### Statistical physics in economics and finance « Econophysics »

## 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 [3], 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[5].

##
*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).

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 :

[1] R.N.
Mantegna and H.E. Stanley (2000).in introduction to Econophysics

[2]P.richmond ,Shuzler, R.colho, P.repetowes: a review of ompirical studies and models of income
distributions in society.

[3] A.A.dragulescu ;V,M.yakovonko.(2000)statistical mechanics of many

[4]Victor
M. Yakovenko(2010)
Statistical mechanics approach to the probability distribution of money.

[5]A. Dr_agulescu and V.M.
Yakovenko (2000)
Statistical mechanics of money.

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