Paper: arXiv 2505.15526

Authors: Marco Menale, Giuseppe Toscani

Abstract

We present a possible approach to measuring inequality in a system of coupled Fokker-Planck-type equations that describe the evolution of distribution densities for two populations interacting pairwise due to social and/or economic factors. The macroscopic dynamics of their mean values follow a Lotka-Volterra system of ordinary differential equations. Unlike classical models of wealth and opinion formation, which tend to converge toward a steady-state profile, the oscillatory behavior of these densities only leads to the formation of local equilibria within the Fokker-Planck system. This makes tracking the evolution of most inequality measures challenging. However, an insightful perspective on the problem is obtained by using the coefficient of variation, a simple inequality measure closely linked to the Gini index. Numerical experiments confirm that, despite the system’s oscillatory nature, inequality initially tends to decrease.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical treatment of non-equilibrium social/economic systems by coupling Fokker-Planck kinetic equations with Lotka-Volterra dynamics to study wealth and loan inequality. However, as a purely theoretical and synthetic numerical simulation paper, empirical rigor with real-world financial data is lacking.

Research Flowchart

  flowchart TD
    A[Goal: Measure Inequality in Oscillatory Lotka-Volterra Kinetic Models] --> B[Model Setup: Coupled Fokker-Planck Equations with Pairwise Interactions]
    B --> C[Derive Macroscopic Dynamics: Mean Values Follow Lotka-Volterra ODEs]
    C --> D[Track Evolution: Local Equilibria & Oscillatory Density Profiles]
    D --> E[Apply Metric: Coefficient of Variation / Gini-related Measure]
    E --> F[Numerical Simulations & Experiments]
    F --> G[Key Outcome: Inequality Initially Decreases Despite System Oscillations]

Cited References (30)

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and 22 more
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0 of 30 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.