**1. Introduction**

12 Stochastic Control

[5] D. Denneberg. (1994). *Non-additive measure and integral*. Kluwer Academic Publishers. [6] S. Doria. (2007). Probabilistic independence with respect to upper and lower conditional probabilities assigned by Hausdorff outer and inner measures. *International Journal of*

[7] S. Doria. (2010). Coherent upper conditional previsions and their integral representation with respect to Hausdorff outer measures, In Combining Soft Computing and Statistical Methods in Data Analysis (C. Borgelt et al. editors), Advances in Intelligent and Soft

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[9] S. Doria. (2012). Characterization of a coherent upper conditional prevision as the Choquet integral with respect to its associated Hausdorff outer measure. *Annals of*

[10] L.E. Dubins. (1975). Finitely additive conditional probabilities, conglomerability and

[12] E. Miranda, M.Zaffalon. (2009). Conditional models: coherence and inference trough sequences of joint mass functions, Journal of Statistical Planning and Inference 140(7),

[13] E. Regazzini. (1985). Finitely additive conditional probabilities, Rend. Sem. Mat.

[14] E. Regazzini. (1987). De Finetti's coherence and statistical inference, The Annals of

[16] T. Seidenfeld, M.J. Schervish, J.B. Kadane. (2001). Improper regular conditional

[17] P. Vicig, M. Zaffalon, F.G. Cozman. (2007). Notes on "Notes on conditional previsions",

[18] P. Walley. (1981). Coherent lower (and upper) probabilities, Statistics Research Report,

[19] P. Walley. (1991). *Statistical Reasoning with Imprecise Probabilities*. Chapman and Hall,

[20] P.M. Williams. (2007). Notes on conditional previsions, International Journal of

[11] K.J. Falconer. (1986). *The geometry of fractals sets*. Cambridge University Press.

[15] C.A. Rogers. (1998). *Hausdorff measures*. Cambridge University Press.

distributions, The Annals of Probability, Vol. 29, No. 4, 1612-1624.

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[2] P. Billingsley. (1986). *Probability and measure*, Wiley, USA.

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62 Stochastic Modeling and Control

1805-1833.

London.

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University of Warwick.

[3] B. de Finetti. (1970). *Teoria della Probabilita'*, Einaudi Editore, Torino. [4] B. de Finetti. (1972). Probability, Induction and Statistics, Wiley, New York.

> 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 equilibria [3, 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 statistical equilibrium.

> 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

©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 work is properly cited. ©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.

multi-agent economic system with an open or closed economy, respectively. The Maxwellian (Gaussian) distribution is derived in Section 3 from geometrical arguments over the volume or the surface of an *N*-sphere [7, 9]. In this case, the motivation will be a multi-particle gas system in contact with a heat reservoir (non-isolated or open system) or with a fixed energy (isolated or closed system), respectively. And finally, in Section 4, the general equilibrium distribution for a set of many identical interacting agents obeying a global additive constraint is also derived [7]. This distribution will be related with the Gamma-like distributions found in several multi-agent economic models. Other two geometrical collateral results, namely the formula for the volume of high-dimensional symmetrical bodies and an alternative image of the canonical ensemble, are proposed in Section 5. And last Section embodies the conclusions.

## **2. Derivation of the Boltzmann-Gibbs distribution**

We proceed to derive here the Boltzmann-Gibbs distribution in two different physical situations with an economic inspiration. The first one considers an ensemble of economic agents that share a variable amount of money (open systems) and the second one deals with the conservative case where the total wealth is fixed (closed systems).

#### **2.1. Multi-agent economic open systems**

Here we assume *N* agents, each one with coordinate *xi*, *i* = 1, . . . , *N*, with *xi* ≥ 0 representing the wealth or money of the agent *i*, and a total available amount of money *E*:

$$\mathbf{x}\_1 + \mathbf{x}\_2 + \dots + \mathbf{x}\_{N-1} + \mathbf{x}\_N \le E. \tag{1}$$

Hence, the volume of the *N*-dimensional pyramid for which the *i*th coordinate is between *xi*

*<sup>f</sup>*(*xi*) = *VN*−1(*<sup>E</sup>* <sup>−</sup> *xi*)

 <sup>1</sup> <sup>−</sup> *xi E*

*<sup>N</sup>*−<sup>1</sup>

The Boltzmann factor *<sup>e</sup>*−*xi*/ is found when *<sup>N</sup>* � 1 but, even for small *<sup>N</sup>*, it can be a good approximation for agents with low wealth. After substituting Eq. (8) into Eq. (7), we obtain the Maxwell-Boltzmann distribution in the asymptotic regime *N* → ∞ (which also implies

> *e*−*x*/

where the index *i* has been removed because the distribution is the same for each agent, and thus the wealth distribution can be obtained by averaging over all the agents. This distribution

This geometrical image of the volume-based statistical ensemble [7] allows us to recover the same result than that obtained from the microcanonical ensemble [8] that we show in the next

Here, we derive the Boltzmann-Gibbs distribution by considering the system in isolation, that is, a closed economy. Without loss of generality, let us assume *N* interacting economic agents, each one with coordinate *xi*, *i* = 1, . . . , *N*, with *xi* ≥ 0, and where *xi* represents an amount of

then this isolated system evolves on the positive part of an equilateral *N*-hyperplane. The

<sup>√</sup>*<sup>N</sup>*

Different rules, deterministic or random, for the exchange of money between agents can be given. Depending on these rules, the system can visit the *N*-hyperplane in an equiprobable manner or not. If the ergodic hypothesis is assumed, each point on the *N*-hyperplane is equiprobable. Then the probability *f*(*xi*)*dxi* of finding agent *i* with money *xi* is proportional to the surface area formed by all the points on the *N*-hyperplane having the *i*th-coordinate

*<sup>x</sup>*<sup>1</sup> + *<sup>x</sup>*<sup>2</sup> + ··· + *xN*−<sup>1</sup> + *xN* = *<sup>E</sup>*, (10)

(*<sup>N</sup>* <sup>−</sup> <sup>1</sup>)! *<sup>E</sup>N*−1. (11)

*<sup>N</sup>*−<sup>1</sup>

Geometrical Derivation of Equilibrium Distributions in Some Stochastic Systems 65

� *<sup>e</sup>*−*xi*/

*VN*(*E*) , (6)

, (7)

. (8)

*dx*, (9)

and *xi* + *dxi* is *VN*−1(*<sup>E</sup>* − *xi*)*dxi*. We normalize it to satisfy Eq. (3), and obtain

*f*(*xi*) = *NE*−<sup>1</sup>

lim *N*�1 <sup>1</sup> <sup>−</sup> *xi E*

If we call the mean wealth per agent, *E* = *N*, then in the limit of large *N* we have

*<sup>f</sup>*(*x*)*dx* <sup>=</sup> <sup>1</sup>

has been found to fit the real distribution of incomes in western societies [10, 11].

money. If we suppose now that the total amount of money *E* is conserved,

surface area *SN*(*E*) of an equilateral *N*-hyperplane of side *E* is given by

*SN*(*E*) =

whose final form, after some calculation is

**2.2. Multi-agent economic closed systems**

*E* → ∞):

section.

Under random or deterministic evolution rules for the exchanging of money among agents, let us suppose that this system evolves in the interior of the *N*-dimensional pyramid given by Eq. (1). The role of a heat reservoir, that in this model supplies money instead of energy, could be played by the state or by the bank system in western societies. The formula for the volume *VN*(*E*) of an equilateral *N*-dimensional pyramid formed by *N* + 1 vertices linked by *N* perpendicular sides of length *E* is

$$V\_N(E) = \frac{E^N}{N!}.\tag{2}$$

We suppose that each point on the *N*-dimensional pyramid is equiprobable, then the probability *f*(*xi*)*dxi* of finding the agent *i* with money *xi* is proportional to the volume formed by all the points into the (*N* − 1)-dimensional pyramid having the *i*th-coordinate equal to *xi*. We show now that *f*(*xi*) is the Boltzmann factor (or the Maxwell-Boltzmann distribution), with the normalization condition

$$\int\_{0}^{E} f(\mathbf{x}\_{i})d\mathbf{x}\_{i} = 1. \tag{3}$$

If the *i*th agent has coordinate *xi*, the *N* − 1 remaining agents share, at most, the money *E* − *xi* on the (*N* − 1)-dimensional pyramid

$$\mathbf{x}\_1 + \mathbf{x}\_2 \cdot \dots + \mathbf{x}\_{i-1} + \mathbf{x}\_{i+1} \cdot \dots + \mathbf{x}\_N \le E - \mathbf{x}\_{i\prime} \tag{4}$$

whose volume is *VN*−1(*<sup>E</sup>* − *xi*). It can be easily proved that

$$V\_N(E) = \int\_0^E V\_{N-1}(E - x\_i) dx\_i. \tag{5}$$

Hence, the volume of the *N*-dimensional pyramid for which the *i*th coordinate is between *xi* and *xi* + *dxi* is *VN*−1(*<sup>E</sup>* − *xi*)*dxi*. We normalize it to satisfy Eq. (3), and obtain

$$f(\mathbf{x}\_i) = \frac{V\_{N-1}(E - \mathbf{x}\_i)}{V\_N(E)},\tag{6}$$

whose final form, after some calculation is

2 Stochastic Control

multi-agent economic system with an open or closed economy, respectively. The Maxwellian (Gaussian) distribution is derived in Section 3 from geometrical arguments over the volume or the surface of an *N*-sphere [7, 9]. In this case, the motivation will be a multi-particle gas system in contact with a heat reservoir (non-isolated or open system) or with a fixed energy (isolated or closed system), respectively. And finally, in Section 4, the general equilibrium distribution for a set of many identical interacting agents obeying a global additive constraint is also derived [7]. This distribution will be related with the Gamma-like distributions found in several multi-agent economic models. Other two geometrical collateral results, namely the formula for the volume of high-dimensional symmetrical bodies and an alternative image of the canonical ensemble, are proposed in Section 5. And last Section embodies the conclusions.

We proceed to derive here the Boltzmann-Gibbs distribution in two different physical situations with an economic inspiration. The first one considers an ensemble of economic agents that share a variable amount of money (open systems) and the second one deals with

Here we assume *N* agents, each one with coordinate *xi*, *i* = 1, . . . , *N*, with *xi* ≥ 0 representing

Under random or deterministic evolution rules for the exchanging of money among agents, let us suppose that this system evolves in the interior of the *N*-dimensional pyramid given by Eq. (1). The role of a heat reservoir, that in this model supplies money instead of energy, could be played by the state or by the bank system in western societies. The formula for the volume *VN*(*E*) of an equilateral *N*-dimensional pyramid formed by *N* + 1 vertices linked by

*VN*(*E*) = *<sup>E</sup><sup>N</sup>*

We suppose that each point on the *N*-dimensional pyramid is equiprobable, then the probability *f*(*xi*)*dxi* of finding the agent *i* with money *xi* is proportional to the volume formed by all the points into the (*N* − 1)-dimensional pyramid having the *i*th-coordinate equal to *xi*. We show now that *f*(*xi*) is the Boltzmann factor (or the Maxwell-Boltzmann distribution),

If the *i*th agent has coordinate *xi*, the *N* − 1 remaining agents share, at most, the money *E* − *xi*

 *E* 0

*N*!

