A kinetic theory of socio-economic structure

Each agent carries a position, a momentum and a wealth. Movement follows an underdamped Langevin equation: a deterministic pull towards valuable resources, a friction term, and a noise term. Being second order in position, the model gives agents inertia, so they carry momentum rather than responding instantly to whatever gradient they happen to be standing on.

The noise is not decoration. Its amplitude is tied to the friction through a fluctuation-dissipation relation, which introduces a social temperature: a single parameter measuring how much of an agent’s behaviour is idiosyncratic rather than optimising. Low temperature means agents follow incentives closely; high temperature means they explore. The paper is explicit that importing this relation from thermal physics into a social system is a substantive modelling choice rather than a neutral one.

Wealth evolves separately, with an intrinsic drift representing proportional growth minus fixed costs, an interaction term representing income earned from resources, and a volatility proportional to current wealth. That last choice makes wealth a geometric Brownian motion, and it is the ingredient that ultimately produces the fat tail.

The state of the whole system is a single point in a space whose dimension grows with the number of agents, which is neither tractable nor interesting. The derivation therefore shifts to the empirical measure: a sum of Dirac deltas, one per agent, treated as a density.

Applying Itô’s lemma to that measure produces an exact equation of fluctuating hydrodynamics, a Dean–Kawasaki-type stochastic PDE. Exact matters here: for any finite number of agents it retains the noise that comes from there being finitely many of them, rather than assuming it away. Taking the mean-field limit under propagation of chaos then removes the martingale term and turns interaction sums into integrals, leaving a deterministic Vlasov–Fokker–Planck system.

Nothing is inserted at the population level. Every term in the macroscopic equations traces back to a term in the individual dynamics.

The application fixes the resources in place, letting their density define a static landscape, and then solves for where people end up and how wealthy they are. The tractability comes from a separation of timescales: position and velocity settle far faster than wealth accumulates, since someone moves daily but grows richer over years. That justifies an ansatz in which the spatial part thermalises for any given level of wealth, leaving a stationary Fokker–Planck equation in wealth alone whose coefficients depend on location.

Spatially, the population follows a Boltzmann factor in the effective potential created by the resources, so people concentrate where resources are dense. How sharply is set by the social temperature: high temperature spreads the population out, low temperature produces tight clusters.

The wealth distribution solves to an inverse gamma distribution, whose large-wealth tail is a power law. The Pareto exponent comes out as one minus twice the growth rate over the squared volatility, which ties an economy-wide statistic directly to two microscopic parameters. Location then modulates the whole distribution: where resource income is high it offsets baseline spending and shifts the distribution upwards, and where it is low the distribution shifts down. Spatial and economic inequality are not two findings but two faces of one solution.

The predictions are checked against a direct simulation of the microscopic equations for 100,000 agents on a two-dimensional periodic domain, under two landscapes: a monocentric one with a single resource well, and a polycentric one with two wells of different intensity.

The simulated wealth distributions match the predicted shape closely in both the resource-rich cores and the peripheries, including the shifts in the body of the distribution and not merely the tail. That is the stronger test, since the tail alone would follow from geometric Brownian motion regardless of the spatial theory.

The headline number is the Gini coefficient. Theory predicts roughly 0.885 in the infinite-population limit; the monocentric simulation gives about 0.796 and the polycentric one about 0.854, with the gap attributable to finite-size effects at 100,000 agents. The direction is the result worth noting: making the landscape more structured raised measured inequality. The spatial arrangement of opportunity is not a backdrop to inequality but one of its drivers.

A fitted distribution tells you what a system looks like now. A derivation tells you which assumptions produced it, and therefore which of them would have to change to get a different answer. Here that is concrete: the Pareto exponent is pinned to growth and volatility, and the strength of clustering to the social temperature, so each macroscopic feature has a microscopic dial behind it.

The machinery generalises beyond this problem. Langevin dynamics and the Fokker–Planck equations that follow from them are the same tools we use for forecasting elsewhere, where the quantity of interest is a price rather than a population. That line runs through our work on reconstructing multivariate Langevin equations from data, and on forecasting with a Langevin equation coupled to a neural ordinary differential equation.