SUPLEMENTO Revista Mexicana de Fısica S58 (1) 110–115 JUNIO 2012

On Ising spin models and statistical wealth condensation: Generating awealth-like distribution

A.R. Hernandez-Montoya∗, H.F. Coronel-Brizio, A. Aguilar-Salas, and N. Bagatella-FloresFacultad de Fısica e Inteligencia Artificial, Universidad Veracruzana, Departamento de Inteligencia Artificial,

Sebastian Camacho 5, Xalapa Veracruz 91000, Mexico.Tel/Fax: 52-228-8172957/8172855∗e-mail: alhernandez@uv.mx

E. ScalasDepartment of Sciences and Advanced Technology,

The University of Eastern Piedmont “Amedeo Avogadro” Viale T. Michel, 11, 15121 Alessandria, Italy.

Recibido el 23 de Marzo de 2010; aceptado el 27 de Abril de 2011

In this paper, the wealth dynamics is investigated by means of the two-dimensional Ising model using Monte Carlo simulations. By mappingagents onto spins and by interpreting their up and down state transitions as the increase and loss of an arbitrary monetary unit, we are ableto generate the universal properties of a wealth-like distribution at the critical temperatureTc. The sub-critical temperature regime is alsoanalyzed. It is shown that log-normal and gamma-like wealth distributions emerge from the simulations.

Keywords: Ising model; critical state; power law; wealth distribution; universatility.

En este artıculo, la dinamica de la acumulacion de riqueza es investigada por medio de una simulacion Monte Carlo del modelo de Isingen dos dimensiones. Haciendo corresponder agentes a los espines e interpretando las transiciones entre sus estados arriba y abajo como elincremento o perdida por el agente de una unidad monetaria arbitraria, podemos generar a temperatura crıtica Tc una distribucion que tienelas mismas propiedades universales que una distribucion del tipo de la distribucion de la riqueza. El regimen subcrıtico tambien es analizado.En este caso se muestra que las distribuciones de la riqueza son compatibles con las distribuciones gamma y log-normal, las cuales emergende nuestras simulaciones.

Descriptores: Modelo de Ising; estado crıtico; ley potencia; distribucion de la riqueza; Universalidad.

PACS: 89.75.-k; 05.65.+b; 05.50.+q; S89.65.Gh

1. Introduction

Multi-agent based models have been successfully used tostudy social and economic phenomena. Recently, many in-teresting multi-agent based models have been proposed to in-vestigate the wealth distribution [1-6] and interesting, empiri-cally confirmed results have been obtained [7,8]. On the otherhand, multi-agent models with a slightly different philoso-phy, the so-called lattice Ising type models, have been alsoused to study social and financial systems and some relatedproblems, such as statistical and dynamical properties ofstock prices, traders behavior, urban segregation, etc. [9-13],however, to the best of our knowledge, this kind of modelshave not been used to study the wealth distribution.

The question: Why are there rich and poor people?is an old question surely raised when civilizations becamesufficiently sophisticated and complex to generate enoughwealth [14]. Possible answers have varied throughout his-tory. Currently, economists and physicists are trying to pro-vide an answer to this important issue and other closely re-lated problems. For example in the global context of the un-equal income distribution of under developed or third worldcountries, many theories and ideas have been proposed to ex-plain why economic and social policies applied successfullyin developed countries have been not very fruitful after theirimplementation in third world countries. Some of these ideas

and theories considered factors such as a surplus of non quali-fied workforce, dependence on external technology, low levelof internal savings, chronic lack of equilibrium between ex-ports and imports, and even low cognitive skills of their pop-ulation, etc. [15-19].

However, the wealth distribution has some very interest-ing, common and universal properties, that are independentof the different countries or societies, developed or under-developed. This fact has been well confirmed by empiricalstudies [7,8,20-22], where [20] is the first reference aboutthe subject. All these and other econophysics studies demon-strate that the shape of the wealth distribution curve for differ-ent countries is a log-normal or gamma distribution for aboutits first 95th percentiles and a Pareto or power law for themost extreme events.

After remarking the theoretical and practical importanceof this problem, is appropriated to cite here the preface of“Econophysics of Wealth Distribution” [21] that results veryillustrative: “Econophysics tried to view this as a natural lawfor a statistical many body - dynamical market system, anal-ogous to gases, liquids or solids: classical or quantum”.