*<sup>x</sup>*<sup>1</sup> + *<sup>x</sup>*<sup>2</sup> + ··· + *xN*−<sup>1</sup> + *xN* ≤ *<sup>E</sup>*. (1)

. (2)

*f*(*xi*)*dxi* = 1. (3)

*VN*−1(*<sup>E</sup>* − *xi*)*dxi*. (5)

*<sup>x</sup>*<sup>1</sup> + *<sup>x</sup>*<sup>2</sup> ··· + *xi*−<sup>1</sup> + *xi*<sup>+</sup><sup>1</sup> ··· + *xN* ≤ *<sup>E</sup>* − *xi*, (4)

**2. Derivation of the Boltzmann-Gibbs distribution**

**2.1. Multi-agent economic open systems**

*N* perpendicular sides of length *E* is

with the normalization condition

on the (*N* − 1)-dimensional pyramid

whose volume is *VN*−1(*<sup>E</sup>* − *xi*). It can be easily proved that

*VN*(*E*) =

 *E* 0

the conservative case where the total wealth is fixed (closed systems).

the wealth or money of the agent *i*, and a total available amount of money *E*:

$$f(\mathbf{x}\_{l}) = NE^{-1} \left( 1 - \frac{\mathbf{x}\_{l}}{E} \right)^{N-1} \text{ . \tag{7}$$

If we call the mean wealth per agent, *E* = *N*, then in the limit of large *N* we have

$$\lim\_{N \gg 1} \left(1 - \frac{\chi\_{\bar{l}}}{E}\right)^{N-1} \simeq e^{-\chi\_{\bar{l}}/\varepsilon}.\tag{8}$$

The Boltzmann factor *<sup>e</sup>*−*xi*/ is found when *<sup>N</sup>* � 1 but, even for small *<sup>N</sup>*, it can be a good approximation for agents with low wealth. After substituting Eq. (8) into Eq. (7), we obtain the Maxwell-Boltzmann distribution in the asymptotic regime *N* → ∞ (which also implies *E* → ∞):

$$f(\mathbf{x})d\mathbf{x} = \frac{1}{\epsilon}e^{-\mathbf{x}/\epsilon}d\mathbf{x},\tag{9}$$

where the index *i* has been removed because the distribution is the same for each agent, and thus the wealth distribution can be obtained by averaging over all the agents. This distribution has been found to fit the real distribution of incomes in western societies [10, 11].

This geometrical image of the volume-based statistical ensemble [7] allows us to recover the same result than that obtained from the microcanonical ensemble [8] that we show in the next section.

#### **2.2. Multi-agent economic closed systems**

Here, we derive the Boltzmann-Gibbs distribution by considering the system in isolation, that is, a closed economy. Without loss of generality, let us assume *N* interacting economic agents, each one with coordinate *xi*, *i* = 1, . . . , *N*, with *xi* ≥ 0, and where *xi* represents an amount of money. If we suppose now that the total amount of money *E* is conserved,

$$\mathbf{x}\_1 + \mathbf{x}\_2 + \dots + \mathbf{x}\_{N-1} + \mathbf{x}\_N = E\_\prime \tag{10}$$

then this isolated system evolves on the positive part of an equilateral *N*-hyperplane. The surface area *SN*(*E*) of an equilateral *N*-hyperplane of side *E* is given by

$$\mathcal{S}\_N(E) = \frac{\sqrt{N}}{(N-1)!} \ E^{N-1}.\tag{11}$$

Different rules, deterministic or random, for the exchange of money between agents can be given. Depending on these rules, the system can visit the *N*-hyperplane in an equiprobable manner or not. If the ergodic hypothesis is assumed, each point on the *N*-hyperplane is equiprobable. Then the probability *f*(*xi*)*dxi* of finding agent *i* with money *xi* is proportional to the surface area formed by all the points on the *N*-hyperplane having the *i*th-coordinate equal to *xi*. We show that *f*(*xi*) is the Boltzmann-Gibbs distribution (the Boltzmann factor), with the normalization condition (3).

If the *i*th agent has coordinate *xi*, the *N* − 1 remaining agents share the money *E* − *xi* on the (*N* − 1)-hyperplane

$$\mathbf{x}\_1 + \mathbf{x}\_2 \cdot \dots + \mathbf{x}\_{i-1} + \mathbf{x}\_{i+1} \cdot \dots + \mathbf{x}\_N = E - \mathbf{x}\_{i\prime} \tag{12}$$

whose surface area is *SN*−1(*<sup>E</sup>* − *xi*). If we define the coordinate *<sup>θ</sup><sup>N</sup>* as satisfying

$$
\sin \theta\_N = \sqrt{\frac{N-1}{N}} \tag{13}
$$

**3. Derivation of the Maxwellian distribution**

*p*2 <sup>1</sup> <sup>+</sup> *<sup>p</sup>*<sup>2</sup>

(closed systems).

define a kinetic energy:

*<sup>E</sup>* <sup>≡</sup> *<sup>R</sup>*2, we have

*p*2

*VN*−1( 

**3.1. Multi-particle open systems**

with the normalization condition

the maximum energy *<sup>R</sup>*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup>

whose volume is *VN*−1(

*<sup>R</sup>*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup>

*p*2 <sup>1</sup> <sup>+</sup> *<sup>p</sup>*<sup>2</sup>

*<sup>R</sup>*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup>

We proceed to derive here the Maxwellian distribution in two different physical situations with inspiration in the theory of ideal gases. The first one considers an ideal gas with a variable energy (open systems) and the second one deals with the case of a gas with a fixed energy

Geometrical Derivation of Equilibrium Distributions in Some Stochastic Systems 67

Let us suppose a one-dimensional ideal gas of *N* non-identical classical particles with masses *mi*, with *i* = 1, . . . , *N*, and total maximum energy *E*. If particle *i* has a momentum *mivi*, we

> *<sup>i</sup>* <sup>≡</sup> <sup>1</sup> 2 *miv*<sup>2</sup>

where *pi* is the square root of the kinetic energy *Ki*. If the total maximum energy is defined as

We see that the system has accessible states with different energy, which can be supplied by a heat reservoir. These states are all those enclosed into the volume of the *N*-sphere given by

where Γ(·) is the gamma function. If we suppose that each point into the *N*-sphere is equiprobable, then the probability *f*(*pi*)*dpi* of finding the particle *i* with coordinate *pi* (energy

*<sup>i</sup>* ) is proportional to the volume formed by all the points on the *N*-sphere having the *i*th-coordinate equal to *pi*. We proceed to show that *f*(*pi*) is the Maxwellian distribution,

If the *i*th particle has coordinate *pi*, the (*N* − 1) remaining particles share an energy less than

*<sup>i</sup>* ). It can be easily proved that

*<sup>R</sup>*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup>

*<sup>R</sup>*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup> *i* )

*VN*−1(

Hence, the volume of the *N*-sphere for which the *i*th coordinate is between *pi* and *pi* + *dpi* is

*VN*−1( 

*<sup>i</sup>*+<sup>1</sup> ··· <sup>+</sup> *<sup>p</sup>*<sup>2</sup>

*<sup>N</sup>*−<sup>1</sup> <sup>+</sup> *<sup>p</sup>*<sup>2</sup>

2 Γ( *<sup>N</sup>* <sup>2</sup> + 1)

*<sup>i</sup>* , (21)

*<sup>N</sup>* <sup>≤</sup> *<sup>R</sup>*2. (22)

*RN*, (23)

*<sup>i</sup>* , (25)

*<sup>i</sup>* )*dpi*. (26)

*VN*(*R*) , (27)

*f*(*pi*)*dpi* = 1. (24)

*<sup>N</sup>* <sup>≤</sup> *<sup>R</sup>*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup>

*Ki* <sup>≡</sup> *<sup>p</sup>*<sup>2</sup>

<sup>2</sup> <sup>+</sup> ··· <sup>+</sup> *<sup>p</sup>*<sup>2</sup>

*VN*(*R*) = *<sup>π</sup> <sup>N</sup>*

 *R* −*R*

*<sup>i</sup>* on the (*N* − 1)-sphere

*<sup>i</sup>*−<sup>1</sup> <sup>+</sup> *<sup>p</sup>*<sup>2</sup>

−*R*

*<sup>i</sup>* )*dpi*. We normalize it to satisfy Eq. (24), and obtain

<sup>2</sup> ··· <sup>+</sup> *<sup>p</sup>*<sup>2</sup>

*VN*(*R*) = *<sup>R</sup>*

*f*(*pi*) =

Eq. (22). The formula for the volume *VN*(*R*) of an *N*-sphere of radius *R* is

it can be easily shown that

$$\mathcal{S}\_N(E) = \int\_0^E \mathcal{S}\_{N-1}(E - \mathbf{x}\_i) \frac{d\mathbf{x}\_i}{\sin \theta\_N}. \tag{14}$$

Hence, the surface area of the *N*-hyperplane for which the *i*th coordinate is between *xi* and *xi* + *dxi* is proportional to *SN*−1(*<sup>E</sup>* − *xi*)*dxi*/ sin *<sup>θ</sup>N*. If we take into account the normalization condition (3), we obtain

$$f(\mathbf{x}\_i) = \frac{1}{\mathcal{S}\_N(E)} \frac{\mathcal{S}\_{N-1}(E - \mathbf{x}\_i)}{\sin \theta\_N} \, \, \, \, \tag{15}$$

whose form after some calculation is

$$f(\mathbf{x}\_i) = (N - 1)E^{-1} \left( 1 - \frac{\mathbf{x}\_i}{E} \right)^{N - 2},\tag{16}$$

If we call *�* the mean wealth per agent, *E* = *N�*, then in the limit of large *N* we have

$$\lim\_{N \gg 1} \left(1 - \frac{\mathbf{x}\_i}{E}\right)^{N-2} \simeq e^{-\mathbf{x}\_i/\varepsilon}.\tag{17}$$

As in the former section, the Boltzmann factor *<sup>e</sup>*−*xi*/*�* is found when *<sup>N</sup>* � 1 but, even for small *N*, it can be a good approximation for agents with low wealth. After substituting Eq. (1) into Eq. (16), we obtain the Boltzmann distribution (9) in the limit *N* → ∞ (which also implies *E* → ∞). This asymptotic result reproduces the distribution of real economic data [10] and also the results obtained in several models of economic agents with deterministic, random or chaotic exchange interactions [6, 12, 13].

Depending on the physical situation, the mean wealth per agent *�* takes different expressions and interpretations. For instance, we can calculate the dependence of *�* on the temperature, which in the microcanonical ensemble is defined by the derivative of the entropy with respect to the energy. The entropy can be written as *<sup>S</sup>* <sup>=</sup> <sup>−</sup>*kN* <sup>∞</sup> <sup>0</sup> *f*(*x*)ln *f*(*x*) *dx*, where *f*(*x*) is given by Eq. (9) and *k* is Boltzmann's constant. If we recall that *�* = *E*/*N*, we obtain

$$S(E) = kN \ln \frac{E}{N} + kN.\tag{18}$$

The calculation of the temperature *T* gives

$$T^{-1} = \left(\frac{\partial S}{\partial E}\right)\_N = \frac{kN}{E} = \frac{k}{\varepsilon}.\tag{19}$$

Thus *�* = *kT*, and the Boltzmann-Gibbs distribution is obtained in its usual form:

$$f(\mathbf{x})d\mathbf{x} = \frac{1}{kT}e^{-\mathbf{x}/kT}d\mathbf{x}.\tag{20}$$

#### **3. Derivation of the Maxwellian distribution**

We proceed to derive here the Maxwellian distribution in two different physical situations with inspiration in the theory of ideal gases. The first one considers an ideal gas with a variable energy (open systems) and the second one deals with the case of a gas with a fixed energy (closed systems).

