[Paper Review] Two kinetic models for non-instantaneous binary alignment collisions

Laura Kanzler, Christian Schmeiser+1 more|arXiv (Cornell University)|Mar 29, 2022

Mathematical Biology Tumor Growth4 citations

TL;DR

This paper introduces two novel kinetic models for non-instantaneous binary collisions, where particle interactions evolve continuously over time rather than as instantaneous jumps. The Stochastic Collision Time Model (SCTM) uses a Poisson process to govern collision durations, while the Deterministic Collision Time Model (DCTM) employs an ODE-driven alignment process. The key contribution is the rigorous proof of existence, uniqueness, and long-time behavior of solutions, with the DCTM recovering standard Boltzmann-type alignment models in the instantaneous limit.

ABSTRACT

A new type of kinetic models with non-instantaneous binary collisions is considered. Collisions are described by a transport process in the joint state space of a pair of particles. The interactions are of alignment type, where the states of the particles approach each other. For two spatially homogeneous models with deterministic or stochastic collision times existence and uniqueness of solutions, the long time behavior, and the instantaneous limit are considered, where the latter leads to standard kinetic models of Boltzmann type.

Motivation & Objective

• To develop a new class of kinetic models that replace instantaneous binary collisions with continuous, finite-time processes.
• To model real-world systems—such as living agents, bacteria, or opinion dynamics—where interactions take measurable time.
• To establish mathematical foundations for non-instantaneous collision models, including existence, uniqueness, and long-time behavior.
• To analyze the formal instantaneous limit of the models, showing convergence to classical Boltzmann-type kinetic equations.
• To provide a rigorous framework for alignment-type interactions in spatially homogeneous settings with deterministic or stochastic collision timing.

Proposed method

• Formulates a coupled system of PDEs for free particles $ f $ and colliding pairs $ g $, with transport dynamics in the joint state space.
• Uses a stochastic collision rate $ ho = \gamma $ governed by a Poisson process in the SCTM, modeling random collision durations.
• Employs a deterministic ODE system with velocity field $ v_2 = \frac{\text{sgn}(\varphi - \varphi_*)}{2}(-1,1)^T $ in the DCTM, leading to finite-time alignment.
• Applies a rescaling $ v_2 \to v_2/\varepsilon $, $ g \to \varepsilon g $ to derive the instantaneous limit, recovering a standard Boltzmann-type collision operator.
• Uses coordinate transformations and integral estimates to prove contraction in $ L^1 $, establishing local and global existence of solutions.
• Imposes symmetry and mass conservation constraints to preserve indistinguishability and physical consistency in the solutions.

Experimental results

Research questions

• RQ1 How can non-instantaneous binary collisions be modeled mathematically using continuous processes in the joint state space of particle pairs?
• RQ2 What are the conditions under which solutions to such models exist and are unique?
• RQ3 How do the long-time behaviors of the stochastic and deterministic models compare?
• RQ4 What is the formal instantaneous limit of the non-instantaneous models, and does it recover known Boltzmann-type kinetic equations?
• RQ5 Does the model preserve key physical quantities such as mass and momentum in the long-time regime?

Key findings

• Solutions to both the Stochastic Collision Time Model (SCTM) and the Deterministic Collision Time Model (DCTM) exist locally and globally in time, with uniqueness established via contraction mapping in $ L^1 $.
• The DCTM exhibits exponential decay of variance, with $ \frac{d}{dt}V_f = -\lambda M_{f_0} V_f $, indicating convergence to equilibrium.
• The instantaneous limit of the DCTM yields a standard Boltzmann-type equation with a gain term $ \lambda \int_{\mathbb{R}} f(\frac{\varphi + \varphi_*}{2}, t) f(\varphi_*, t) \, d\varphi_* $, corresponding to midpoint alignment.
• The limiting equation conserves mass and momentum, with $ M_f(t) = M_{f_0} $ and $ I_f(t) = M_{f_0} \varphi_\infty $, and satisfies the weak formulation $ \int Q_2(f,f) h \, d\varphi = 2\lambda \int_{\mathbb{R}^2} f f_* \left( h(\frac{\varphi + \varphi_*}{2}) - \frac{h(\varphi) + h(\varphi_*)}{2} \right) d\varphi_* d\varphi $.
• The DCTM solution satisfies the indistinguishability property $ g(\varphi, \varphi_*, t) = g(\varphi_*, \varphi, t) $ for all $ t > 0 $, preserving symmetry under particle exchange.
• The SCTM model is shown to be well-posed under the same conditions, with existence and uniqueness proven via similar contraction arguments in the $ L^1 $-based functional space.