**Author details**

Serena Doria *Department of Engineering and Geology, University G.d'Annunzio, Chieti-Pescara, Italy*

## **8. References**

[1] P. Artzner, F. Delbaen, J. Elber, D. Heath.(1999) Coherent measures of risk. Mathematical Finance, 9, 203-228.

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**1. Introduction**

http://dx.doi.org/10.5772/46152

equilibria [3, 4].

statistical equilibrium.

Classical statistical physics deals with statistical systems in equilibrium. The ensemble theory offers a useful framework that allows to characterize and to work out the properties of this type of systems [1]. Two fundamental distributions to describe situations in equilibrium are the Boltzmann-Gibbs (exponential) distribution and the Maxwellian (Gaussian) distribution. The first one represents the distribution of the energy states of a system and the second one fits the distribution of velocities in an ideal gas. They can be explained from different perspectives. In the physics of equilibrium, they are usually obtained from the principle of maximum entropy [2]. In the physics out of equilibrium, there have recently been proposed two nonlinear models that explain the decay of any initial distribution to these asymptotic

**Geometrical Derivation of Equilibrium** 

Ricardo López-Ruiz and Jaime Sañudo

Additional information is available at the end of the chapter

**Distributions in Some Stochastic Systems** 

**Chapter 4**

In this chapter, these distributions are alternatively obtained from a geometrical interpretation of different multi-agent systems evolving in phase space under the hypothesis of equiprobability. Concretely, an economic context is used to illustrate this derivation. Thus, from a macroscopic point of view, we consider that markets have an intrinsic stochastic ingredient as a consequence of the interaction of an undetermined ensemble of agents that trade and perform an undetermined number of commercial transactions at each moment. A kind of models considering this unknowledge associated to markets are the gas-like models [5, 6]. These random models interpret economic exchanges of money between agents similarly to collisions in a gas where particles share their energy. In order to explain the two before mentioned statistical behaviors, the Boltzmann-Gibbs and Maxwellian distributions, we will not suppose any type of interaction between the agents. The geometrical constraints and the hypothesis of equiprobability will be enough to explain these distributions in a situation of

Thus, the Boltzmann-Gibbs (exponential) distribution is derived in Section 2 from the geometrical properties of the volume of an *N*-dimensional pyramid or from the properties of the surface of an *N*-dimensional hyperplane [7, 8]. In both cases, the motivation will be a

work is properly cited.

©2012 López-Ruiz and Sañudo , licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original

©2012 López-Ruiz and Sañudo licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