#### **3.1. Multi-particle open systems**

4 Stochastic Control

equal to *xi*. We show that *f*(*xi*) is the Boltzmann-Gibbs distribution (the Boltzmann factor),

If the *i*th agent has coordinate *xi*, the *N* − 1 remaining agents share the money *E* − *xi* on the

*<sup>N</sup>* <sup>−</sup> <sup>1</sup>

*SN*−1(*<sup>E</sup>* <sup>−</sup> *xi*) *dxi*

*SN*−1(*<sup>E</sup>* − *xi*) sin *θ<sup>N</sup>*

> <sup>1</sup> <sup>−</sup> *xi E*

*<sup>N</sup>*−<sup>2</sup>

As in the former section, the Boltzmann factor *<sup>e</sup>*−*xi*/*�* is found when *<sup>N</sup>* � 1 but, even for small *N*, it can be a good approximation for agents with low wealth. After substituting Eq. (1) into Eq. (16), we obtain the Boltzmann distribution (9) in the limit *N* → ∞ (which also implies *E* → ∞). This asymptotic result reproduces the distribution of real economic data [10] and also the results obtained in several models of economic agents with deterministic, random or

Depending on the physical situation, the mean wealth per agent *�* takes different expressions and interpretations. For instance, we can calculate the dependence of *�* on the temperature, which in the microcanonical ensemble is defined by the derivative of the entropy with respect

by Eq. (9) and *k* is Boltzmann's constant. If we recall that *�* = *E*/*N*, we obtain

*T*−<sup>1</sup> =

*<sup>S</sup>*(*E*) = *kN* ln *<sup>E</sup>*

 *∂S ∂E N*

Thus *�* = *kT*, and the Boltzmann-Gibbs distribution is obtained in its usual form: *<sup>f</sup>*(*x*)*dx* <sup>=</sup> <sup>1</sup>

<sup>=</sup> *kN <sup>E</sup>* <sup>=</sup> *<sup>k</sup> �*

*kT <sup>e</sup>*

Hence, the surface area of the *N*-hyperplane for which the *i*th coordinate is between *xi* and *xi* + *dxi* is proportional to *SN*−1(*<sup>E</sup>* − *xi*)*dxi*/ sin *<sup>θ</sup>N*. If we take into account the normalization

*SN*(*E*)

whose surface area is *SN*−1(*<sup>E</sup>* − *xi*). If we define the coordinate *<sup>θ</sup><sup>N</sup>* as satisfying

*SN*(*E*) =

sin *θ<sup>N</sup>* =

 *E* 0

*<sup>f</sup>*(*xi*) = <sup>1</sup>

*<sup>f</sup>*(*xi*)=(*<sup>N</sup>* <sup>−</sup> <sup>1</sup>)*E*−<sup>1</sup>

lim *N*�1 <sup>1</sup> <sup>−</sup> *xi E*

If we call *�* the mean wealth per agent, *E* = *N�*, then in the limit of large *N* we have

*<sup>x</sup>*<sup>1</sup> + *<sup>x</sup>*<sup>2</sup> ··· + *xi*−<sup>1</sup> + *xi*<sup>+</sup><sup>1</sup> ··· + *xN* = *<sup>E</sup>* − *xi*, (12)

sin *θ<sup>N</sup>*

*<sup>N</sup>*−<sup>2</sup>

*<sup>N</sup>* , (13)

. (14)

, (15)

, (16)

� *<sup>e</sup>*−*xi*/*�*. (17)

<sup>0</sup> *f*(*x*)ln *f*(*x*) *dx*, where *f*(*x*) is given

. (19)

*<sup>N</sup>* <sup>+</sup> *kN*. (18)

<sup>−</sup>*x*/*kTdx*. (20)

with the normalization condition (3).

(*N* − 1)-hyperplane

it can be easily shown that

condition (3), we obtain

whose form after some calculation is

chaotic exchange interactions [6, 12, 13].

The calculation of the temperature *T* gives

to the energy. The entropy can be written as *<sup>S</sup>* <sup>=</sup> <sup>−</sup>*kN* <sup>∞</sup>

Let us suppose a one-dimensional ideal gas of *N* non-identical classical particles with masses *mi*, with *i* = 1, . . . , *N*, and total maximum energy *E*. If particle *i* has a momentum *mivi*, we define a kinetic energy:

$$K\_{\rm i} \equiv p\_{\rm i}^2 \equiv \frac{1}{2} m\_{\rm i} v\_{\rm i}^2 \tag{21}$$

where *pi* is the square root of the kinetic energy *Ki*. If the total maximum energy is defined as *<sup>E</sup>* <sup>≡</sup> *<sup>R</sup>*2, we have

$$p\_1^2 + p\_2^2 + \dots + p\_{N-1}^2 + p\_N^2 \le \mathbb{R}^2. \tag{22}$$

We see that the system has accessible states with different energy, which can be supplied by a heat reservoir. These states are all those enclosed into the volume of the *N*-sphere given by Eq. (22). The formula for the volume *VN*(*R*) of an *N*-sphere of radius *R* is

$$V\_N(R) = \frac{\pi^{\frac{N}{2}}}{\Gamma(\frac{N}{2} + 1)} R^N \, , \tag{23}$$

where Γ(·) is the gamma function. If we suppose that each point into the *N*-sphere is equiprobable, then the probability *f*(*pi*)*dpi* of finding the particle *i* with coordinate *pi* (energy *p*2 *<sup>i</sup>* ) is proportional to the volume formed by all the points on the *N*-sphere having the *i*th-coordinate equal to *pi*. We proceed to show that *f*(*pi*) is the Maxwellian distribution, with the normalization condition

$$\int\_{-\mathbb{R}}^{\mathbb{R}} f(p\_i) dp\_i = 1. \tag{24}$$

If the *i*th particle has coordinate *pi*, the (*N* − 1) remaining particles share an energy less than the maximum energy *<sup>R</sup>*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup> *<sup>i</sup>* on the (*N* − 1)-sphere

$$p\_1^2 + p\_2^2 \cdots + p\_{i-1}^2 + p\_{i+1}^2 \cdots + p\_N^2 \le \mathbb{R}^2 - p\_{i\prime}^2 \tag{25}$$

whose volume is *VN*−1( *<sup>R</sup>*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup> *<sup>i</sup>* ). It can be easily proved that

$$V\_N(R) = \int\_{-R}^{R} V\_{N-1}(\sqrt{R^2 - p\_i^2}) dp\_i. \tag{26}$$

Hence, the volume of the *N*-sphere for which the *i*th coordinate is between *pi* and *pi* + *dpi* is *VN*−1( *<sup>R</sup>*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup> *<sup>i</sup>* )*dpi*. We normalize it to satisfy Eq. (24), and obtain

$$f(p\_i) = \frac{V\_{N-1}(\sqrt{R^2 - p\_i^2})}{V\_N(R)},\tag{27}$$

whose final form, after some calculation is

$$f(p\_i) = C\_N R^{-1} \left(1 - \frac{p\_i^2}{R^2}\right)^{\frac{N-1}{2}},\tag{28}$$

with

$$\mathbb{C}\_{N} = \frac{1}{\sqrt{\pi}} \frac{\Gamma(\frac{N+2}{2})}{\Gamma(\frac{N+1}{2})}.\tag{29}$$

coordinate *pi* (energy *p*<sup>2</sup>

on the (*N* − 1)-sphere

then

with

whose surface area is *SN*−1(

It can be easily proved that

whose final form, after some calculation is

Eq. (24), and obtain

*<sup>i</sup>* ) is proportional to the surface area formed by all the points on the

Geometrical Derivation of Equilibrium Distributions in Some Stochastic Systems 69

*<sup>N</sup>* <sup>=</sup> *<sup>R</sup>*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup>

*<sup>i</sup>* ). If we define the coordinate *θ* as satisfying

*i*

*<sup>i</sup>* , (35)

*<sup>i</sup>* , (36)

. (37)

, (39)

, (40)

. (41)

<sup>2</sup> . (42)

*<sup>i</sup>* /2*�* is found when *<sup>N</sup>* � 1 but, even for

. (43)

*SN*−1(*<sup>R</sup>* cos *<sup>θ</sup>*)*Rdθ*. (38)

*N*-sphere having the *i*th-coordinate equal to *pi*. Our objective is to show that *f*(*pi*) is the

If the *<sup>i</sup>*th particle has coordinate *pi*, the (*<sup>N</sup>* <sup>−</sup> <sup>1</sup>) remaining particles share the energy *<sup>R</sup>*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup>

*<sup>R</sup>*<sup>2</sup> cos2 *<sup>θ</sup>* <sup>=</sup> *<sup>R</sup>*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup>

*Rd<sup>θ</sup>* <sup>=</sup> *dpi* (<sup>1</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup> *i <sup>R</sup>*<sup>2</sup> )1/2

−*π*/2

*SN*(*R*)

*f*(*pi*) = *CNR*−<sup>1</sup>

*CN* <sup>=</sup> <sup>1</sup> <sup>√</sup>*<sup>π</sup>*

> *CN* � <sup>1</sup> <sup>√</sup>*<sup>π</sup>*

If we call *�* the mean energy per particle, *E* = *R*<sup>2</sup> = *N�*, then in the limit of large *N* we have

 *<sup>N</sup>*−<sup>3</sup> 2

small *N*, it can be a good approximation for particles with low energies. After substituting

For *N* � 1, Stirling's approximation can be applied to Eq. (41), leading to

lim *N*�1

As in the former section, the Boltzmann factor *e*−*p*<sup>2</sup>

lim *N*�1

 <sup>1</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup> *i R*2

Hence, the surface area of the *N*-sphere for which the *i*th coordinate is between *pi* and *pi* + *dpi* is *SN*−1(*<sup>R</sup>* cos *<sup>θ</sup>*)*Rdθ*. We rewrite the surface area as a function of *pi*, normalize it to satisfy

> *SN*−1(

> > <sup>1</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup> *i R*2

(<sup>1</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup> *i <sup>R</sup>*<sup>2</sup> )1/2

Γ( *<sup>N</sup>* 2 ) Γ( *<sup>N</sup>*−<sup>1</sup> <sup>2</sup> )

*N*

� *e* <sup>−</sup>*p*<sup>2</sup> *<sup>i</sup>* /2*�*

*<sup>R</sup>*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup> *i* )

> *<sup>N</sup>*−<sup>3</sup> 2

*<sup>i</sup>*+<sup>1</sup> ··· <sup>+</sup> *<sup>p</sup>*<sup>2</sup>

*<sup>i</sup>*−<sup>1</sup> <sup>+</sup> *<sup>p</sup>*<sup>2</sup>

Maxwellian distribution, with the normalization condition (24).

<sup>2</sup> ··· <sup>+</sup> *<sup>p</sup>*<sup>2</sup>

*<sup>R</sup>*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup>

*SN*(*R*) = *<sup>π</sup>*/2

*<sup>f</sup>*(*pi*) = <sup>1</sup>

*p*2 <sup>1</sup> <sup>+</sup> *<sup>p</sup>*<sup>2</sup>

For *N* � 1, Stirling's approximation can be applied to Eq. (29), leading to

$$\lim\_{N \gg 1} \mathbb{C}\_N \simeq \frac{1}{\sqrt{\pi}} \sqrt{\frac{N}{2}}.\tag{30}$$

If we call *�* the mean energy per particle, *E* = *R*<sup>2</sup> = *N�*, then in the limit of large *N* we have

$$\lim\_{N \gg 1} \left( 1 - \frac{p\_i^2}{R^2} \right)^{\frac{N-1}{2}} \simeq e^{-p\_i^2/2\varepsilon}.\tag{31}$$

The factor *e*−*p*<sup>2</sup> *<sup>i</sup>* /2*�* is found when *<sup>N</sup>* � 1 but, even for small *<sup>N</sup>*, it can be a good approximation for particles with low energies. After substituting Eqs. (30)–(31) into Eq. (28), we obtain the Maxwellian distribution in the asymptotic regime *N* → ∞ (which also implies *E* → ∞):

$$f(p)dp = \sqrt{\frac{1}{2\pi\epsilon}}e^{-p^2/2\varepsilon}dp\_\prime \tag{32}$$

where the index *i* has been removed because the distribution is the same for each particle, and thus the velocity distribution can be obtained by averaging over all the particles.

This newly shows that the geometrical image of the volume-based statistical ensemble [7] allows us to recover the same result than that obtained from the microcanonical ensemble [9] that it is presented in the next section.

