Somewhere in the gap between a person deciding where to live and a nation’s wealth clustering in a handful of cities, something gets lost. Economists model the aggregate; sociologists study the individual; neither, quite, manages to speak the other’s language. The equations that govern a gas of molecules, however, turn out to say something rather precise about how both come to be, and a new paper by physicist Miguel Durán-Olivencia at Vortico Tech may be the most rigorous attempt yet to build that bridge from scratch.
The paper, published in the Journal of Statistical Mechanics in April 2026, treats human beings the way a physicist treats particles: with momentum, friction, and an internal variable representing wealth, all evolving together through the mathematics of non-equilibrium statistical mechanics.
The starting point is a set of equations borrowed from the physics of Brownian motion, known as Langevin dynamics. Each agent in Durán-Olivencia’s model has a position, a velocity, and a wealth level. Deterministic forces pull agents toward resources (fertile land, mineral deposits, a natural harbour), while a stochastic term captures the irreducible unpredictability of human decision-making. This randomness isn’t a flaw in the model; it’s the whole point. Classical economics long assumed hyper-rational agents with perfect information. The Langevin formalism treats bounded rationality as a temperature, something to be measured rather than argued about.
That phrase, “social temperature,” does a lot of work in this framework. At low temperatures, agents behave like near-perfect optimizers, clustering tightly around the best available resources. Crank up the temperature and behavior becomes more exploratory, the population more diffuse. A peculiar kind of metaphor, perhaps, but also a precise one.
From Particles to Populations
From this microscopic description, Durán-Olivencia then derives equations governing the collective behavior of the whole population using a mathematical object called the empirical measure, essentially a continuous approximation of where everyone is and how wealthy they are. The derivation runs through two intermediate levels: a mesoscopic “fluctuating hydrodynamics” description that remains valid for any finite number of agents, and then a macroscopic deterministic system, the Vlasov-Fokker-Planck equations, recovered in the limit when the number of agents becomes very large. The procedure is formally analogous to deriving the laws of fluid mechanics from the jostling of individual molecules.
None of this would be especially noteworthy if it merely reproduced known results in new notation. What makes the paper striking is what falls out analytically when the model is applied to a simple scenario: a population of agents interacting with a fixed, spatially uneven resource landscape. No policy, no history, no cultural variable. Just physics and geography.
The stationary solution to the equations predicts, first, that population will cluster around resource-rich areas, with the degree of clustering controlled by social temperature (a sensible enough result, consistent with centuries of urban history). But it also predicts the shape of the wealth distribution within any given location. At any fixed point in space, the distribution of individual wealth follows an Inverse Gamma form. For large wealth values, this distribution has a power-law tail: the probability of finding someone with wealth above a threshold falls off as a power of that threshold, with an exponent tied directly to the microscopic parameters governing how fast wealth grows and how volatile that growth is. In other words, the Pareto distribution, the famous 80-20 rule, the long tail that shows up in income statistics in virtually every country ever studied, emerges here not as an assumption or a fitted parameter, but as a consequence. It drops out of the physics.
Geography and Wealth as a Single System
There is a further wrinkle, maybe the most consequential one. The shape of the wealth distribution is not constant across space; it shifts depending on how resource-rich the local environment is. In areas with abundant resources, the entire distribution is pushed toward higher values. In resource-poor regions, it collapses toward lower ones. Spatial inequality and economic inequality, in this framework, are not two separate phenomena requiring two separate explanations. They are coupled outputs of a single dynamical system, seeded by nothing more exotic than an uneven landscape.
Durán-Olivencia validated these predictions with simulations of 100,000 agents, testing both a single resource concentration and a more complex two-well landscape. The Gini coefficients recovered from simulation (roughly 0.80 for the single-well case and 0.85 for the double-well) tracked the analytical predictions well, with the discrepancies attributable to finite-size effects that shrink as agent numbers grow. Intriguingly, the more spatially structured landscape produced higher overall inequality, suggesting that polycentric geographies may concentrate wealth more severely than monocentric ones.
The model has limitations its author is candid about. The mean-field approximation works best when interactions are long-range, and it neglects the kind of correlations between specific agents that network effects can produce. Demographic change (births, deaths, migration by discrete jump rather than continuous drift) isn’t yet in the equations. And the assumption linking individual randomness to social friction, a direct borrowing from the fluctuation-dissipation theorem of equilibrium thermodynamics, is, as the paper acknowledges, “a profound and non-trivial modeling choice” for an inherently non-equilibrium system.
Still, the framework does something that phenomenological models can’t quite manage: it guarantees that the aggregate behaviors it predicts actually follow from specific agent-level rules, rather than being assumed at the macroscopic level and worked backward. Policy interventions that change the effective resource landscape can, in principle, be traced through the equations to their population-level effects. That makes the model testable in ways that purely descriptive accounts of inequality are not. Whether Pareto exponents measured in the real world can be connected to the microscopic parameters Durán-Olivencia identifies, rates of wealth growth, volatility, resource extraction efficiency, is an empirical question that, for the first time, has a theoretical framework specific enough to be asked of it.
The paper was completed, the author notes, while accompanying his son during a hospital stay. Physics, it turns out, is not always done in the physics department.
Frequently Asked Questions
Does this mean wealth inequality is inevitable, just built into the laws of physics?
Not quite, though the model does show that power-law wealth distributions can emerge from very basic physical principles without any intentional design. What drives inequality in this framework is the spatial unevenness of resources combined with how wealth grows proportionally to existing wealth. Changing either of those inputs, redistributing access to resources or altering the volatility and growth rate of wealth accumulation, would shift the distribution. The physics doesn’t say inequality is fixed; it says where it comes from.
Why use particle physics to model people? Don’t humans make choices that molecules don’t?
That’s exactly the critique this approach is designed to answer. The randomness in the Langevin equations isn’t a simplification that ignores human choice; it’s a formal way of representing bounded rationality, the fact that people make decisions under incomplete information with idiosyncratic preferences. The “social temperature” parameter captures how much random variation exists in behavior across a population. Higher temperature means more unpredictable, exploratory behavior; lower temperature means agents hew more closely to the rational optimum, and the framework includes choice by modeling its unpredictability statistically.
What’s the Pareto distribution and why does it keep showing up in wealth data?
The Pareto distribution describes situations where a small fraction of a population holds a disproportionately large share of the total. In wealth data, something close to it appears in virtually every economy ever measured: the top 20% typically hold around 80% of wealth, though the exact numbers vary. This paper offers a mechanistic account: when wealth grows proportionally to existing wealth and that growth is subject to random fluctuations, a power-law tail emerges naturally in the stationary distribution, with the steepness of the tail set by the ratio of growth rate to volatility.
Could this framework actually be used to predict what happens when a city gains or loses a major industry?
In principle, yes, though the model is still at the foundational stage. Because the spatial wealth distribution is derived analytically from the resource landscape, a change to that landscape (a mine closing, a port opening) would shift the effective potential field and, through the equations, alter both where people live and how wealthy they tend to be in each place. Whether the model is precise enough to make quantitative predictions for real geographies would require fitting its parameters to empirical data, which the paper doesn’t yet attempt; that’s the next step.
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