Thus, the observed wealth distribution could be an emer-gent and universal property of systems with a very high num-ber of particles or agents that interact and exchange “some-thing”. Trying to show this, we reduce the study of this

ON ISING SPIN MODELS AND STATISTICAL WEALTH CONDENSATION: GENERATING A WEALTH-LIKE DISTRIBUTION 111

problem to answer the more modest question: Which is thesimplest way to generate a wealth-like distribution using theIsing Model?

We answer this question in the present paper which is or-ganized as follows: In the next section we make a brief in-troduction to the Ising Model and to the algorithm used tosimulate it. Then, in the subsequent Sec. 3 it is shown how toobtain a wealth-like distribution from the simulations; in Sec.4, the results of the fits applied to the generated distributionsare presented, showing that they have the universal proper-ties of a wealth distribution; finally, in the last section, someremarks and conclusions about the analysis are presented.

2. The Ising model

The Ising model is a simple but nontrivial, very well stud-ied, microscopical lattice model of a ferromagnetic material,able to reproduce in two or more than two dimensional lat-tices, critical and phase transition phenomena. Historicallyhas its origin when, during his Ph. D research, E. Ising stud-ied a simple one-dimensional model of ferromagnetic inter-actions. In his 1924 Ph.D thesis [23], Ising presented the ex-act solution of this one-dimensional model showing that nophase transition was possible at finite temperature. In 1944L. Onsager calculated the exact partition function of the IsingModel in two dimensions in zero magnetic field [24]. Thetwo-dimensional Ising Model in a magnetic field or general-izations to upper dimensions are still unsolved.

The Ising model is also used to model non-magnetic sys-tems, such as lattice gases, binary alloys, or neuronal net-works [25]. In other areas of research outside physics, aswe have mentioned before it has wide applications [9-13].Bibliography about the simple Ising model and its latticespin models extensions (Potts, Baxter, Baxter-Wu, Ising-SpinGlass, Heisenberg, etc.) can be found elsewhere [26-31].

Here, we describe briefly the two-dimensional IsingModel, which corresponds to the one simulated and used inthe generation of the data analyzed in this work.

The Ising model consists of a square lattice withN = M × M cells, each cell representing an “agent” orparticle with spin 1/2 where:

Each celli, i = 1, 2, . . . , N has two possible states:

1. Si = +1, or white color which means celli has its spinup.

2. Si = −1, or black color which means spin celli hasits spin down.

The two-dimensional Ising model Hamiltonian can bewritten as follows:

H = −∑

i, j

JSiSj −∑

i

HSi.

Where:

• For eachi, the sum is performed up to its first neigh-bors in a Von Neumann neighborhood, see Fig. 1 for aexplanation of this term.

• Si = +1 or -1, depending on whether spin is up or down,respectively.

• J is the interaction energy between two neighboringspins and is−J when they are parallel, andJ if theyare anti-parallel.

• H is an external magnetic field.

The system behaves macroscopically as a ferromagnetwhenJ > 0 and as an anti-ferromagnet ifJ < 0.

The above lattice model withJ > 0, as was already men-tioned, presents a continuous phase transition depending ontemperature with two phases. The corresponding order pa-rameter is the mean magnetization, defined as:

m =1N

∑

i

Si,

and the two phases are a Ferromagnetic one forT < Tc andthe other a Paramagnetic phase forT > Tc, where the criticaltemperatureTc, is in the two-dimensional case:

Tc ≈ 2.2692

(in units of the nearest-neighbor interaction energykJ).

FIGURE 1. Von Neumann neighborhoods, wherer represents therange of the interaction between the non-central cells inside theneighborhood and the central one. a)r = 1 or to first or nearestneighbors of the central cell. b) A Von Neumann neighborhoodwith r = 2 includes the 12 cells displayed, surrounding the centralcell. c) The case forr = 3 is self-explanatory. We have used in thiswork ther = 1 Von Neumann neighborhood shown in a).

Rev. Mex. Fis. S58 (1) (2012) 110–115

112 A.R. HERNANDEZ-MONTOYA, H.F. CORONEL-BRIZIO, A. AGUILAR-SALAS, N. BAGATELLA-FLORES, AND E. SCALAS

2.1. Monte Carlo Method

The Ising Model was simulated using the Monte CarloMethod with a lattice sizeM = 300 × 300, with peri-odic boundary conditions, with a null external magnetic fieldH = 0 by using the Metropolis Algorithm, a well known, ef-ficient, ergodic and Markovian method [29,32-35], which sat-isfies detailed balance and whose general steps are describedbelow:

1. Randomly, generate an initial state for each spin,Si,i = 1, 2, 3, . . . , M (here, in order to reach the equi-librium faster, we have selected an initial stateSi=1,i = 1, 2, . . . ,M ).