#### **3.2. Multi-particle closed systems**

We start by assuming a one-dimensional ideal gas of *N* non-identical classical particles with masses *mi*, with *i* = 1, . . . , *N*, and total energy *E*. If particle *i* has a momentum *mivi*, newly we define a kinetic energy *Ki* given by Eq. (21), where *pi* is the square root of *Ki*. If the total energy is defined as *<sup>E</sup>* <sup>≡</sup> *<sup>R</sup>*2, we have

$$p\_1^2 + p\_2^2 + \dots + p\_{N-1}^2 + p\_N^2 = R^2. \tag{33}$$

We see that the isolated system evolves on the surface of an *N*-sphere. The formula for the surface area *SN*(*R*) of an *N*-sphere of radius *R* is

$$S\_N(R) = \frac{2\pi^{\frac{N}{2}}}{\Gamma(\frac{N}{2})} R^{N-1} \, \Big|\,\,\,\,\tag{34}$$

where Γ(·) is the gamma function. If the ergodic hypothesis is assumed, that is, each point on the *N*-sphere is equiprobable, then the probability *f*(*pi*)*dpi* of finding the particle *i* with coordinate *pi* (energy *p*<sup>2</sup> *<sup>i</sup>* ) is proportional to the surface area formed by all the points on the *N*-sphere having the *i*th-coordinate equal to *pi*. Our objective is to show that *f*(*pi*) is the Maxwellian distribution, with the normalization condition (24).

If the *<sup>i</sup>*th particle has coordinate *pi*, the (*<sup>N</sup>* <sup>−</sup> <sup>1</sup>) remaining particles share the energy *<sup>R</sup>*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup> *i* on the (*N* − 1)-sphere

$$p\_1^2 + p\_2^2 \cdots + p\_{i-1}^2 + p\_{i+1}^2 \cdots + p\_N^2 = \mathbb{R}^2 - p\_{i\prime}^2 \tag{35}$$

whose surface area is *SN*−1( *<sup>R</sup>*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup> *<sup>i</sup>* ). If we define the coordinate *θ* as satisfying

$$R^2 \cos^2 \theta = R^2 - p\_{i\prime}^2 \tag{36}$$

then

6 Stochastic Control

 <sup>1</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup> *i R*2

Γ( *<sup>N</sup>*+<sup>2</sup> <sup>2</sup> ) Γ( *<sup>N</sup>*+<sup>1</sup> <sup>2</sup> )

*N*

� *e* <sup>−</sup>*p*<sup>2</sup> *<sup>i</sup>* /2*�*

*<sup>i</sup>* /2*�* is found when *<sup>N</sup>* � 1 but, even for small *<sup>N</sup>*, it can be a good approximation

 *<sup>N</sup>*−<sup>1</sup> 2

, (28)

. (29)

<sup>2</sup> . (30)

<sup>−</sup>*p*2/2*�dp*, (32)

*<sup>N</sup>* = *<sup>R</sup>*2. (33)

*RN*−1, (34)

. (31)

*f*(*pi*) = *CNR*−<sup>1</sup>

*CN* <sup>=</sup> <sup>1</sup> <sup>√</sup>*<sup>π</sup>*

> *CN* � <sup>1</sup> <sup>√</sup>*<sup>π</sup>*

If we call *�* the mean energy per particle, *E* = *R*<sup>2</sup> = *N�*, then in the limit of large *N* we have

 *<sup>N</sup>*−<sup>1</sup> 2

for particles with low energies. After substituting Eqs. (30)–(31) into Eq. (28), we obtain the Maxwellian distribution in the asymptotic regime *N* → ∞ (which also implies *E* → ∞):

> 1 2*π� e*

where the index *i* has been removed because the distribution is the same for each particle, and

This newly shows that the geometrical image of the volume-based statistical ensemble [7] allows us to recover the same result than that obtained from the microcanonical ensemble [9]

We start by assuming a one-dimensional ideal gas of *N* non-identical classical particles with masses *mi*, with *i* = 1, . . . , *N*, and total energy *E*. If particle *i* has a momentum *mivi*, newly we define a kinetic energy *Ki* given by Eq. (21), where *pi* is the square root of *Ki*. If the total

We see that the isolated system evolves on the surface of an *N*-sphere. The formula for the

where Γ(·) is the gamma function. If the ergodic hypothesis is assumed, that is, each point on the *N*-sphere is equiprobable, then the probability *f*(*pi*)*dpi* of finding the particle *i* with

*<sup>N</sup>*−<sup>1</sup> <sup>+</sup> *<sup>p</sup>*<sup>2</sup>

2 Γ( *<sup>N</sup>* 2 )

<sup>2</sup> <sup>+</sup> ··· <sup>+</sup> *<sup>p</sup>*<sup>2</sup>

*SN*(*R*) = <sup>2</sup>*<sup>π</sup> <sup>N</sup>*

For *N* � 1, Stirling's approximation can be applied to Eq. (29), leading to

lim *N*�1

lim *N*�1

 <sup>1</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup> *i R*2

*f*(*p*)*dp* =

thus the velocity distribution can be obtained by averaging over all the particles.

whose final form, after some calculation is

with

The factor *e*−*p*<sup>2</sup>

that it is presented in the next section.

**3.2. Multi-particle closed systems**

energy is defined as *<sup>E</sup>* <sup>≡</sup> *<sup>R</sup>*2, we have

surface area *SN*(*R*) of an *N*-sphere of radius *R* is

*p*2 <sup>1</sup> <sup>+</sup> *<sup>p</sup>*<sup>2</sup>

$$Rd\theta = \frac{dp\_i}{(1 - \frac{p\_i^2}{R^2})^{1/2}}.\tag{37}$$

It can be easily proved that

$$\mathcal{S}\_N(\mathbb{R}) = \int\_{-\pi/2}^{\pi/2} \mathcal{S}\_{N-1}(\mathbb{R}\cos\theta) \mathrm{Rd}\theta. \tag{38}$$

Hence, the surface area of the *N*-sphere for which the *i*th coordinate is between *pi* and *pi* + *dpi* is *SN*−1(*<sup>R</sup>* cos *<sup>θ</sup>*)*Rdθ*. We rewrite the surface area as a function of *pi*, normalize it to satisfy Eq. (24), and obtain

$$f(p\_i) = \frac{1}{S\_N(\mathbb{R})} \frac{S\_{N-1}(\sqrt{\mathbb{R}^2 - p\_i^2})}{(1 - \frac{p\_i^2}{\mathbb{R}^2})^{1/2}},\tag{39}$$

whose final form, after some calculation is

$$f(p\_i) = \mathbb{C}\_N \mathbb{R}^{-1} \left( 1 - \frac{p\_i^2}{\mathbb{R}^2} \right)^{\frac{N-3}{2}},\tag{40}$$

with

$$\mathbb{C}\_{N} = \frac{1}{\sqrt{\pi}} \frac{\Gamma(\frac{N}{2})}{\Gamma(\frac{N-1}{2})}.\tag{41}$$

For *N* � 1, Stirling's approximation can be applied to Eq. (41), leading to

$$\lim\_{N \gg 1} \mathbb{C}\_N \simeq \frac{1}{\sqrt{\pi}} \sqrt{\frac{N}{2}}.\tag{42}$$

If we call *�* the mean energy per particle, *E* = *R*<sup>2</sup> = *N�*, then in the limit of large *N* we have

$$\lim\_{N \gg 1} \left( 1 - \frac{p\_i^2}{R^2} \right)^{\frac{N-3}{2}} \simeq e^{-p\_i^2/2\varepsilon}.\tag{43}$$

As in the former section, the Boltzmann factor *e*−*p*<sup>2</sup> *<sup>i</sup>* /2*�* is found when *<sup>N</sup>* � 1 but, even for small *N*, it can be a good approximation for particles with low energies. After substituting

Eqs. (42)–(43) into Eq. (40), we obtain the Maxwellian distribution (32) in the asymptotic regime *N* → ∞ (which also implies *E* → ∞).

Depending on the physical situation the mean energy per particle *�* takes different expressions. For an isolated one-dimensional gas we can calculate the dependence of *�* on the temperature, which in the microcanonical ensemble is defined by differentiating the entropy with respect to the energy. The entropy can be written as *<sup>S</sup>* <sup>=</sup> <sup>−</sup>*kN* <sup>∞</sup> <sup>−</sup><sup>∞</sup> *<sup>f</sup>*(*p*)ln *<sup>f</sup>*(*p*) *dp*, where *f*(*p*) is given by Eq. (32) and *k* is the Boltzmann constant. If we recall that *�* = *E*/*N*, we obtain

$$S(E) = \frac{1}{2}kN\ln\left(\frac{E}{N}\right) + \frac{1}{2}kN(\ln(2\pi) - 1). \tag{44}$$

The calculation of the temperature *T* gives

$$T^{-1} = \left(\frac{\partial S}{\partial E}\right)\_N = \frac{kN}{2E} = \frac{k}{2\epsilon}.\tag{45}$$

Taking into account the normalization condition *<sup>E</sup>*1/*<sup>b</sup>*

*VN*((*<sup>E</sup>* <sup>−</sup> *<sup>x</sup>b*)1/*b*) = *gb*(*N*)

*�* =< *x<sup>b</sup>* >. If *N* tends toward infinity, it results:

thermodynamic limit (*N*, *E* → ∞):

Hence, the conjecture (48) is proved.

where it has been used that *�* =< *x<sup>b</sup>* >= <sup>∞</sup>

The calculation of the temperature *T* gives

recall that *�* = *E*/*N*, we obtain

*<sup>f</sup>*(*x*) = *VN*−1((*<sup>E</sup>* <sup>−</sup> *<sup>x</sup>b*)1/*b*)

The *N*-dimensional volume, *VN*(*b*, *ρ*), of a *b*-symmetrical body with side of length *ρ* is

The parameter *b* indicates the original equation (47) that defines the boundaries of the volume

Coming back to Eq. (50), we can manipulate *VN*((*<sup>E</sup>* <sup>−</sup> *<sup>x</sup>b*)1/*b*) to obtain (the index *<sup>b</sup>* is omitted

(*<sup>E</sup>* <sup>−</sup> *<sup>x</sup>b*)1/*<sup>b</sup>*

If we suppose *E* = *N�*, then *�* represents the mean value of *x<sup>b</sup>* in the collectivity, that is,

 *<sup>N</sup> b*

1/*b*) = *VN*(*E*1/*b*) *e*

Substituting this last expression in formula (50), the exact form for *f*(*x*) is found in the

*cb* <sup>=</sup> *gb*(*<sup>N</sup>* <sup>−</sup> <sup>1</sup>)

Doing a thermodinamical simile, we can calculate the dependence of *�* on the temperature by differentiating the entropy with respect to the energy. The entropy can be written as *S* =

<sup>0</sup> *f*(*x*)ln *f*(*x*) *dx*, where *f*(*x*) is given by Eq. (55) and *k* is the Boltzmann constant. If we

<sup>0</sup> *<sup>x</sup><sup>b</sup> <sup>f</sup>*(*x*)*dx*.

<sup>=</sup> *kN bE* <sup>=</sup> *<sup>k</sup> b�*

*<sup>N</sup>*

= *e*−*xb*/*b�*

<sup>=</sup> *gb*(*N*) *<sup>E</sup> <sup>N</sup>*

<sup>−</sup>*xb*/*b�*

*<sup>f</sup>*(*x*)*dx* <sup>=</sup> *cb �*−1/*<sup>b</sup> <sup>e</sup>*−*xb*/*b�dx*, (55)

*gb*(*N*) *<sup>N</sup>*1/*<sup>b</sup>* . (56)

*<sup>b</sup>* (<sup>1</sup> <sup>−</sup> *<sup>b</sup>* ln *cb*), (57)

. (58)

*b* <sup>1</sup> <sup>−</sup> *<sup>x</sup><sup>b</sup> E*

 *<sup>N</sup> b*

. (53)

. (54)

. (52)

proportional to the term *ρ<sup>N</sup>* and to a coefficient *gb*(*N*) that depends on *N*:

*VN*(*b*, *ρ*). Thus, for instance, from Eq. (2), we have *gb*<sup>=</sup>1(*N*) = 1/*N*!.

lim *N*�1

*VN*((*<sup>E</sup>* <sup>−</sup> *<sup>x</sup>b*)

*<sup>S</sup>*(*E*) = *kN*

*T*−<sup>1</sup> =

*<sup>b</sup>* ln *<sup>E</sup> N* + *kN*

> *∂S ∂E N*

 <sup>1</sup> <sup>−</sup> *<sup>x</sup><sup>b</sup> E*

obtained:

in the formule of *VN*):

Thus,

with *cb* given by

−*kN* <sup>∞</sup>

<sup>0</sup> *f*(*x*)*dx* = 1, the expression for *f*(*x*) is

*VN*(*E*1/*b*) . (50)

*VN*(*b*, *ρ*) = *gb*(*N*) *ρN*. (51)

Geometrical Derivation of Equilibrium Distributions in Some Stochastic Systems 71

Thus *�* = *kT*/2, consistent with the equipartition theorem. If *p*<sup>2</sup> is replaced by <sup>1</sup> <sup>2</sup>*mv*2, the Maxwellian distribution is a function of particle velocity, as it is usually given in the literature:

$$g(v)dv = \sqrt{\frac{m}{2\pi kT}}e^{-mv^2/2kT}dv.\tag{46}$$

#### **4. General derivation of the equilibrium distribution**

In this section, we are interested in the same problem above presented but in a general way. We address this question in the volume-based statistical framework.

