Complex Systems with Trivial Dynamics
aa r X i v : . [ n li n . AO ] O c t Complex Systems with Trivial Dynamics
Ricardo L´opez-Ruiz
Dept. of Computer Science, Faculty of Science and Bifi,Universidad de Zaragoza, 50009 - Zaragoza, Spain, [email protected]
Abstract.
In this communication, complex systems with a near trivialdynamics are addressed. First, under the hypothesis of equiprobabilityin the asymptotic equilibrium, it is shown that the (hyper) planar geom-etry of an N -dimensional multi-agent economic system implies the ex-ponential (Boltzmann-Gibss) wealth distribution and that the sphericalgeometry of a gas of particles implies the Gaussian (Maxwellian) dis-tribution of velocities. Moreover, two non-linear models are proposed toexplain the decay of these statistical systems from an out-of-equilibriumsituation toward their asymptotic equilibrium states. Keywords:
Statistical models, Equilibrium distributions, Decay towardequilibrium, Nonlinear models.
In this paper, different classical results [1,2] are recalled. They are obtained froma geometrical interpretation of different multi-agent systems evolving in phasespace under the hypothesis of equiprobability [3,4]. Two nonlinear models thatexplain the decay of these statistical systems to their asymptotic equilibriumstates are also collected [5,6].We sketch in section 2 the derivation of the Boltzmann-Gibbs (exponen-tial) distribution [4] by means of the geometrical properties of the volume ofan N -dimensional pyramid. The same result is obtained when the calculationis performed over the surface of a such N -dimensional body. In both cases, themotivation is a multi-agent economic system with an open or closed economy,respectively.Also, a continuous version of an homogeneous economic gas-like model [5] isgiven in the section 2. This model explains the appearance, independently of theinitial wealth distribution given to the system, of the exponential (Boltzmann-Gibbs) distribution as the asymptotic equilibrium in random markets, and ingeneral in many other natural phenomena with the same type of interactions.The Maxwellian (Gaussian) distribution is derived in section 3 from geo-metrical arguments over the volume or the surface of an N -sphere [3]. Here,the motivation is 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.he ubiquity of the Maxwellian velocity distribution in ideal gases is alsoexplained in the section 3 with a nonlinear mapping acting in the space of velocitydistributions [6]. This mapping is an operator that gives account of the decay ofany initial velocity distribution toward the Gaussian (Maxwellian) distribution.Last section contains the conclusions. Here we assume N agents, each one with coordinate x i , i = 1 , . . . , N , with x i ≥ i , and a total available amountof money E : x + x + · · · + x N − + x N ≤ E. (1)Under random or deterministic evolution rules for the exchanging of moneyamong 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 inthis model supplies money instead of energy, could be played by the state orby the bank system in western societies. The formula for the volume V N ( E ) ofan equilateral N -dimensional pyramid formed by N + 1 vertices linked by N perpendicular sides of length E is V N ( E ) = E N N ! . (2)We suppose that each point on the N -dimensional pyramid is equiprobable, thenthe probability f ( x i ) dx i of finding the agent i with money x i is proportional tothe volume formed by all the points into the ( N − i th-coordinate equal to x i . Then, it can be shown that the Boltzmann factor(or the Maxwell-Boltzmann distribution), f ( x i ), is given by f ( x i ) = V N − ( E − x i ) V N ( E ) , (3)that verifies the normalization condition Z E f ( x i ) dx i = 1 . (4)The final form of f ( x ), in the asymptotic regime N → ∞ (which implies E → ∞ )and taking the mean wealth ǫ = E/N , is: f ( x ) dx = 1 ǫ e − x/ǫ dx, (5)where the index i has been removed because the distribution is the same foreach agent, and thus the wealth distribution can be obtained by averaging overall the agents. This distribution has been found to fit the real distribution ofincomes in western societies [7]. .2 Multi-Agent Economic Closed Systems We derive now the Boltzmann-Gibbs distribution by considering the system inisolation, that is, a closed economy. Without loss of generality, let us assume N interacting economic agents, each one with coordinate x i , i = 1 , . . . , N , with x i ≥
