Interacting Particle Systems and Kinetic Equations

Introduction:

Complex systems and phenomena in the natural and applied sciences are frequently modeled using (stochastically) interacting particle systems at the microscopic level and (non-local) partial differential equations at the macroscopic level. Gaining insight into these models is essential for understanding and controlling such complex behaviors.

This workshop aims to bring together researchers from both areas to encourage collaboration and cross-disciplinary exchange. By combining expertise, we seek to develop new methods and tools to deepen our understanding of complex systems across physics, biology, and the social sciences.

Location and Date: WATN-LT C (G24), 30 July 2025.

Speakers:

·      Ethan Baker (University of Birmingham)

·      Chuyi Chen (University of Birmingham)

·      Zihui He (Bielefeld University)

·      Wenxuan Tao (University of Birmingham)

·      Peiyuan Zhao (University of Birmingham)

Schedule:

13:00-13:30: Ethan Baker, White noise and Newtonian limits for the Generalised Relativistic Langevin Equation

13:30-14:00: Peiyuan Zhao, Structure-preserving neural networks for GENERIC formulation

14:00-14:30: coffee break

14:30-15:00: Chuyi Chen, Model Reduction of Stochastic Differential Equations

15:00-15:30: Zihui He, A variational approach to a fuzzy Boltzmann equation

15:30-16:00: Wenxuan Tao, Ergodicity and asymptotic limits for Langevin interacting systems with singular forces and multiplicative noise

Title and Abstract

Ethan Baker (University of Birmingham)

Title: White noise and Newtonian limits for the Generalised Relativistic Langevin Equation

Abstract: The Langevin dynamics models the motion of a particle obeying Newton’s second law subject to a random noise, typically caused by collisions with surrounding particles. The classical underdamped and overdamped Langevin equations model such motion as a Markov process with white noise, whereas the generalised Langevin equation considers the memory of the process and a stationary Gaussian noise. Recently, relativistic variants of the underdamped Langevin equation and generalised Langevin equation have been considered in the literature, where we consider the motion of particles with relativistic kinetic energy as opposed to classical kinetic energy. As with the generalised Langevin equation, given the memory kernel is a finite sum of exponentials, the process described by the generalised relativistic Langevin equation has the Markov property. This formulation can be used to derive both the generalised Langevin equation and relativistic Langevin equation. In this presentation, we state and prove these derivations via a white noise and Newtonian limit for the generalised Langevin equation respectively. We also describe recent developments in deriving the Langevin equations from microscopic Newtonian and relativistic models, and from each other.

Chuyi Chen (University of Birmingham):

Title: Model Reduction of Stochastic Differential Equations

Abstract: We investigate the coarse-graining of linear stochastic differential equations, particularly focusing on Ornstein–Uhlenbeck processes. For the one-dimensional case, we derive an explicit expression for the stationary covariance matrix and analyze the convergence rates of both the exact and effective dynamics. We employ conditional expectation and invariant manifold methods for model reduction. To quantify long-time behavior, we use Wasserstein distance and relative entropy to measure convergence to equilibrium. Our results provide a comparative analysis of the original, effective, and reduced dynamics, supported by theoretical estimates and numerical results.

For a Two-State Quantum System, We present a reduction framework for multivariate geometric Brownian motions (GBMs), widely used in finance, biology, and physics. Our method combines invariant manifold techniques with adiabatic elimination to derive closed-form reduced equations for the deterministic part, and uses an extended fluctuation-dissipation theorem to handle the stochastic part. We also use numerical results to illustrate the differences between the original and reduced dynamics.

Zihui He (Bielefeld University)

Title: A variational approach to a fuzzy Boltzmann equation

Abstract: We study a fuzzy Boltzmann equation where the particles interact via delocalised collisions. We discuss the existence and uniqueness of solutions, as well as their convergence to the solutions of classical Boltzmann equations. We also provide a variational characterisation, casting the fuzzy Boltzmann equation into the framework of GENERIC (General Equations for Non-Equilibrium Reversible-Irreversible Coupling) systems. The grazing limit of fuzzy Boltzmann equations is also discussed.

This talk is based on joint work with Matthias Erbar (Bielefeld).

Peiyuan Zhao (University of Birmingham)

Title: Structure-preserving neural networks for GENERIC formulation:

Abstract:  GENERIC (general equation for the nonequilibrium reversible-irreversible coupling) formulation is an efficient way of formulating the dynamics of mixed reversible and irreversible systems, which was given by a time evolution equation and supplemented by two degeneracy conditions (conservation of total energy and non-negative entropy production rate). There has been significant amount of work put into numerical integration of reversible Hamiltonian dynamics in the past few decades, and more recently there were also some success in the irreversible dynamic. However, there were very few attempts on data-driven approach that aim to fulfilling the degeneracy conditions.  In this study we will be focusing on methods to study the GENERIC formulation by neural networks while fulfilling  the physical properties, leading to a more realistic and  hence more accurate prediction of an unknown system.

Wenxuan Tao (University of Birmingham): 

Title: Ergodicity and asymptotic limits for Langevin interacting systems with singular forces and multiplicative noise

Abstract: This talk addresses two classes of Langevin systems with singular interaction potentials driven by state-dependent multiplicative white noise. In the first case, the kinetic energy takes the classical form 1/2mv^2, and the friction—bounded and position-dependent—varies with the displacement x. We prove that the system's solution converges exponentially fast to the invariant Gibbs measure. Furthermore, we analyze the small-mass limit (m→0) of the displacement and the limit is given by a SDE with a noise-induced diffusion term.

In the second case, the kinetic energy is described by the relativistic form c\sqrt{m^2c^2+p^2}. We investigate a physically motivated, unbounded friction matrix that depends on the momentum. For this system, we establish convergence to equilibrium at a polynomial rate. Additionally, we derive the Newtonian limit as c→∞.