Let *b* be a positive real constant (cases *b* = 1, 2 have been indicated in the former sections). If we have a set of positive variables (*x*1, *x*2,..., *xN*) verifying the constraint

$$\mathbf{x}\_1^b + \mathbf{x}\_2^b + \dots + \mathbf{x}\_{N-1}^b + \mathbf{x}\_N^b \le E \tag{47}$$

with an adequate mechanism assuring the equiprobability of all the possible states (*x*1, *x*2,..., *xN*) into the volume given by expression (47), will we have for the generic variable *x* the distribution

$$f(\mathbf{x})d\mathbf{x} \sim \epsilon^{-1/b} e^{-\mathbf{x}^b/b\varepsilon} d\mathbf{x},\tag{48}$$

when we average over the ensemble in the limit *N*, *E* → ∞, with *E* = *N�*, and constant *�*?. Now it is shown that the answer is affirmative. Similarly, we claim that if the weak inequality (47) is transformed in equality the result will be the same, as it has been proved for the cases *b* = 1, 2 in Refs. [8, 9].

From the cases *b* = 1, 2, (see Eqs. (6) and (27)), we can extrapolate the general formula that will give us the statistical behavior *f*(*x*) of the generic variable *x*, when the system runs equiprobably into the volume defined by a constraint of type (47). The probability *f*(*x*)*dx* of finding an agent with generic coordinate *<sup>x</sup>* is proportional to the volume *VN*−1((*<sup>E</sup>* <sup>−</sup> *<sup>x</sup>b*)1/*b*) formed by all the points into the (*N* − 1)-dimensional symmetrical body limited by the constraint (*<sup>E</sup>* <sup>−</sup> *<sup>x</sup>b*). Thus, the *<sup>N</sup>*-dimensional volume can be written as

$$V\_N(\mathbb{E}^{1/b}) = \int\_0^{\mathbb{E}^{1/b}} V\_{N-1}((\mathbb{E} - \mathbb{x}^b)^{1/b}) \, d\mathbf{x}.\tag{49}$$

Taking into account the normalization condition *<sup>E</sup>*1/*<sup>b</sup>* <sup>0</sup> *f*(*x*)*dx* = 1, the expression for *f*(*x*) is obtained:

$$f(\mathbf{x}) = \frac{V\_{N-1}((E - \mathbf{x}^b)^{1/b})}{V\_N(E^{1/b})}.\tag{50}$$

The *N*-dimensional volume, *VN*(*b*, *ρ*), of a *b*-symmetrical body with side of length *ρ* is proportional to the term *ρ<sup>N</sup>* and to a coefficient *gb*(*N*) that depends on *N*:

$$V\_N(b,\rho) = \mathcal{g}\_b(N)\,\rho^N. \tag{51}$$

The parameter *b* indicates the original equation (47) that defines the boundaries of the volume *VN*(*b*, *ρ*). Thus, for instance, from Eq. (2), we have *gb*<sup>=</sup>1(*N*) = 1/*N*!.

Coming back to Eq. (50), we can manipulate *VN*((*<sup>E</sup>* <sup>−</sup> *<sup>x</sup>b*)1/*b*) to obtain (the index *<sup>b</sup>* is omitted in the formule of *VN*):

$$N\_N((E-\mathbf{x}^b)^{1/b}) = g\_b(N)\left[(E-\mathbf{x}^b)^{1/b}\right]^N = g\_b(N)\,\mathbb{E}^{\frac{N}{\mathsf{P}}}\left(1-\frac{\mathsf{x}^b}{E}\right)^{\frac{N}{\mathsf{P}}}.\tag{52}$$

If we suppose *E* = *N�*, then *�* represents the mean value of *x<sup>b</sup>* in the collectivity, that is, *�* =< *x<sup>b</sup>* >. If *N* tends toward infinity, it results:

$$\lim\_{N \gg 1} \left( 1 - \frac{\mathbf{x}^b}{E} \right)^{\frac{N}{b}} = e^{-\mathbf{x}^b / b\varepsilon}. \tag{53}$$

Thus,

8 Stochastic Control

Eqs. (42)–(43) into Eq. (40), we obtain the Maxwellian distribution (32) in the asymptotic

Depending on the physical situation the mean energy per particle *�* takes different expressions. For an isolated one-dimensional gas we can calculate the dependence of *�* on the temperature, which in the microcanonical ensemble is defined by differentiating the entropy

*f*(*p*) is given by Eq. (32) and *k* is the Boltzmann constant. If we recall that *�* = *E*/*N*, we obtain

<sup>=</sup> *kN* <sup>2</sup>*<sup>E</sup>* <sup>=</sup> *<sup>k</sup>* 2*�*

 *E N* + 1 2

 *∂S ∂E N*

Thus *�* = *kT*/2, consistent with the equipartition theorem. If *p*<sup>2</sup> is replaced by <sup>1</sup>

*m*

Maxwellian distribution is a function of particle velocity, as it is usually given in the literature:

In this section, we are interested in the same problem above presented but in a general way.

Let *b* be a positive real constant (cases *b* = 1, 2 have been indicated in the former sections). If

with an adequate mechanism assuring the equiprobability of all the possible states (*x*1, *x*2,..., *xN*) into the volume given by expression (47), will we have for the generic variable

when we average over the ensemble in the limit *N*, *E* → ∞, with *E* = *N�*, and constant *�*?. Now it is shown that the answer is affirmative. Similarly, we claim that if the weak inequality (47) is transformed in equality the result will be the same, as it has been proved for the cases

From the cases *b* = 1, 2, (see Eqs. (6) and (27)), we can extrapolate the general formula that will give us the statistical behavior *f*(*x*) of the generic variable *x*, when the system runs equiprobably into the volume defined by a constraint of type (47). The probability *f*(*x*)*dx* of finding an agent with generic coordinate *<sup>x</sup>* is proportional to the volume *VN*−1((*<sup>E</sup>* <sup>−</sup> *<sup>x</sup>b*)1/*b*) formed by all the points into the (*N* − 1)-dimensional symmetrical body limited by the

*VN*−1((*<sup>E</sup>* <sup>−</sup> *<sup>x</sup>b*)

*<sup>N</sup>*−<sup>1</sup> <sup>+</sup> *<sup>x</sup><sup>b</sup>*

<sup>2</sup> <sup>+</sup> ··· <sup>+</sup> *<sup>x</sup><sup>b</sup>*

<sup>−</sup><sup>∞</sup> *<sup>f</sup>*(*p*)ln *<sup>f</sup>*(*p*) *dp*, where

*kN*(ln(2*π*) − 1). (44)

<sup>2</sup>*πkT <sup>e</sup>*−*mv*2/2*kTdv*. (46)

*<sup>f</sup>*(*x*)*dx* <sup>∼</sup> *�*−1/*<sup>b</sup> <sup>e</sup>*−*xb*/*b�dx*, (48)

. (45)

*<sup>N</sup>* ≤ *E* (47)

1/*b*) *dx*. (49)

<sup>2</sup>*mv*2, the

with respect to the energy. The entropy can be written as *<sup>S</sup>* <sup>=</sup> <sup>−</sup>*kN* <sup>∞</sup>

*T*−<sup>1</sup> =

*g*(*v*)*dv* =

**4. General derivation of the equilibrium distribution**

We address this question in the volume-based statistical framework.

*xb* <sup>1</sup> <sup>+</sup> *<sup>x</sup><sup>b</sup>*

we have a set of positive variables (*x*1, *x*2,..., *xN*) verifying the constraint

constraint (*<sup>E</sup>* <sup>−</sup> *<sup>x</sup>b*). Thus, the *<sup>N</sup>*-dimensional volume can be written as

 *<sup>E</sup>*1/*<sup>b</sup>* 0

*VN*(*E*1/*b*) =

*<sup>S</sup>*(*E*) = <sup>1</sup> 2 *kN* ln

regime *N* → ∞ (which also implies *E* → ∞).

The calculation of the temperature *T* gives

*x* the distribution

*b* = 1, 2 in Refs. [8, 9].

$$V\_N((E - \mathbf{x}^b)^{1/b}) = V\_N(E^{1/b}) \, e^{-\mathbf{x}^b/b\varepsilon}. \tag{54}$$

Substituting this last expression in formula (50), the exact form for *f*(*x*) is found in the thermodynamic limit (*N*, *E* → ∞):

$$f(\mathbf{x})d\mathbf{x} = \mathbf{c}\_b \,\epsilon^{-1/b} \, e^{-\mathbf{x}^b/b\varepsilon} d\mathbf{x} \,\tag{55}$$

with *cb* given by

$$\mathcal{L}\_b = \frac{g\_b(N-1)}{g\_b(N)N^{1/b}}.\tag{56}$$

Hence, the conjecture (48) is proved.

Doing a thermodinamical simile, we can calculate the dependence of *�* on the temperature by differentiating the entropy with respect to the energy. The entropy can be written as *S* = −*kN* <sup>∞</sup> <sup>0</sup> *f*(*x*)ln *f*(*x*) *dx*, where *f*(*x*) is given by Eq. (55) and *k* is the Boltzmann constant. If we recall that *�* = *E*/*N*, we obtain

$$S(E) = \frac{kN}{b} \ln\left(\frac{E}{N}\right) + \frac{kN}{b} (1 - b \ln c\_b) \tag{57}$$

where it has been used that *�* =< *x<sup>b</sup>* >= <sup>∞</sup> <sup>0</sup> *<sup>x</sup><sup>b</sup> <sup>f</sup>*(*x*)*dx*.

