer the difficulties which arise in the economy and finance.
as an approach of analogy to statistical systems balance, we consider a simple model of a closed economic system whose exchange the money between economic agents is modeled by a gas of atoms in binary collisions and elastic where the money and middle value of it per agent respectively consider as equal of energy and temperature of gas.
Next to we study the probability distribution of the money for the various types of monetary transactions between agents where the money total is preserved and the number of economic agents is fixed. the probability of distribution takes the exponential shape of Boltzmann-gibbs characterized by an effective temperature equal to the middle value of the amount of moneys by economic agent at the balance.the emergence of the distribution of Boltzmann-gibbs is shown by the means of numerical simulations.
Key words: Econophysics, dynamics of money, probability distribution of Boltzmann-gibbs, simulation, monetary transaction.

## Introduction:

An economic system is a big statistical system characterized by the interaction of several economic agents, the applications of statistical physics promise new insights into the problems associated with analyzing complex economic and financial phenomena.
The econophysiciens arrive at predictions verified. for example the distribution of income and wealth or the stock market fluctuations [1,2].
The principle of basis of this draft is to model, simulate and study the behavior of the interactions of agents in an economic system based on the kinetic theory by the fact that there is a remarkable similarity and an important analogy (see table above) with physical systems , that is to say in comparing the economic agents to simple physical objects (molecules, particles…) which all behave the same way. From there you can do statistical analysis and derive economic laws, as they do physical models .

## Correspondance between physics and economics:

Statistical physics distribution the energy in a gas of atoms.

## Economic transactions between agents

-Conservation of money: m1 + m2 = m1′+ m2
-Detailed balance:w12®1’2’P(m1) P(m2) =w1’2’®12P(m1′)P(m2′)
-Boltzmann-Gibbs probability distribution P(m) µ exp(−m/T)
of money m, where T = ámñ is the money temperature.

## Numerical Simulation Model:

Figure 1: Histogram and points: stationary probability distribution of money P(m). Solid curves: fits to the Boltzmann-Gibbs law P(m) ∝ exp(−m/T). Vertical lines: the initial distribution of money.

The stationary distribution of money m is exponential: P(m) µ exp(−m/T)

Figure2: Time evolution of entropy. Top curve: for the exchange of a random fractionν of the average money in the system: Dm = ν M/N. Bottom curve: for the exchange of a small constant amount Dm = 1. The time scale for the bottom curve is 500 times greater than indicated, so it actually ends at the time 106.

## Conclusion:

This work is part of a very special line of research that is growing remarkably in recent years in this case study of economic figures using tools and techniques borrowed from physics(econophysics).
In this work, we studied the behavior of economic systems by analogy approach and implementing the models of statistical physics, and for that we have focused on Boltzmann-Gibbs description of the distribution money balance in a perfect economic system (case of an ideal gas).

## Bibliographie :

 R.N. Mantegna and H.E. Stanley (2000).in introduction to Econophysics
P.richmond ,Shuzler, R.colho, P.repetowes: a review of ompirical studies and models of income distributions in society.
 A.A.dragulescu ;V,M.yakovonko.(2000)statistical mechanics of many
Victor M. Yakovenko(2010) Statistical mechanics approach to the probability distribution of money.
A. Dr_agulescu and V.M. Yakovenko (2000) Statistical mechanics of money.