2. Randomly, chooseSj , j = 1, 2, 3, . . . , N and calculate∆E = −2SjJ

∑j, i Si −HSj (the sum is performed

over a Von Neumann neighborhood ofSj).

3. If ∆E ≥ 0 change direction of the spin,i.e.Sj=−Sj ; otherwise, selectra, a random number uni-formly distributed in the interval[0, 1], such as if

ra < exp(∆E/KBT

), put Si = −Si; the spin is

left unchanged otherwise.

4. Repeat steps 2 and 3.

The Ising Model was simulated for three different trialtemperatures; one below the critical temperatureTc, settingup T = 0.7 Tc (ferromagnetic phase); one equal to the crit-ical temperatureT = Tc (critical state) and the last abovethe critical temperature,T = 1.3 Tc (paramagnetic phase).Images of the overall corresponding spin equilibrium statesfrom the simulations are shown in Fig. 2, sub-Fig. 2(a) tosub-Fig. 2(c).

As a final note for this section, it is remarked that, forsimplicity, we have chosen to perform the simulations usingthe Metropolis Algorithm instead of the “heat bath” or theWolff Algorithms [29,35-37]: The “heat bath” algorithm re-quires a larger amount of random numbers in order to reach

thermal equilibrium; on the other hand, although Wolff al-gorithm could be a good option since that we set up in oursimulationH = 0, it is computationally more intensive.

In next section it will be explained how to use these sim-ulations to generate the corresponding data samples.

3. Generating the Wealth-Like Distribution

For simplicity, let us change here the language, and insteadof cells, we name them “agents”. We address the problemof generating three wealth like-distributions, one for eachMonte Carlo simulation under different temperatures fromthe most direct point of view: Each agent wins an arbitrarymonetary unit whenever its corresponding spin transition fin-ishes in an up state and loses one “monetary” unit whenits final state is down. We keep record of the accumulated“wealth” of each agentCi(t), i = 1, 2, . . . , M at time stept.

The procedure for generating the sample data can be sum-marized as follows:

• Run simulation of the Ising model (N = M × M =300× 300 agents).

• Select the temperature and wait until the systemreaches equilibrium.

• For each time stept, ifSi(t) = 1 → Ci(t) = Ci(t− 1) + 1,Si(t) = −1 → Ci(t) = Ci(t− 1)− 1.

• Accumulate enough statistics. In this case and for eachone of our three trial temperatures, we have recordedthree thousand times 90000 random flips, after reach-ing equilibrium.

• Calculate the wealth of the poorest agent. Let us de-note it withWmin. This number will be only used totranslate the distribution minimum bin value to the ori-gin, as explained below.

FIGURE 2. Overall spins equilibrium states for our three trial temperaturesT = 0.7 Tc, T = 1.0 Tc andT = 1.3 Tc

Rev. Mex. Fis. S58 (1) (2012) 110–115

ON ISING SPIN MODELS AND STATISTICAL WEALTH CONDENSATION: GENERATING A WEALTH-LIKE DISTRIBUTION 113

FIGURE 3. Wealth probability density function for each one of thethree trial temperatures.

FIGURE 4. Generated distributions in a log-log plot. Each his-togram has 9000 entries.

The distribution ofWi := Ci − Wmin, i = 1, 2, . . . , Nwill be the generated “Wealth Distribution”.

Figure 3 shows the wealth probability density functionsgenerated under the three trial temperatures and Fig. 4 showstheir corresponding log-log plot histograms.

In next section we will discuss the analysis of these dis-tributions.

4. Data analysis and fits

In this section we present the fits performed on the data sam-ples for the three different trial temperatures. In order to havean unified notation, parameters or constants to be estimatedby the fitting procedures, are all denoted by lower case Greekletters,i.e. α, β, γ..., etc.

Results from simulation show that forT = 0.7 Tc, thebehaviour of the sample density can be closely described bythe probability density function of the gamma distribution,

fg(w) = wα−1e−wβ (Γ (α))−1 (βα)−1

,

FIGURE 5. Gamma fit for temperature0.7 Tc.

FIGURE 6. Lognormal1.0 Tc.

whereas forT = Tc, and without regard of the extreme dataevents, the distribution behaviour is best described by the log-normal model

fln(w) = 1/2 e−1/2(ln(w)−µ)2

σ2√

2w−1σ−1 1√π

.

The best fit is obtained when the Gaussian model is usedfor the data corresponding toT = 1.3 Tc.