The calculation of the temperature *T* gives

$$T^{-1} = \left(\frac{\partial S}{\partial E}\right)\_N = \frac{kN}{bE} = \frac{k}{b\epsilon}.\tag{58}$$

Thus *�* = *kT*/*b*, a result that recovers the theorem of equipartition of energy for the quadratic case *b* = 2. The distribution for all *b* is finally obtained:

$$f(\mathbf{x})d\mathbf{x} = c\_b \left(\frac{b}{kT}\right)^{1/b} e^{-\mathbf{x}^b/kT} d\mathbf{x}.\tag{59}$$

**Figure 1.** Normalization constant *cb* versus *b*, calculated from Eq. (63). The asymptotic behavior is: lim*b*→<sup>0</sup> *cb* = ∞, and lim*b*→<sup>∞</sup> *cb* = 1. This last asymptote is represented by the dotted line. The minimum

Geometrical Derivation of Equilibrium Distributions in Some Stochastic Systems 73

Let us now recall two interesting statistical economic models that display a statistical behavior given by distributions nearly to the form (61), that is, the standard Gamma distributions with

*<sup>b</sup>* <sup>−</sup><sup>1</sup> *e*−*<sup>z</sup> dz*. (66)

, hence in this model the economy

<sup>1</sup> <sup>−</sup> *<sup>λ</sup>* , (69)

*<sup>i</sup>* = *λui* + *�*(1 − *λ*)(*ui* + *uj*), (67)

*<sup>j</sup>* = *λuj* + *�*¯(1 − *λ*)(*ui* + *uj*), (68)

*<sup>i</sup>* + *u*� *j*

Γ( <sup>1</sup> *b* ) *z* 1

**ECONOMIC MODEL A:** The first one is the saving propensity model introduced by Chakraborti and Chakrabarti [11]. In this model a set of *N* economic agents, having each agent *i* (with *i* = 1, 2, ··· , *N*) an amount of money, *ui*, exchanges it under random binary (*i*, *j*)

), by the following the exchange rule:

with *�*¯ = (1 − *�*), and *�* a random number in the interval (0, 1). The parameter *λ*, with 0 < *λ* < 1, is fixed, and represents the fraction of money saved before carrying out the transaction. Let

*<sup>n</sup>*(*λ*) = <sup>1</sup> <sup>+</sup> <sup>2</sup>*<sup>λ</sup>*

and scaling the wealth of the agents as *z*¯ = *nu*/ < *u* >, with < *u* > representing the average money over the ensemble of agents, it is found that the asymptotic wealth distribution in this

*<sup>f</sup>*(*z*)*dz* <sup>=</sup> <sup>1</sup>

of *cb* is reached for *b* = 3.1605, taking the value *cb* = 0.7762.

shape parameter 1/*b*,

interactions, (*ui*, *uj*) → (*u*�

*i* , *u*� *j*

us observe that money is conserved, i.e., *ui* + *uj* = *u*�

is closed. Defining the parameter *n*(*λ*) as

*u*�

*u*�

#### **4.1. General relationship between geometry and economic gas models**

If we perform the change of variables *<sup>y</sup>* <sup>=</sup> *�*−1/*bx* in the normalization condition of *<sup>f</sup>*(*x*), <sup>∞</sup> <sup>0</sup> *f*(*x*)*dx* = 1, where *f*(*x*) is expressed in (55), we find that

$$c\_{\mathfrak{b}} = \left[ \int\_0^\infty e^{-y^{\mathfrak{b}}/b} \, dy \right]^{-1} \,. \tag{60}$$

If we introduce the new variable *z* = *yb*/*b*, the distribution *f*(*x*) as function of *z* reads:

$$f(z)dz = \frac{c\_b}{b^{1-\frac{1}{b}}} z^{\frac{1}{b}-1} e^{-z} dz. \tag{61}$$

Let us observe that the Gamma function appears in the normalization condition,

$$\int\_0^\infty f(z)dz = \frac{c\_b}{b^{1-\frac{1}{b}}} \int\_0^\infty z^{\frac{1}{b}-1} e^{-z} dz = \frac{c\_b}{b^{1-\frac{1}{b}}} \Gamma\left(\frac{1}{b}\right) = 1. \tag{62}$$

This implies that

$$c\_b = \frac{b^{1-\frac{1}{b}}}{\Gamma\left(\frac{1}{b}\right)}.\tag{63}$$

By using Mathematica the positive constant *cb* is plotted versus *b* in Fig. 1. We see that lim*b*→<sup>0</sup> *cb* = <sup>∞</sup>, and that lim*b*→<sup>∞</sup> *cb* = 1. The minimum of *cb* is reached for *<sup>b</sup>* = 3.1605, taking the value *cb* = 0.7762. Still further, we can calculate from Eq. (63) the asymptotic dependence of *cb* on b:

$$\lim\_{b \to 0} c\_b = \sqrt{\frac{1}{2\pi}} \sqrt{b} \, e^{1/b} \left( 1 - \frac{b}{12} + \dotsb \right),\tag{64}$$

$$\lim\_{b \to \infty} c\_b = b^{-1/b} \left( 1 + \frac{\gamma}{b} + \dotsb \right),\tag{65}$$

where *γ* is the Euler constant, *γ* = 0.5772. The asymptotic function (64) is obtained after substituting in (63) the value of Γ(1/*b*) by (1/*b* − 1)!, and performing the Stirling approximation on this last expression, knowing that 1/*b* → ∞. The function (65) is found after looking for the first Taylor expansion terms of the Gamma function around the origin *x* = 0. They can be derived from the Euler's reflection formula, Γ(*x*)Γ(1 − *x*) = *π*/ sin(*πx*). We obtain <sup>Γ</sup>(*<sup>x</sup>* <sup>→</sup> <sup>0</sup>) = *<sup>x</sup>*−<sup>1</sup> <sup>+</sup> <sup>Γ</sup>� (1) + ··· . From here, recalling that Γ� (1) = −*γ*, we get Γ(1/*b*) = *b* − *γ* + ··· , when *b* → ∞. Although this last term of the Taylor expansion, −*γ*, is negligible we maintain it in expression (65). The only minimum of *cb* is reached for the solution *b* = 3.1605 of the equation *ψ*(1/*b*) + log *b* + *b* − 1 = 0, where *ψ*(·) is the digamma function (see Fig. 1).

**Figure 1.** Normalization constant *cb* versus *b*, calculated from Eq. (63). The asymptotic behavior is: lim*b*→<sup>0</sup> *cb* = ∞, and lim*b*→<sup>∞</sup> *cb* = 1. This last asymptote is represented by the dotted line. The minimum of *cb* is reached for *b* = 3.1605, taking the value *cb* = 0.7762.

10 Stochastic Control

Thus *�* = *kT*/*b*, a result that recovers the theorem of equipartition of energy for the quadratic

If we perform the change of variables *<sup>y</sup>* <sup>=</sup> *�*−1/*bx* in the normalization condition of *<sup>f</sup>*(*x*), <sup>∞</sup>

*e*−*yb*/*<sup>b</sup> dy*

−<sup>1</sup>

*<sup>b</sup>* <sup>−</sup><sup>1</sup> *<sup>e</sup>*−*<sup>z</sup> dz* <sup>=</sup> *cb*

*b*

*<sup>b</sup>* <sup>+</sup> ···

*b*1<sup>−</sup> <sup>1</sup> *b* Γ 1 *b* 

<sup>12</sup> <sup>+</sup> ···

(1) + ··· . From here, recalling that Γ�

1/*<sup>b</sup>*

*e*−*xb*/*kTdx*. (59)

. (60)

= 1. (62)

, (64)

(1) = −*γ*, we get

, (65)

*<sup>b</sup>* <sup>−</sup><sup>1</sup> *e*−*<sup>z</sup> dz*. (61)

. (63)

 *b kT*

case *b* = 2. The distribution for all *b* is finally obtained:

<sup>0</sup> *f*(*x*)*dx* = 1, where *f*(*x*) is expressed in (55), we find that

*<sup>f</sup>*(*z*)*dz* <sup>=</sup> *cb*

lim *b*→0

lim

We obtain <sup>Γ</sup>(*<sup>x</sup>* <sup>→</sup> <sup>0</sup>) = *<sup>x</sup>*−<sup>1</sup> <sup>+</sup> <sup>Γ</sup>�

function (see Fig. 1).

*cb* =

*<sup>b</sup>*→<sup>∞</sup> *cb* <sup>=</sup> *<sup>b</sup>*−1/*<sup>b</sup>*

 1 2*π* √ *b e*1/*<sup>b</sup>* <sup>1</sup> <sup>−</sup> *<sup>b</sup>*

> <sup>1</sup> <sup>+</sup> *<sup>γ</sup>*

where *γ* is the Euler constant, *γ* = 0.5772. The asymptotic function (64) is obtained after substituting in (63) the value of Γ(1/*b*) by (1/*b* − 1)!, and performing the Stirling approximation on this last expression, knowing that 1/*b* → ∞. The function (65) is found after looking for the first Taylor expansion terms of the Gamma function around the origin *x* = 0. They can be derived from the Euler's reflection formula, Γ(*x*)Γ(1 − *x*) = *π*/ sin(*πx*).

Γ(1/*b*) = *b* − *γ* + ··· , when *b* → ∞. Although this last term of the Taylor expansion, −*γ*, is negligible we maintain it in expression (65). The only minimum of *cb* is reached for the solution *b* = 3.1605 of the equation *ψ*(1/*b*) + log *b* + *b* − 1 = 0, where *ψ*(·) is the digamma

*b*1<sup>−</sup> <sup>1</sup> *b*

 ∞ 0

This implies that

of *cb* on b:

*f*(*x*)*dx* = *cb*

*cb* =

**4.1. General relationship between geometry and economic gas models**

 ∞ 0

If we introduce the new variable *z* = *yb*/*b*, the distribution *f*(*x*) as function of *z* reads:

*b*1<sup>−</sup> <sup>1</sup> *b z* 1

*cb* <sup>=</sup> *<sup>b</sup>*1<sup>−</sup> <sup>1</sup>

Γ 1 *b*

By using Mathematica the positive constant *cb* is plotted versus *b* in Fig. 1. We see that lim*b*→<sup>0</sup> *cb* = <sup>∞</sup>, and that lim*b*→<sup>∞</sup> *cb* = 1. The minimum of *cb* is reached for *<sup>b</sup>* = 3.1605, taking the value *cb* = 0.7762. Still further, we can calculate from Eq. (63) the asymptotic dependence

*<sup>f</sup>*(*z*)*dz* <sup>=</sup> *cb*

Let us observe that the Gamma function appears in the normalization condition,

 ∞ 0 *z* 1

Let us now recall two interesting statistical economic models that display a statistical behavior given by distributions nearly to the form (61), that is, the standard Gamma distributions with shape parameter 1/*b*,

$$f(z)dz = \frac{1}{\Gamma(\frac{1}{\mathfrak{D}})} z^{\frac{1}{\mathfrak{T}}-1} e^{-z} dz. \tag{66}$$

**ECONOMIC MODEL A:** The first one is the saving propensity model introduced by Chakraborti and Chakrabarti [11]. In this model a set of *N* economic agents, having each agent *i* (with *i* = 1, 2, ··· , *N*) an amount of money, *ui*, exchanges it under random binary (*i*, *j*) interactions, (*ui*, *uj*) → (*u*� *i* , *u*� *j* ), by the following the exchange rule:

$$
\mu\_i' = \lambda \mu\_i + \varepsilon (1 - \lambda)(\mu\_i + \mu\_j),
\tag{67}
$$

$$
\mu\_j' = \lambda \mu\_j + \bar{\mathfrak{e}} (1 - \lambda) (\mu\_i + \mathfrak{u}\_j), \tag{68}
$$

with *�*¯ = (1 − *�*), and *�* a random number in the interval (0, 1). The parameter *λ*, with 0 < *λ* < 1, is fixed, and represents the fraction of money saved before carrying out the transaction. Let us observe that money is conserved, i.e., *ui* + *uj* = *u*� *<sup>i</sup>* + *u*� *j* , hence in this model the economy is closed. Defining the parameter *n*(*λ*) as

*<sup>n</sup>*(*λ*) = <sup>1</sup> <sup>+</sup> <sup>2</sup>*<sup>λ</sup>* <sup>1</sup> <sup>−</sup> *<sup>λ</sup>* , (69)

and scaling the wealth of the agents as *z*¯ = *nu*/ < *u* >, with < *u* > representing the average money over the ensemble of agents, it is found that the asymptotic wealth distribution in this system is nearly obeying the standard Gamma distribution [14, 15]

$$f(\bar{z})d\bar{z} = \frac{1}{\Gamma(n)}\bar{z}^{n-1}e^{-\bar{z}}d\bar{z}.\tag{70}$$

The case *n* = 1, which means a null saving propensity, *λ* = 0, recovers the model of Dragulescu and Yakovenko [10] in which the Gibbs distribution is observed. If we compare Eqs. (70) and (66), a close relationship between this economic model and the geometrical problem solved in the former section can be established. It is enough to make

$$n = 1/b\_{\prime} \tag{71}$$

values 0 or 1, we have the model studied by Angle [17], that gives very different asymptotic results. The exchange parameter, *ω*, represents the maximum fraction of wealth lost by one of the two interacting agents (0 < *ω* < 1). Whether the agent who is going to loose part of the money is the *i*-th or the *j*-th agent, depends nonlinearly on (*xi* − *xj*), and this is decided by the random dichotomous function *η*(*t*): *η*(*t* > 0) = 1 (with additional probability 1/2) and *η*(*t* < 0) = 0 (with additional probability 1/2). Hence, when *xi* > *xj*, the value *η* = 1 produces a wealth transfer from agent *i* to agent *j* with probability 1/2, and when *xi* < *xj*, the value *η* = 0 produces a wealth transfer from agent *j* to agent *i* with probability 1/2. Defining

