Modeling Wealth Inequality with Kinetic Theory- Physics Research project

Introduction to kinetic theory:

Kinetic theory explains how gases behave by looking at the movement of tiny particles: atoms or molecules. The theory, instead of tracking each particle one by one, uses probability to understand the average behavior of millions or billions of them. This means that, even though a basic physics lesson suggests that individual particles move randomly, their collective behavior follows predictable patterns. The pressure and temperature in a gas can be explained using kinetic theory, based on how often and how hard particles collide with a wall of a container [1].

The basic principle is that energy is transferred via (elastic) collisions. When two particles bump into each other, they exchange energy, but the total energy stays the same: conservation of energy. Over time, the energy spreads out evenly across the system, leading to thermal equilibrium: when there is no net transfer of thermal energy between them [2]. This concept was developed in the 19th century by scientists like James Clerk Maxwell and Ludwig Boltzmann. Boltzmann distribution describes how energy is spread among particles in a gas [3].

What’s I found particularly interesting is that the same mathematical ideas used to study particles in gases can also be applied to systems outside of physics. My research will look at how physicists and economists have explored how kinetic theory can model wealth distribution in a society. Instead of energy, the quantity being exchanged is wealth (the stock of assets), and instead of particles, the system is made up of people [4].

The Analogy: People as Particles

In the same way that we see gas particles collide randomly in the commonly represented cube model, people in an economy interact randomly through trades, purchases, or exchanges. Each person acts like a particle, and their money is like the energy they carry. When we see two people interact, they exchange money just like particles exchange energy during collisions. As with any good economics model these interactions are usually modeled as random and fair, meaning no one cheats or has extra asymmetric information.

Suppose I give everyone £100 worth of shares in Tesla (a randomly chosen company that has no relevance in itself with the model). In each “collision,” you can expect to see two people are randomly selected and share their total wealth equally, or slightly inequally. Over time, just like in a gas where energy spreads unevenly, money becomes unevenly distributed. Some people end up with more, and others with less, even though the rules were fair and random.

During my research, I came across the work proposed by physicists Adrian Dragulescu and Victor Yakovenko in 2000 [5]. They showed that even if everyone starts equally and trades randomly, wealth tends to concentrate over time, with a small number of people becoming much richer than the rest [5]. In the UK, where the richest 10% of households own 43% of the nation’s wealth while the bottom 50% share just 9%, Robert Frank’s ideas in his book Success and Luck help explain this divergence. Whilst reading the book I began to understand how skills and opportunities are often exchanged within networks of highly skilled individuals, reinforcing existing advantages and contributing to the concentration of wealth [6].

The foundations of this model work as statistical mechanics [7] (the main-ish part of physics behind kinetic theory) doesn’t depend on what the particles are, but only how they interact. That’s why it can be applied to large systems of people too. The same math used to explain heat in gases can show how wealth can “flow” through society, eventually settling into a stable, but often unequal, distribution.

Simple Wealth Models using Kinetic Theory

Once we accept the idea that people can be modeled like gas particles and money like energy, we can build simple simulations/models to see what happens to wealth over time. The wise words of George box: “All models are wrong, but some are useful” [8] are worthwhile to consider when looking at the proposed three models.

1. The Basic Random Exchange Model

In the simplest model, everyone starts with the same amount of money say, £100. Then, at each step:

• Two people are randomly selected.

• They pool their money together.

• They split it evenly or randomly.

  This model is known as the random exchange model, and it behaves like an ideal gas: money (like energy) spreads out due to random “collisions.” It was surprising to find out that this model leads to extreme inequality. Over time, most people end up with little or no money, while one person ends up with almost everything, just like how energy can "condense" into one particle in certain physics systems.

