**6. Conclusion**

16 Stochastic Control

**Figure 3.** The factor *gb* (*N*) versus *N* for *b* = 10, 40, 100, calculated from Eq. (91). Observe that *gb* (*N*) = 1

From Section 2, here we offer a different image of the usual presentation that can be found in

Let us suppose that a system with mean energy *E*¯, and in thermal equilibrium with a heat reservoir, is observed during a very long period *τ* of time. Let *Ei* be the energy of the system

If we repeat this process of observation a huge number (toward infinity) of times, the different vectors of measurements, (*E*1, *<sup>E</sup>*2,..., *<sup>E</sup>τ*−1, *<sup>E</sup>τ*), with 0 <sup>≤</sup> *Ei* <sup>≤</sup> *<sup>τ</sup>* · *<sup>E</sup>*¯, will finish by covering equiprobably the whole surface of the *τ*-dimensional hyperplane given by Eq. (93). If it is now taken the limit *τ* → ∞, the asymptotic probability *p*(*E*) of finding the system with an

*p*(*E*) ∼ *e*

thermodynamic simile, the temperature *T* can also be calculated. It is obtained that

The *stamp* of the canonical ensemble, namely, the Boltzmann factor,

is finally recovered from this new image of the canonical ensemble.

is found by means of the geometrical arguments exposed in Section 2 [8]. Doing a

<sup>−</sup>*E*/*E*¯

*<sup>E</sup>*<sup>1</sup> <sup>+</sup> *<sup>E</sup>*<sup>2</sup> <sup>+</sup> ··· <sup>+</sup> *<sup>E</sup>τ*−<sup>1</sup> <sup>+</sup> *<sup>E</sup><sup>τ</sup>* <sup>=</sup> *<sup>τ</sup>* · *<sup>E</sup>*¯. (93)

, (94)

*E*¯ = *kT*. (95)

*<sup>p</sup>*(*E*) <sup>∼</sup> *<sup>e</sup>*−*E*/*kT*, (96)

**5.2. A microcanonical image of the canonical ensemble**

the literature [1] about the canonical ensemble.

energy *E* (where the index *i* has been removed),

at time *i*. Then we have:

for *N* = 1, and lim*N*→<sup>∞</sup> *gb* (*N*) = 0.

In summary, this work has presented a straightforward geometrical argument that in a certain way recalls us the equivalence between the canonical and the microcanonical ensembles in the thermodynamic limit for the particular context of physical sciences. In the more general context of homogeneous multi-agent systems, we conclude by highlighting the statistical equivalence of the volume-based and surface-based calculations in this type of systems.

Thus, we have shown that the Boltzmann factor or the Maxwellian distribution describe the general statistical behavior of each small part of a multi-component system in equilibrium whose components or parts are given by a set of random linear or quadratic variables, respectively, that satisfy an additive constraint, in the form of a conservation law (closed systems) or in the form of an upper limit (open systems), and that reach the equiprobability when they decay to equilibrium.

Let us remark that these calculations do not need the knowledge of the exact or microscopic randomization mechanisms of the multi-agent system in order to attain the equiprobability. In some cases, it can be reached by random forces [6], in other cases by chaotic [13, 19] or deterministic [12] causes. Evidently, the proof that these mechanisms generate equiprobability is not a trivial task and it remains as a typical challenge in this kind of problems.

The derivation of the equilibrium distribution for open systems in a general context has also been presented by considering a general multi-agent system verifying an additive constraint. Its statistical behavior has been derived from geometrical arguments. Thus, the Maxwellian and the Boltzmann-Gibbs distributions are particular cases of this type of systems. Also, other multi-agent economy models, such as the Dragalescu and Yakovenko's model [10], the Chakraborti and Chakrabarti's model [11] and the modified Angle's model [16], show similar statistical behaviors than our general geometrical system. This fact has fostered our particular geometrical interpretation of all those models.

We hope that this framework can be useful to establish other possible relationships between the statistics of multi-agent systems and the geometry associated to such systems in equilibrium.