*<sup>n</sup>*(*ω*) = <sup>3</sup> <sup>−</sup> <sup>2</sup>*<sup>ω</sup>*

and scaling the wealth of the agents as *z*¯ = *nu*/ < *u* >, with < *u* > representing the average money over the ensemble of agents, it is found that the asymptotic wealth distribution in this

<sup>Γ</sup>(*n*) *<sup>z</sup>*¯

The case *n* = 1, which means an exchange parameter *ω* = 3/4, recovers the model of Dragulescu and Yakovenko [10] in which the Gibbs distribution is observed. If we compare Eqs. (79) and (66), a close relationship between this economic model and the geometrical

to have two equivalent systems. This means that, from Eq. (80), we can calculate *b* from the

*<sup>b</sup>* <sup>=</sup> <sup>2</sup>*<sup>ω</sup>*

As *ω* takes its values in the interval (0, 1), then the parameter *b* runs in the interval (0, 2). It is curious to observe that in this model the interval *ω* ∈ (3/4, 1) maps on *b* ∈ (1, 2), a fact that does not occur in MODEL A. On the other hand, recalling that *z* = *xb*/*b�*, we can get the

where *�* is a free parameter that determines the mean value of *x<sup>b</sup>* in the equivalent geometrical system. Formula (83) means to perform the change of variables *ui* → *xi*, with *i* = 1, 2, ··· , *N*, for all the particles/agents of the ensemble. Then, also in this case, we conjecture that the economic system represented by the generic pair (*λ*, *u*), when it is transformed in the geometrical system given by the generic pair (*b*, *x*), as indicated by the rules (82) and (83), runs in an equiprobable form on the surface defined by the relationship (47), where the inequality has been transformed in equality. As explained above, this last detail is due to the fact the economic system is closed, and then it conserves the total money, whose equivalent quantity in the geometrical problem is *E*. If the economic system were open, with an upper limit in the wealth, then the transformed system would evolve in an equiprobable way over the volume defined by the inequality (47), although its statistical behavior would continue to be the same

*u* 1/*<sup>b</sup>*

*<sup>n</sup>*−<sup>1</sup> *e*

*<sup>f</sup>*(*z*¯)*dz*¯ <sup>=</sup> <sup>1</sup>

problem solved in the last section can be established. It is enough to make

*x* =

as it has been proved for the cases *b* = 1, 2 in Refs. [8, 9].

 *�* < *u* >

system is nearly to fit the standard Gamma distribution [15, 16]

<sup>2</sup>*<sup>ω</sup>* , (78)

Geometrical Derivation of Equilibrium Distributions in Some Stochastic Systems 75

<sup>−</sup>*z*¯ *dz*¯. (79)

*n* = 1/*b*, (80) *z*¯ = *z*, (81)

<sup>3</sup> <sup>−</sup> <sup>2</sup>*<sup>ω</sup>* . (82)

. (83)

in this case the parameter *n*(*ω*) as

exchange parameter *ω* with the formula

equivalent variable *x* from Eq. (81),

$$
\mathbf{z} = \mathbf{z}\_\prime \tag{72}
$$

to have two equivalent systems. This means that, from Eq. (71), we can calculate *b* from the saving parameter *λ* with the formula

$$b = \frac{1 - \lambda}{1 + 2\lambda}.\tag{73}$$

As *λ* takes its values in the interval (0, 1), then the parameter *b* also runs in the same interval (0, 1). On the other hand, recalling that *z* = *xb*/*b�*, we can get the equivalent variable *x* from Eq. (72),

$$\mathbf{x} = \left[\frac{\epsilon}{<\mu>}\,\mu\right]^{1/b}\mathbf{y} \tag{74}$$

where *�* is a free parameter that determines the mean value of *x<sup>b</sup>* in the equivalent geometrical system. Formula (74) means to perform the change of variables *ui* → *xi*, with *i* = 1, 2, ··· , *N*, for all the particles/agents of the ensemble. Then, we conjecture that the economic system represented by the generic pair (*λ*, *u*), when it is transformed in the geometrical system given by the generic pair (*b*, *x*), as indicated by the rules (73) and (74), runs in an equiprobable form on the surface defined by the relationship (47), where the inequality has been transformed in equality. This last detail is due to the fact the economic system is closed, and then it conserves the total money, whose equivalent quantity in the geometrical problem is *E*. If the economic system were open, with an upper limit in the wealth, then the transformed system would evolve in an equiprobable way over the volume defined by the inequality (47), although its statistical behavior would continue to be the same as it has been proved for the cases *b* = 1, 2 in Refs. [8, 9].

**ECONOMIC MODEL B:** The second one is a model introduced in [16]. In this model a set of *N* economic agents, having each agent *i* (with *i* = 1, 2, ··· , *N*) an amount of money, *ui*, exchanges it under random binary (*i*, *j*) interactions, (*ui*, *uj*) → (*u*� *i* , *u*� *j* ), by the following the exchange rule:

$$
\mu\_i' = \mu\_i - \Delta\mu\_\prime \tag{75}
$$

$$
\mu\_j' = \mu\_j + \Delta\mu\_\prime \tag{76}
$$

where

$$
\Delta \mu = \eta (\mathbf{x}\_i - \mathbf{x}\_j) \,\epsilon \omega \mathbf{x}\_i - \left[1 - \eta (\mathbf{x}\_i - \mathbf{x}\_j)\right] \epsilon \omega \mathbf{x}\_j \tag{77}
$$

with *�* a continuous uniform random number in the interval (0, 1). When this variable is transformed in a Bernouilli variable, i.e. a discrete uniform random variable taking on the values 0 or 1, we have the model studied by Angle [17], that gives very different asymptotic results. The exchange parameter, *ω*, represents the maximum fraction of wealth lost by one of the two interacting agents (0 < *ω* < 1). Whether the agent who is going to loose part of the money is the *i*-th or the *j*-th agent, depends nonlinearly on (*xi* − *xj*), and this is decided by the random dichotomous function *η*(*t*): *η*(*t* > 0) = 1 (with additional probability 1/2) and *η*(*t* < 0) = 0 (with additional probability 1/2). Hence, when *xi* > *xj*, the value *η* = 1 produces a wealth transfer from agent *i* to agent *j* with probability 1/2, and when *xi* < *xj*, the value *η* = 0 produces a wealth transfer from agent *j* to agent *i* with probability 1/2. Defining in this case the parameter *n*(*ω*) as

12 Stochastic Control

<sup>Γ</sup>(*n*) *<sup>z</sup>*¯

The case *n* = 1, which means a null saving propensity, *λ* = 0, recovers the model of Dragulescu and Yakovenko [10] in which the Gibbs distribution is observed. If we compare Eqs. (70) and (66), a close relationship between this economic model and the geometrical

to have two equivalent systems. This means that, from Eq. (71), we can calculate *b* from the

*<sup>b</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>λ</sup>*

As *λ* takes its values in the interval (0, 1), then the parameter *b* also runs in the same interval (0, 1). On the other hand, recalling that *z* = *xb*/*b�*, we can get the equivalent variable *x* from

where *�* is a free parameter that determines the mean value of *x<sup>b</sup>* in the equivalent geometrical system. Formula (74) means to perform the change of variables *ui* → *xi*, with *i* = 1, 2, ··· , *N*, for all the particles/agents of the ensemble. Then, we conjecture that the economic system represented by the generic pair (*λ*, *u*), when it is transformed in the geometrical system given by the generic pair (*b*, *x*), as indicated by the rules (73) and (74), runs in an equiprobable form on the surface defined by the relationship (47), where the inequality has been transformed in equality. This last detail is due to the fact the economic system is closed, and then it conserves the total money, whose equivalent quantity in the geometrical problem is *E*. If the economic system were open, with an upper limit in the wealth, then the transformed system would evolve in an equiprobable way over the volume defined by the inequality (47), although its statistical behavior would continue to be the same as it has been proved for the cases *b* = 1, 2

**ECONOMIC MODEL B:** The second one is a model introduced in [16]. In this model a set of *N* economic agents, having each agent *i* (with *i* = 1, 2, ··· , *N*) an amount of money, *ui*,

with *�* a continuous uniform random number in the interval (0, 1). When this variable is transformed in a Bernouilli variable, i.e. a discrete uniform random variable taking on the

*u* 1/*<sup>b</sup>*

 *�* < *u* > *<sup>n</sup>*−<sup>1</sup> *e*−*z*¯ *dz*¯. (70)

*n* = 1/*b*, (71) *z*¯ = *z*, (72)

<sup>1</sup> <sup>+</sup> <sup>2</sup>*λ*. (73)

*i* , *u*� *j*

*<sup>i</sup>* = *ui* − Δ*u*, (75)

*<sup>j</sup>* = *uj* + Δ*u*, (76)

Δ*u* = *η*(*xi* − *xj*) *�ωxi* − [1 − *η*(*xi* − *xj*)] *�ωxj*, (77)

), by the following the

, (74)

system is nearly obeying the standard Gamma distribution [14, 15]

saving parameter *λ* with the formula

Eq. (72),

in Refs. [8, 9].

exchange rule:

where

*<sup>f</sup>*(*z*¯)*dz*¯ <sup>=</sup> <sup>1</sup>

problem solved in the former section can be established. It is enough to make

*x* =

exchanges it under random binary (*i*, *j*) interactions, (*ui*, *uj*) → (*u*�

*u*�

*u*�

$$m(\omega) = \frac{3 - 2\omega}{2\omega},\tag{78}$$

and scaling the wealth of the agents as *z*¯ = *nu*/ < *u* >, with < *u* > representing the average money over the ensemble of agents, it is found that the asymptotic wealth distribution in this system is nearly to fit the standard Gamma distribution [15, 16]

$$f(\overline{z})d\overline{z} = \frac{1}{\Gamma(n)} \overline{z}^{n-1} e^{-\overline{z}} d\overline{z}.\tag{79}$$

The case *n* = 1, which means an exchange parameter *ω* = 3/4, recovers the model of Dragulescu and Yakovenko [10] in which the Gibbs distribution is observed. If we compare Eqs. (79) and (66), a close relationship between this economic model and the geometrical problem solved in the last section can be established. It is enough to make

$$n = 1/b,\tag{80}$$

$$
\mathbf{z} = \mathbf{z}\_\prime \tag{81}
$$

to have two equivalent systems. This means that, from Eq. (80), we can calculate *b* from the exchange parameter *ω* with the formula

$$b = \frac{2\omega}{3 - 2\omega}.\tag{82}$$

As *ω* takes its values in the interval (0, 1), then the parameter *b* runs in the interval (0, 2). It is curious to observe that in this model the interval *ω* ∈ (3/4, 1) maps on *b* ∈ (1, 2), a fact that does not occur in MODEL A. On the other hand, recalling that *z* = *xb*/*b�*, we can get the equivalent variable *x* from Eq. (81),

$$\mathfrak{x} = \left[\frac{\mathfrak{e}}{<\mathfrak{u}>} \mathfrak{u}\right]^{1/b}. \tag{83}$$

where *�* is a free parameter that determines the mean value of *x<sup>b</sup>* in the equivalent geometrical system. Formula (83) means to perform the change of variables *ui* → *xi*, with *i* = 1, 2, ··· , *N*, for all the particles/agents of the ensemble. Then, also in this case, we conjecture that the economic system represented by the generic pair (*λ*, *u*), when it is transformed in the geometrical system given by the generic pair (*b*, *x*), as indicated by the rules (82) and (83), runs in an equiprobable form on the surface defined by the relationship (47), where the inequality has been transformed in equality. As explained above, this last detail is due to the fact the economic system is closed, and then it conserves the total money, whose equivalent quantity in the geometrical problem is *E*. If the economic system were open, with an upper limit in the wealth, then the transformed system would evolve in an equiprobable way over the volume defined by the inequality (47), although its statistical behavior would continue to be the same as it has been proved for the cases *b* = 1, 2 in Refs. [8, 9].