0, and where x i represents an amount of money. If we suppose that thetotal amount of money E is conserved, x + x + · · · + x N − + x N = E, (6)then this isolated system evolves on the positive part of an equilateral N -hyperplane. The surface area S N ( E ) of an equilateral N -hyperplane of side E isgiven by S N ( E ) = √ N ( N − E N − . (7)If the ergodic hypothesis is assumed, each point on the N -hyperplane is equiprob-able. Then the probability f ( x i ) dx i of finding agent i with money x i is propor-tional to the surface area formed by all the points on the N -hyperplane havingthe i th-coordinate equal to x i . It can be shown that f ( x i ) is the Boltzmannfactor (Boltzmann-Gibbs distribution), with the normalization condition (4). Ittakes the form, f ( x i ) = 1 S N ( E ) S N − ( E − x i )sin θ N , (8)where the coordinate θ N satisfies sin θ N = q N − N . After some calculation theBoltzmann distribution is newly recovered: f ( x ) dx = 1 kτ e − x/kτ dx, (9)with ǫ = kτ , being k the Boltzmann constant and τ the temperature of thestatistical system. We consider an ensemble of economic agents trading with money in a randommanner [7]. This is one of the simplest gas-like models, in which an initial amountof money is given to each agent, let us suppose the same to each one. Then, pairsof agents are randomly chosen and they exchange their money also in a randomway. When the gas evolves under these conditions, the exponential distributionappears as the asymptotic wealth distribution. In this model, the microdynamicsis conservative because the local interactions conserve the money. Hence, themacrodynamics is also conservative and the total amount of money is constantin time.The discrete version of this model is as follows [7]. The trading rules for eachinteracting pair ( m i , m j ) of the ensemble of N economic agents can be writtens m ′ i = σ ( m i + m j ) ,m ′ j = (1 − σ )( m i + m j ) , (10) i, j = 1 . . . N, where σ is a random number in the interval (0 , i, j ) are randomlychosen. Their initial money ( m i , m j ), at time t , is transformed after the interac-tion in ( m ′ i , m ′ j ) at time t + 1. The asymptotic distribution p f ( m ), obtained bynumerical simulations, is the exponential (Boltzmann-Gibbs) distribution, p f ( m ) = β exp( − β m ) , with β = 1 / < m > gas , (11)where p f ( m )d m denotes the PDF ( probability density function ), i.e. the proba-bility of finding an agent with money (or energy in a gas system) between m and m + d m . Evidently, this PDF is normalized, || p f || = R ∞ p f ( m )d m = 1. Themean value of the wealth, < m > gas , can be easily calculated directly from thegas by < m > gas = P i m i /N .The continuous version of this model [8] considers the evolution of an initialwealth distribution p ( m ) at each time step n under the action of an operator T . Thus, the system evolves from time n to time n + 1 to asymptotically reachthe equilibrium distribution p f ( m ), i.e.lim n →∞ T n ( p ( m )) → p f ( m ) . (12)In this particular case, p f ( m ) is the exponential distribution with the sameaverage value < p f > than the initial one < p > , due to the local and totalrichness conservation.The derivation of the operator T is as follows [8]. Suppose that p n is thewealth distribution in the ensemble at time n . The probability to have a quantityof money x at time n + 1 will be the sum of the probabilities of all those pairsof agents ( u, v ) able to produce the quantity x after their interaction, that is,all the pairs verifying u + v > x . Thus, the probability that two of these agentswith money ( u, v ) interact between them is p n ( u ) ∗ p n ( v ). Their exchange istotally random and then they can give rise with equal probability to any value x comprised in the interval (0 , u + v ). Therefore, the probability to obtain aparticular x (with x < u + v ) for the interacting pair ( u, v ) will be p n ( u ) ∗ p n ( v ) / ( u + v ). Then, T has the form of a nonlinear integral operator, p n +1 ( x ) = T p n ( x ) = Z Z u + v>x p n ( u ) p n ( v ) u + v d u d v . (13)If we suppose T acting in the PDFs space, it has been proved [5] that T conserves the mean wealth of the system, < T p > = < p > . It also conservesthe norm ( || · || ), i.e. T maintains the total number of agents of the system, || T p || = || p || = 1, that by extension implies the conservation of the total richnessof the system. We have also shown that the exponential distribution p f ( x ) withhe right average value is the only steady state of T , i.e. T p f = p f . Computationsalso seem to suggest that other high period orbits do not exist. In consequence,it can be argued that the relation (12) is true. This decaying behavior toward theexponential distribution is essentially maintained in the extension of this modelfor more general random markets. Let us suppose a one-dimensional ideal gas of N non-identical classical particleswith masses m i , with i = 1 , . . . , N , and total maximum energy E . If particle i has a momentum m i v i , we define a kinetic energy: K i ≡ p i ≡ m i v i , (14)where p i is the square root of the kinetic energy K i . If the total maximum energyis defined as E ≡ R , we have p + p + · · · + p N − + p N ≤ R . (15)We see that the system has accessible states with different energy, which can besupplied by a heat reservoir. These states are all those enclosed into the volumeof the N -sphere given by Eq. (15). The formula for the volume V N ( R ) of an N -sphere of radius R is V N ( R ) = π N Γ ( N + 1) R N , (16)where Γ ( · ) is the gamma function. If we suppose that each point into the N -sphere is equiprobable, then the probability f ( p i ) dp i of finding the particle i with coordinate p i (energy p i ) is proportional