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Income Inequality and Geometrical Frustration

March 20, 2017

We've all been frustrated by events in our personal lives. As children, we never received that one special toy that we really wanted. As adults, we wonder if that red light will ever change; or, if we'll ever have enough money saved for a down payment on a house. Physicists are likewise frustrated by things like peer review, but they have a more scientific understanding of frustration.

Just as in personal frustration, when a goal is blocked by some situation, geometrical frustration involves a goal, typically descent into the lowest energy state, that's blocked by a competing requirement of a necessary spatial arrangement of objects. One simple example of this is found when trying to establish an antiferromagnetic ordering of spins on a triangular lattice.

In 1924, German physicist, Ernst Ising (1900-1998), did his thesis on a model of magnetism involving nearest neighbor interaction between spins arrayed in a straight line. This model was subsequently generalized to a two-dimensional lattice by Lars Onsager (1903-1976), who was awarded a Nobel Prize in Chemistry in 1968 for his contributions to the thermodynamics of irreversible processes.

In antiferromagnetic ordering, near-neighbor spins should be antiparallel, and this is easy to do on a square lattice. As the figure shows, when we try to do this on a triangular lattice, we're frustrated, since one spin is necessarily parallel to a neighbor spin in any triangle. G.H. Wannier (1911-1983), known especially among condensed matter physicists for his eponymous Wannier functions, calculated the residual entropy at absolute zero of spins on such a triangular lattice as 0.3383R, or about 0.67 cal/K/mol.

Attempted antiferromagnetic ordering on a triangular lattice

Attempted antiferromagnetic ordering on a triangular lattice. Only two of the three spin pairs can be aligned antiparallel on any triangular unit cell. (Created using Inkscape.)

Much earlier than that, Linus Pauling used geometrical frustration to explain the residual entropy of ice.[2] Solid water H2O, exists as oxygen anions surrounded by four protons, any two of which can be in closer proximity to form the water molecule, an arrangement commonly known as the ice rules. Pauling found the number of possible arrangements Ω of particles in the ground state of an ensemble of N ice molecules to be
Ω < 22N(6/16)N.
Calculating the Boltzmann entropy from this yields the following:
S = kBln(Ω) = NkBln(3/2) = 0.81 cal/K/mol,
where kB is the Boltzmann constant. This value agrees closely with an experimental value of 0.82 ± 0.05 cal/K/mol.[3]

Ludwig Boltzmann (1844-1906) applied such statistics to thermodynamics in 1875, and he used statistics to explain the probability distribution of speeds of molecules in a gas, now called the Maxwell–Boltzmann distribution, that was first investigated by James Clerk Maxwell (1831-1879) in 1860. What's surprising is that the distribution of incomes follows this same curve. I reviewed this idea that "people are like gas" in an earlier article (The Entropy of Nations, January 22, 2014)

Such a correspondence would be just an analogy without a mechanism; and, just as gas molecules exchange energy when they meet each other, people exchange wealth. The precise distribution of wealth is generated by fine-tuning the degree of wealth transferred at random meetings. The idea that we're being guided by such an "invisible hand" is exemplified by the following graph that shows the partitioning of wealth between the many and the few in the United States for the year 2000. Such studies are now a routine part of a field known as econophysics.

US wealth distribution, year 2000

US wealth distribution for the year 2000.

Such regularity in data suggests a simple mechanism.

(Data from ref. 4, graphed using Gnumeric.)

Two scientists affiliated with Los Alamos National Laboratory (Los Alamos, New Mexico), and a visiting scientist from the Université Paris Saclay (Gif-sur-Yvette, France) have recently published a statistical model in which they've applied geometrical frustration to a simple social network. They show how income inequality can arise in such a system, but also how constraints on interactions between agents can lead to greater income equality.[5-6]

The model relates to the traditional Horatio Alger rags-to-riches story, since it incorporates channels of wealth transfer that transcend social stratification.[6] Says Cristiano Nisoli, a member of the Physics of Condensed Matter and Complex Systems group at Los Alamos and lead author of the study,
"Most theories of wealth inequality rely on social stratification due to income inequality and inheritance... We consider, however, the possibility that in our more economically fluid world, novel, direct channels for wealth transfer could be available, especially for financial wealth."[6]

Income equality (a.k.a., "fairness in wealth distribution") is easily quantified by the Lorenz curve and the Gini coefficient. In the LANL model, an ensemble of agents has the ability to acquire wealth through transfer from other agents; and, having wealth gives an advantage of gaining more wealth, which is an obvious feature of nearly all societies. Setting these agents loose in their computer world leads to different wealth distributions depending on applied constraints.[6]

The "law-of-the-jungle" scenario, illustrated below where the allocation of opportunities does not change and wealth is acquired in a meeting of agents without any rules, leads to gross inequality. As Nisoli explains, "If driven by power alone, the market evolution reaches a static equilibrium characterized by the most savage inequality." The opportunities to acquire wealth are concentrated in just a few agents, and these amass all the wealth of the society.[6]

Acquisition of wealth according to the 'law of the jungle'

The rich get richer. An example of the acquisition of wealth when the "law of the jungle" in which every agent (blue) can grab wealth (red circles) to fill available opportunities (white circles). (Fig. 1 of ref. 5, modified, licensed under a Creative Commons Attribution License)[5)]

When transactions of wealth are regulated such that people can gain or lose wealth from only neighbors in the network, fairness is considerably increased. With some tuning, there's an emergence of three social classes, lower, middle, and upper.[6] Just as in real life, an unexpected economic event shifts the equilibrium to distort a previously stable class structure.[6] This research was funded by the US Department of Energy.[5]


  1. G. H. Wannier, "Antiferromagnetism. The Triangular Ising Net," Phys. Rev., vol. 79, no. 2 (July 15, 1950), pp. 357ff., DOI:https://doi.org/10.1103/PhysRev.79.357.
  2. Linus Pauling, "The Structure and Entropy of Ice and of Other Crystals with Some Randomness of Atomic Arrangement," J. Am. Chem. Soc., vol. 57, no. 12 (December, 1935), pp. 2680-2684, DOI: 10.1021/ja01315a102.
  3. W. F. Giauque and J. W. Stout, "The Entropy of Water and the Third Law of Thermodynamics. The Heat Capacity of Ice from 15 to 273K," J. Am. Chem. Soc., vol. 58, no. 7 (July 1, 1936), pp. 1144-1150, DOI: 10.1021/ja01298a023.
  4. James B. Davies, Anthony Shorrocks, Susanna Sandstrom and Edward N. Wolff, "The World Distribution of Household Wealth," escholarship.org web site (The University of California), July, 2007, p. 26.
  5. Benoit Mahault, Avadh Saxena, and Cristiano Nisoli, "Emergent inequality and self-organized social classes in a network of power and frustration," PLoS ONE, vol. 12, no. 2 (February 17, 2017), Article No. e0171832, doi:10.1371/journal.pone.0171832. This is an open access publication with a PDF file here.
  6. Science versus the Horatio Alger myth, DOE/Los Alamos National Laboratory Press Release, February 22, 2017.

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