#### **5. Other additional geometrical questions**

As two collateral results, we address two additional problems in this section. The first one presents the finding of the general formula for the volume of a high-dimensional symmetrical body and the second one offers an alternative presentation of the canonical ensemble.

#### **5.1. Formula for the volume of a high-dimensional body**

We are concerned now with the asymptotic formula (*N* → ∞) for the volume of the *N*-dimensional symmetrical body enclosed by the surface

$$\mathbf{x}\_1^b + \mathbf{x}\_2^b + \dots + \mathbf{x}\_{N-1}^b + \mathbf{x}\_N^b = E. \tag{84}$$

**Figure 2.** The factor *gb* (*N*) versus *b* for *N* = 10, 40, 100, calculated from Eq. (91). Observe that *gb* (*N*) = 0

Geometrical Derivation of Equilibrium Distributions in Some Stochastic Systems 77

that recovers the exact results for *b* = 1, 2. The behavior of *a* is monotonous decreasing when *b* is varied from *b* = 0, where *a* diverges as *a* ∼ 1/*b* + ··· , up to the limit *b* → ∞, where *a*

> Γ <sup>1</sup> *<sup>b</sup>* + 1 *<sup>N</sup>*

Γ *<sup>N</sup> <sup>b</sup>* + 1

It would be also possible to multiply this last expression (91) by a general polynomial *K*(*N*) in the variable *N*, and all the derivation done from Eq. (88) would continue to be correct. We omit this possibility in our calculations. For a fixed *N*, we have that *gb*(*N*) increases monotonously from *gb*(*N*) = 0, for *b* = 0, up to *gb*(*N*) = 1, in the limit *b* → ∞ (see Fig. 2). For a fixed *b*, we have that *gb*(*N*) decreases monotonously from *gb*(*N*) = 1, for *N* = 1, up to *gb*(*N*) = 0, in the

The final result, that has been shown to be valid for any *N* [18], for the volume of an *N*-dimensional symmetrical body of characteristic *b* given by the boundary (84) reads:

> Γ <sup>1</sup> *<sup>b</sup>* + 1 *<sup>N</sup>*

Γ *<sup>N</sup> <sup>b</sup>* + 1 , (91)

*<sup>ρ</sup>N*, (92)

*gb*(*N*) =

*VN*(*b*, *ρ*) =

decays asymptotically toward the value *a*<sup>∞</sup> = *e*−*<sup>γ</sup>* = 0.5614.

Hence, the formula for *gb*(*N*) is obtained:

limit *N* → ∞ (see Fig. 3).

with *<sup>ρ</sup>* <sup>∼</sup> *<sup>E</sup>*1/*b*.

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

The linear dimension *<sup>ρ</sup>* of this volume, i.e., the length of one of its sides verifies *<sup>ρ</sup>* <sup>∼</sup> *<sup>E</sup>*1/*b*. As argued in Eq. (51), the *N*-dimensional volume, *VN*(*b*, *ρ*), is proportional to the term *ρ<sup>N</sup>* and to a coefficient *gb*(*N*) that depends on *N*. Thus,

$$V\_N(b,\rho) = \mathcal{g}\_b(N)\,\rho^N \,. \tag{85}$$

where the characteristic *b* indicates the particular boundary given by equation (84).

For instance, from Equation (2), we can write in a formal way:

$$g\_{b=1}(N) = \frac{1\stackrel{N}{\dagger}}{\Gamma(\frac{N}{\dagger} + 1)}.\tag{86}$$

From Eq. (23), if we take the diameter, *ρ* = 2*R*, as the linear dimension of the *N*-sphere, we obtain:

$$g\_{b=2}(\mathbf{N}) = \frac{\left(\frac{\pi}{4}\right)^{\frac{N}{2}}}{\Gamma\left(\frac{N}{2} + 1\right)}.\tag{87}$$

These expressions (86) and (87) suggest a possible general formula for the factor *gb*(*N*), let us say

$$g\_b(N) = \frac{a^{\frac{N}{b}}}{\Gamma\left(\frac{N}{b} + 1\right)}\,\tag{88}$$

where *a* is a *b*-dependent constant to be determined. For example, *a* = 1 for *b* = 1 and *a* = *π*/4 for *b* = 2.

In order to find the dependence of *a* on the parameter *b*, the regime *N* → ∞ is supposed. Applying Stirling approximation for the factorial ( *<sup>N</sup> <sup>b</sup>* )! in the denominator of expression (88), and inserting it in expression (56), it is straightforward to find out the relationship:

$$c\_b = (ab)^{-1/b}.\tag{89}$$

From here and formula (63), we get:

$$a = \left[\Gamma\left(\frac{1}{b} + 1\right)\right]^b \,\tag{90}$$

**Figure 2.** The factor *gb* (*N*) versus *b* for *N* = 10, 40, 100, calculated from Eq. (91). Observe that *gb* (*N*) = 0 for *b* = 0, and lim*b*→<sup>∞</sup> *gb* (*N*) = 1.

that recovers the exact results for *b* = 1, 2. The behavior of *a* is monotonous decreasing when *b* is varied from *b* = 0, where *a* diverges as *a* ∼ 1/*b* + ··· , up to the limit *b* → ∞, where *a* decays asymptotically toward the value *a*<sup>∞</sup> = *e*−*<sup>γ</sup>* = 0.5614.

Hence, the formula for *gb*(*N*) is obtained:

$$\lg\_b(N) = \frac{\Gamma\left(\frac{1}{b} + 1\right)^N}{\Gamma\left(\frac{N}{b} + 1\right)},\tag{91}$$

It would be also possible to multiply this last expression (91) by a general polynomial *K*(*N*) in the variable *N*, and all the derivation done from Eq. (88) would continue to be correct. We omit this possibility in our calculations. For a fixed *N*, we have that *gb*(*N*) increases monotonously from *gb*(*N*) = 0, for *b* = 0, up to *gb*(*N*) = 1, in the limit *b* → ∞ (see Fig. 2). For a fixed *b*, we have that *gb*(*N*) decreases monotonously from *gb*(*N*) = 1, for *N* = 1, up to *gb*(*N*) = 0, in the limit *N* → ∞ (see Fig. 3).

The final result, that has been shown to be valid for any *N* [18], for the volume of an *N*-dimensional symmetrical body of characteristic *b* given by the boundary (84) reads:

$$V\_N(b,\rho) = \frac{\Gamma\left(\frac{1}{b} + 1\right)^N}{\Gamma\left(\frac{N}{b} + 1\right)} \rho^N \,\,\_{\prime}\tag{92}$$

with *<sup>ρ</sup>* <sup>∼</sup> *<sup>E</sup>*1/*b*.

14 Stochastic Control

As two collateral results, we address two additional problems in this section. The first one presents the finding of the general formula for the volume of a high-dimensional symmetrical body and the second one offers an alternative presentation of the canonical ensemble.

We are concerned now with the asymptotic formula (*N* → ∞) for the volume of the

The linear dimension *<sup>ρ</sup>* of this volume, i.e., the length of one of its sides verifies *<sup>ρ</sup>* <sup>∼</sup> *<sup>E</sup>*1/*b*. As argued in Eq. (51), the *N*-dimensional volume, *VN*(*b*, *ρ*), is proportional to the term *ρ<sup>N</sup>* and to

*<sup>N</sup>*−<sup>1</sup> <sup>+</sup> *<sup>x</sup><sup>b</sup>*

*N* 1 Γ( *<sup>N</sup>* <sup>1</sup> + 1)

*<sup>N</sup>* = *E*. (84)

. (86)

. (87)

, (88)

*<sup>b</sup>* )! in the denominator of expression (88),

. (89)

, (90)

*VN*(*b*, *<sup>ρ</sup>*) = *gb*(*N*) *<sup>ρ</sup>N*, (85)

<sup>2</sup> <sup>+</sup> ··· <sup>+</sup> *<sup>x</sup><sup>b</sup>*

where the characteristic *b* indicates the particular boundary given by equation (84).

*gb*<sup>=</sup>2(*N*) =

*gb*<sup>=</sup>1(*N*) = <sup>1</sup>

From Eq. (23), if we take the diameter, *ρ* = 2*R*, as the linear dimension of the *N*-sphere, we

Γ *<sup>N</sup>* <sup>2</sup> + 1

These expressions (86) and (87) suggest a possible general formula for the factor *gb*(*N*), let us

Γ *<sup>N</sup> <sup>b</sup>* + 1

where *a* is a *b*-dependent constant to be determined. For example, *a* = 1 for *b* = 1 and *a* = *π*/4

In order to find the dependence of *a* on the parameter *b*, the regime *N* → ∞ is supposed.

*cb* = (*ab*)−1/*<sup>b</sup>*

*<sup>b</sup>*

*gb*(*N*) = *<sup>a</sup>*

and inserting it in expression (56), it is straightforward to find out the relationship:

*a* = Γ 1 *<sup>b</sup>* <sup>+</sup> <sup>1</sup>

 *π* 4 *N* 2

> *N b*

**5. Other additional geometrical questions**

**5.1. Formula for the volume of a high-dimensional body**

*xb* <sup>1</sup> <sup>+</sup> *<sup>x</sup><sup>b</sup>*

For instance, from Equation (2), we can write in a formal way:

Applying Stirling approximation for the factorial ( *<sup>N</sup>*

From here and formula (63), we get:

*N*-dimensional symmetrical body enclosed by the surface

a coefficient *gb*(*N*) that depends on *N*. Thus,

obtain:

say

for *b* = 2.

#### 16 Stochastic Control 78 Stochastic Modeling and Control Geometrical Derivation of Equilibrium Distributions in some Stochastic Systems <sup>17</sup>

**Figure 3.** The factor *gb* (*N*) versus *N* for *b* = 10, 40, 100, calculated from Eq. (91). Observe that *gb* (*N*) = 1 for *N* = 1, and lim*N*→<sup>∞</sup> *gb* (*N*) = 0.

**6. Conclusion**

when they decay to equilibrium.

geometrical interpretation of all those models.

equilibrium.

**Author details** López-Ruiz Ricardo

*Zaragoza, Spain* Sañudo Jaime

*Zaragoza, Spain*

**7. References**

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.

Geometrical Derivation of Equilibrium Distributions in Some Stochastic Systems 79

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

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

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

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

*Department of Computer Science, Faculty of Science, Universidad de Zaragoza, Zaragoza, Spain Also at BIFI, Institute for Biocomputation and Physics of Complex Systems, Universidad de Zaragoza,*

*Also at BIFI, Institute for Biocomputation and Physics of Complex Systems, Universidad de Zaragoza,*

[1] Huang K (1987) Statistical Mechanics. John Wiley & Sons, New York; Munster A (1969) Statistical Thermodynamics (volume I). Springer-Verlag, Berlin; Ditlevsen PD (2004)

*Department of Physics, Faculty of Science, Universidad de Extremadura, Badajoz, Spain*

Turbulence and Climate Dynamics. Frydendal, Copenhagen.

is not a trivial task and it remains as a typical challenge in this kind of problems.

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

From Section 2, here we offer a different image of the usual presentation that can be found in the literature [1] about the canonical ensemble.

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 at time *i*. Then we have:

$$E\_1 + E\_2 + \dots + E\_{\tau - 1} + E\_\tau = \tau \cdot \vec{E}.\tag{93}$$

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

$$p(E) \sim e^{-E/E},\tag{94}$$

is found by means of the geometrical arguments exposed in Section 2 [8]. Doing a thermodynamic simile, the temperature *T* can also be calculated. It is obtained that

$$
\vec{E} = kT.\tag{95}
$$

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

$$p(E) \sim e^{-E/kT},\tag{96}$$

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