to the volume formed by all thepoints on the N -sphere having the i th-coordinate equal to p i . It can be shownthat f ( p i ) = V N − ( p R − p i ) V N ( R ) , (17)which is normalized, R R − R f ( p i ) dp i = 1. The Maxwellian distribution is obtainedin the asymptotic regime N → ∞ (which implies E → ∞ ): f ( p ) dp = r πǫ e − p / ǫ dp, (18)with ǫ = E/N being the mean energy per particle and where the index i hasbeen removed because the distribution is the same for each particle. Then theequilibrium velocity distribution can also be obtained by averaging over all theparticles. .2 Multi-particle closed systems We start by assuming a one-dimensional ideal gas of N non-identical classicalparticles with masses m i , with i = 1 , . . . , N , and total energy E . If particle i has a momentum m i v i , newly we define a kinetic energy K i given by Eq. (14),where p i is the square root of K i . If the total energy is defined as E ≡ R , wehave p + p + · · · + p N − + p N = R . (19)We see that the isolated system evolves on the surface of an N -sphere. Theformula for the surface area S N ( R ) of an N -sphere of radius R is S N ( R ) = 2 π N Γ ( N ) R N − , (20)where Γ ( · ) is the gamma function. If the ergodic hypothesis is assumed, thatis, each point on the N -sphere is equiprobable, then the probability f ( p i ) dp i offinding the particle i with coordinate p i (energy p i ) is proportional to the surfacearea formed by all the points on the N -sphere having the i th-coordinate equalto p i . It can be shown that f ( p i ) = 1 S N ( R ) S N − ( p R − p i )(1 − p i R ) / , (21)which is normalized. Replacing p by mv , f ( p ) takes the following form g ( v )in the asymptotic limit N → ∞ , g ( v ) dv = r m πkτ e − mv / kτ dv. (22)This is the typical form of the Maxwellian distribution, with ǫ = kτ / Here, as we have done in the anterior case of economic systems, we present anew model to explain the Maxwellian distribution as a limit point in the spaceof velocity distributions for a gas system evolving from any initial condition [6].Consider an ideal gas with particles of unity mass in the three-dimensional(3 D ) space. As long as there is not a privileged direction in the equilibrium,we can take any direction in the space and study the discrete time evolution ofthe velocity distribution in that direction. Let us call this direction U . We cancomplete a Cartesian system with two additional orthogonal directions V, W . If p n ( u )d u represents the probability of finding a particle of the gas with velocitycomponent in the direction U comprised between u and u + d u at time n , thenthe probability to have at this time n a particle with a 3 D velocity ( u, v, w ) willbe p n ( u ) p n ( v ) p n ( w ).he particles of the gas collide between them, and after a number of interac-tions of the order of system size, a new velocity distribution is attained at time n + 1. Concerning the interaction of particles with the bulk of the gas, we maketwo simplistic and realistic assumptions in order to obtain the probability of hav-ing a velocity x in the direction U at time n + 1: (1) Only those particles with anenergy bigger than x at time n can contribute to this velocity x in the direction U , that is, all those particles whose velocities ( u, v, w ) verify u + v + w ≥ x ;(2) The new velocities after collisions are equally distributed in their permittedranges, that is, particles with velocity ( u, v, w ) can generate maximal velocities ± U max = ±√ u + v + w , then the allowed range of velocities [ − U max , U max ]measures 2 | U max | , and the contributing probability of these particles to the ve-locity x will be p n ( u ) p n ( v ) p n ( w ) / (2 | U max | ). Taking all together we finally getthe expression for the evolution operator T . This is: p n +1 ( x ) = T p n ( x ) = Z Z Z u + v + w ≥ x p n ( u ) p n ( v ) p n ( w )2 √ u + v + w d u d v d w . (23)Let us remark that we have not made any supposition about the type ofinteractions or collisions between the particles and, in some way, the equivalentof the Boltzmann hypothesis of molecular chaos [9] would be the two simplisticassumptions we have stated on the interaction of particles with the bulk ofthe gas. Then, an alternative framework than those usually presented in theliterature [10] appears now on the scene. In fact, it is possible to show thatthe operator T conserves in time the energy and the null momentum of thegas. Moreover, for any initial velocity distribution, the system tends towards itsequilibrium, i.e. towards the Maxwellian velocity distribution (1D case). Thismeans that lim n →∞ T n ( p ( x )) → p α ( x ) = r απ e − αx (24)with α = (2 < x , p > ) − . In physical terms, it means that for any initialvelocity distribution of the gas, it decays to the Maxwellian distribution, whichis just the fixed point of the dynamics. Recalling that in the equilibrium
Acknowledgements
Several collaborators have participated in the development of different aspectsof this line of research. Concretely, X. Calbet, J. Sa˜nudo, J.L. Lopez and E.Shivanian. See the references.