Figure 1: A histogram showing exponential wealth distribution [9]

Figure 1 shows the result of such a simulation. The blue curve represents the probability of people having a certain amount of money after many random exchanges. It forms an exponential distribution, meaning most people have small amounts of money, and only a few have a lot. The red line is the theoretical prediction based on statistical physics, which matches the simulation very closely. The small graph inside (which is called an inset) uses a logarithmic scale to show that the exponential pattern holds even for large amounts of money. This simple yet powerful model demonstrates how inequality can emerge not from unfair rules, but from the mathematics of randomness: just like how energy spreads unevenly among particles in a gas [5].

2. Adding Saving Behavior (The Chakraborti Model) [9]

In the next model, we have a look at how people trade and understand that real people don’t trade all their money; most save a portion. To reflect this, we can add a saving factor to the model:

• Essentially, each person saves a percentage of their wealth before each exchange.

• The rest is pooled and split between the two.
  When people save money, the results change dramatically:

• Wealth no longer collapses to one person.

• A more realistic distribution appears: some people are rich, many are average, some are poor.

  3. Emergence of Inequality

  Further looking into this, even though the system starts fairly (equal money, random trades), inequality emerges naturally. This mirrors the second law of thermodynamics in physics: systems tend toward disorder unless external forces (like taxes or regulations) change the rules [10].

  These models show that inequality isn't always the result of unfairness, it can arise just from randomness and basic rules or as Robert Frank may put it, due to luck [6]. I found this pretty cool, as it connects two topics that I am currently doing at A-level, in both physics and economics, adds some empirical research and provides reasoning, and doesn’t simply state facts.

Concluding thoughts

This was an interesting project for me to be able to research. With regards to these models, they are significant because they really show that inequality can emerge naturally, even when everyone starts equally, and trades are random and fair. The basic model explains how wealth can become concentrated purely through chance, and the saving model adds realism by showing that people who save money tend to keep and build wealth over time. These ideas, although borrowed from physics, give us a new way to think about economics and inequality using simple rules and simulations.

However, the models have limits: they assume people behave in very simplified ways, ignore income from work or investment, and don’t include inheritance, taxes, or social advantages. They’re useful because they help explain big patterns with small rules, just like in physics, and give us a new way to understand how inequality can grow, even in fair systems.

This project, although it was meant to be 3 pages long was so interesting. I found it interesting to see how concepts from physics that we learn at A-level could help explain wealth inequality. Reading an actual academic paper for the first time was very challenging, especially because of some of the wording and math confused me. I really liked how the random nature of wealth distribution as well, and how it complimented my wider reading in economics, reading the book Success and Luck; concepts such as luck having such an important factor in success.

Bibliography

1. [1]  K. Academy, "The kinetic molecular theory of gases".

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3. [3]  Cercignani, "The Man Who Trusted Atoms".

4. [4]  C. h. (. D. a. b. t. O. o. t. m. C. S. a. R. F. r. K. e. m. o. markets, "Wikipedia," [Online]. Available: https://en.wikipedia.org/wiki/Kinetic_exchange_models_of_markets. [Accessed 22 June 2025].

5. [5]  V. M. Y. Adrian Dragulescu, "Statistical mechanics of money," [Online]. Available: https://arxiv.org/abs/cond-mat/0001432. [Accessed 22 June 2025].

6. [6]  Robert H. Frank, Sucess and luck, pp. 20-22.

7. [7]  G. S. University, "Statistical Mechanics Concepts," [Online]. Available: http://hyperphysics.phy- astr.gsu.edu. [Accessed 22 June 2025].

8. [8]  G. E. P. Box, Interviewee, [Interview].

9. [9]  A. C. a. B. K. Chakrabarti, "How saving propensity asects its distribution," The European Physical Journal.

10. [10]  Wikepedia,"Secondlawofthermodynamics,"[Online].Available: https://en.wikipedia.org/wiki/Second_law_of_thermodynamics. [Accessed 22 June 23].

11. [11]  A.&.Y.Dragulescu,"Therandomexchangemodel,"Statisticalmechanicsofmoney.,p.723–729.