Geometrical derivation of the Boltzmann factor

The Maxwellian distribution was derived in Ref. 1 by geometrical arguments over an Nsphere. Following this insight, here we explain the origin of the Boltzmann factor by means of the geometrical properties of an N-hyperplane. This alternative view can be thought as an a priori theoretical argument that explains why the probability of a magnitude, constrained by a conservation law, is given by the exponential distribution when the ensemble or system over which that magnitude is measured is in thermodinamical equilibrium.

By thermodinamical equilibrium we mean the macroscopic stationary regime where all the microscopic accessible states of the system are equiprobable. This implies that the deterministic or random time evolution of the system verifies the ergodic hypothesis. This has been proved in very few systems 2 , and hence, in general, it is supposed as an hypothesis in classical statistical mechanics 3 . If the system do not explore its own phase space with equiprobable results for all the accessible states then the system exhibits some kind of ergodicity breaking and the final distribution will not be the exponential distribution. We can call it a non-Boltzmann equilibrium.

Without loss of generality, here we obtain analytically the Boltzmann factor (or Maxwell-Boltzmann distribution) in a model where the ergodic hypothesis has been computationally proved. The method was developed in Ref. 1 and it is grounded on geometrical arguments.

The model was proposed by Dragulescu and Yakovenko 4 in order to explain the distribution of wealth in human society. Thus it is known that the incomes of 90% of the population in western societies can be fitted by a exponential distribution. 5 Supposing initially equity in a set of many agents, that is, all of them have the same initial quantity of money, the model establishes random binary interactions in which agents exchange money but conserve its total amount. It is computationally found that the system asymptotically tends toward the exponential distribution. This means that, in this case, the evolution mechanisms proposed by Dragulescu and Yakovenko for the money exchange among agents verify the ergodic hypothesis and, then, its final distribution is the exponential distribution. (Let us say that other non-Boltzmann random mechanisms for the money exchange are also proposed in Ref. 4. These mechanisms imply the breaking of the ergodic hypothesis and therefore the asymptotic evolution of the system toward non-Boltzmann equilibria).

We start by assuming N agents, each one with coordinate (money) x i , i = 1, . . . , N, with x i ≥ 0, and a total amount of money E that is conserved:

(1)

Under the evolution rules proposed in Ref. 4, this isolated system evolves on the positive part of an equilateral N-hyperplane (i.e., the surface formed for all those points with x i ≥ 0, for all i). The formula for the surface area S N (E) of an equilateral N-hyperplane of side E is

(See appendix A for the derivation of this formula). If the ergodic hypothesis is assumed, each state of the microcanonical ensemble (that is, each point on the N-hyperplane) is equiprobable, then the probability f (x i )dx i of finding the agent i with money x i is proportional to the surface area formed by all the points on the N-hyperplane having the ith-coordinate equal to x i . Our objective is to show that f (x i ) is the Boltzmann factor (or the Maxwell-Bolztamnn distribution), with the normalization condition

If the ith agent has coordinate x i , the N -1 remaining agents share the money Ex i on the (N -1)-hyperplane

whose surface area is S N -1 (Ex i ). If we define the coordinate θ N (see appendix A) as

it can be easily proved that

Hence, the surface area of the N-hyperplane for which the ith coordinate is between x i and x i + dx i is S N -1 (Ex i )dx i / sin θ N . We rewrite the surface area as a function of x i , normalize it to satisfy Eq. ( 3), and obtain

whose final form, after some calculation is

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

The Boltzmann factor e -x i /ǫ is found when N ≫ 1 but, even for small N, it can be a good approximation for agents with low wealth. After substituting Eq. ( 9) into Eq. ( 8), we obtain the Maxwell-Boltzmann distribution in the asymptotic regime N → ∞ (which also implies

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.

Depending on the physical situation the mean wealth per agent ǫ takes different expressions and interpretations. For instance, in the case of an isolated one-dimensional gas, the x i variable represents the energy of the particle i, thus we can calculate the dependence of ǫ 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 S = -kN ∞ -∞ f (x) ln f (x) dx, w

This content is AI-processed based on open access ArXiv data.