Finally, for T = Tc and considering the extreme events,a power law distribution:

fp(w) = βw−α,

whereβ constant, was found to provide a good fit.These results are summarized in Figs. 5 to 8 and Table I,

where the estimated values of the parameters are included.

TABLE I. Fitted “Wealth” distributions to simulated data.

Data Sample Fitted Distribution

Parameters

D 0.7 Tc fg(w) (Gamma) α = 9.23, β = 5.69

D 1.0 Tc fln(w) (Log-normal) µ = 6.45, σ = 0.55

D 1.0 Tc fp(w) (asymptotic α ≈ 3.4

Power law)

D 1.3 Tc Normal µ = 703.8, σ = 158.16

Rev. Mex. Fis. S58 (1) (2012) 110–115

114 A.R. HERNANDEZ-MONTOYA, H.F. CORONEL-BRIZIO, A. AGUILAR-SALAS, N. BAGATELLA-FLORES, AND E. SCALAS

FIGURE 7. Normal fit for temperature1.3 Tc.

FIGURE 8. Left Panel: Overall wealth distribution forT = Tc.Right Panel: Same as before, but corrected for noise and finite lat-tice size effects. The last 294 largest values forming the baselinestep, were excluded.

The parameters appearing in Table I, were estimated bymaximum likelihood. The corresponding asymptotic theoryestablishes that the the variances are of the estimators are oforderO(n−1) ' 10−5, wheren denotes the number of sam-ple values used to compute the estimates, in our case 90000.If required, the asymptotic exact expressions for the standarderrors of the estimators can be obtained by computingn timesthe inverse of Fisher’s Information Matrix and substitutingthe estimates to produce a more precise approximation.

5. Summary

Using the Ising Model and interpreting their spins states+1and−1 as a “gain” or a “loss” respectively of an arbitrary“Monetary Unit”, we are able of reproduce the followingwealth-like distributions generated below critical, critical andabove critical temperatures:

• A Gamma distribution fits well the generated wealth-like distribution toT = 0.7 Tc.

• WhenT = Tc, the mean body of the wealth-like dis-tribution is well described by a log-normal distribu-tion and the part corresponding to the high incomepopulation decays as a power law with an exponent

α ≈ 3.4 (Wealth condensation case).The Power Lawfit was performed using maximum likelihood estima-tion for a left censored sample as described in Ref. 38.

• For the caseT = 1.3 Tc, a Gaussian distribution fitswell our data.

And then, we have reproduced the universal properties ofthe wealth distribution. Now, although we are aware that theway we reproduce the properties of the wealth distribution isvery simplistic, has many limitations, and accepting that wemust be very careful in order to make analogies relating Sta-tistical Physical Systems with Economic Systems, is possibleto say some interesting things as consequence of the resultsobtained and about the relation between Statistical Physicsand Economics:

• Macroscopic properties of Economical Complex Sys-tems, in particular, those produced by the underlyingmechanism of wealth production and distribution aresubject of being studied by means of Statistical Physicsand multi-agent simulations methodologies.

• The results obtained here are an evidence, that inany enough complex system, with a large number ofexchanging agents, universal properties, such as theshape of wealth distribution emerge, and StatisticalPhysics is well suited to complement Economics in thestudy of these kind of phenomena presented in Eco-nomical Complex Systems.

• In order to be accepted, any new proposed marketmulti-agent model must reproduce the universal prop-erties discussed in this work.

• Criticality (as in this work), or Self-Organized Critical-ity, make not necessary to introduce any asymmetry onagents interactions in market multi-agent models to re-produce the universal properties of wealth distributionas suggested by some authors [4].

However further research must be still be done to clarifymore the above mentioned points.

Acknowledgments

To the Memory of our teacher Augusto Garcıa, a great scien-tist and human being (ARHM, NBF). This work has beensupported by Conacyt-Mexico under project grant number155492 and by Conacyt-Mexico and MAE-Italy under grant146498 A.R. Hernandez-Montoya thanks to S. Jimenez fortheir useful suggestions and time. Also, H.F. Coronel-Brizioand A.R. Hernandez-Montoya acknowledge support from theSistema Nacional de Investigadores (SNI), CONACyT, Mex-ico. Part of the present work has been carried out usingROOT [39]. Finally we thank the anonymous referees fortheir very useful suggestions that improved greatly the qual-ity of this paper.

Rev. Mex. Fis. S58 (1) (2012) 110–115

ON ISING SPIN MODELS AND STATISTICAL WEALTH CONDENSATION: GENERATING A WEALTH-LIKE DISTRIBUTION 115

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