Research

https://doi.org/10.1088/1742-5468/ae5c8d

Bridging the gap between individual agent behavior and macroscopic societal patterns is a central challenge in the social sciences. In this work, we propose a solution to this problem via a kinetic theory formulation. We demonstrate that complex, empirically-observed phenomena, such as the concentration of populations in cities and the emergence of power-law wealth distributions, can be derived directly from a microscopic model of agents governed by underdamped Langevin dynamics. Our multi-scale derivation yields the exact mesoscopic fluctuating (Dean–Kawasaki) dynamics and the macroscopic Vlasov–Fokker–Planck system of equations. The analytical solution of this system reveals how a heterogeneous resource landscape alone is sufficient to generate the coupled structures of spatial and economic inequality, thus providing a formal link between micro-level stochasticity and macro-level deterministic order.

https://doi.org/10.1186/s12862-025-02485-6

Models of evolutionary dynamics have long been dominated by a paradigm of gradualism, yet the fossil record consistently points to a history defined by punctuation. This disconnect between theory and data has left major macroevolutionary events, such as punctuated equilibria, explosive radiations, and mass extinctions, without a unified first-principles explanation. We argue that this gap stems from a subtle ubiquitous assumption in theoretical models: that the underlying fitness landscapes are mathematically smooth (Lipschitz continuous). Our framework provides a new generative engine for macroevolutionary theory. It makes specific quantitative predictions for paleontological patterns, including the decoupling of disparity and diversity during adaptive radiations, and for the genomic signatures of lineages that undergo rapid evolution. By replacing the assumption of smoothness with non-Lipschitz continuity, we offer a rigorous mathematical reconciliation between gradualist models and the punctuated nature of the fossil record.

https://doi.org/10.1103/zncf-n4y3

Obtaining an accurate description of complex systems is challenging, particularly when their elements exhibit intricate interactions. We propose a data-driven multivariate Langevin equation (LE) to approximate real-world complex systems observables. By reconstructing the drift and diffusion terms of the LE through a nonparametric technique following the definition of the Kramers-Moyal coefficients, our approach unravels the main features of a complex system without requiring a priori knowledge about the underlying governing mechanisms. We illustrate our framework adaptability, reliability, and capability to extract pertinent information through three case studies. First, we benchmark the framework with a simple prototypical example from mechanics, a particle confined by a bistable potential energy well. We then turn to two more involved examples from financial markets, the electricity day-ahead prices and currency-exchange rates, where the nonparametric multivariate LE has not previously been applied. In all cases, our framework accurately identifies the equilibrium values, metastabilty regions, and distinct diffusion behaviors, in a functional agnostic manner, as opposed to price-equation models that require specific domain knowledge.

https://doi.org/10.1063/5.0189402

Accurate prediction of electricity day-ahead prices is essential in competitive electricity markets. Although stationary electricity-price forecasting techniques have received considerable attention, research on non-stationary methods is comparatively scarce, despite the common prevalence of non-stationary features in electricity markets. Specifically, existing non-stationary techniques will often aim to address individual non-stationary features in isolation, leaving aside the exploration of concurrent multiple non-stationary effects. Our overarching objective here is the formulation of a framework to systematically model and forecast non-stationary electricity-price time series, encompassing the broader scope of non-stationary behavior. For this purpose, we develop a data-driven model that combines an N-dimensional Langevin equation (LE) with a neural-ordinary differential equation (NODE). The LE captures fine-grained details of the electricity-price behavior in stationary regimes but is inadequate for non-stationary conditions. To overcome this inherent limitation, we adopt a NODE approach to learn, and at the same time predict, the difference between the actual electricity-price time series and the simulated price trajectories generated by the LE. By learning this difference, the NODE reconstructs the non-stationary components of the time series that the LE is not able to capture. We exemplify the effectiveness of our framework using the Spanish electricity day-ahead market as a prototypical case study. Our findings reveal that the NODE nicely complements the LE, providing a comprehensive strategy to tackle both stationary and non-stationary electricity-price behavior. The framework’s dependability and robustness is demonstrated through different non-stationary scenarios by comparing it against a range of basic naïve methods.

https://doi.org/10.1063/5.0146920

The swift progression and expansion of machine learning (ML) have not gone unnoticed within the realm of statistical mechanics. In particular, ML techniques have attracted attention by the classical density-functional theory (DFT) community, as they enable automatic discovery of free-energy functionals to determine the equilibrium-density profile of a many-particle system. Within classical DFT, the external potential accounts for the interaction of the many-particle system with an external field, thus, affecting the density distribution. In this context, we introduce a statistical-learning framework to infer the external potential exerted on a classical many-particle system. We combine a Bayesian inference approach with the classical DFT apparatus to reconstruct the external potential, yielding a probabilistic description of the external-potential functional form with inherent uncertainty quantification. Our framework is exemplified with a grand-canonical one-dimensional classical particle ensemble with excluded volume interactions in a confined geometry. The required training dataset is generated using a Monte Carlo (MC) simulation where the external potential is applied to the grand-canonical ensemble. The resulting particle coordinates from the MC simulation are fed into the learning framework to uncover the external potential. This eventually allows us to characterize the equilibrium density profile of the system by using the tools of DFT. Our approach benchmarks the inferred density against the exact one calculated through the DFT formulation with the true external potential. The proposed Bayesian procedure accurately infers the external potential and the density profile. We also highlight the external-potential uncertainty quantification conditioned on the amount of available simulated data. The seemingly simple case study introduced in this work might serve as a prototype for studying a wide variety of applications, including adsorption, wetting, and capillarity, to name a few.