The mean-field limit for particle systems with uniform full-rank constraints

Steffen Plunder; Bernd Simeon

doi:10.3934/krm.2023012

Article Contents

2023, Volume 16, Issue 6: 884-912. Doi: 10.3934/krm.2023012

This issue Previous Article Solutions of the 2D radially symmetric Vlasov-Maxwell system with an initial focusing phase Next Article Hybrid kinetic/fluid numerical method for the Vlasov-BGK equation in the diffusive scaling

The mean-field limit for particle systems with uniform full-rank constraints

Steffen Plunder^1, , and
Bernd Simeon^2,

1.
Institute for the Advanced Study of Human Biology (ASHBi), Kyoto University Institute for Advanced Study, Kyoto University, Kyoto 606-8315, Japan
2.
Department of Mathematics, RPTU Kaiserslautern, Paul-Ehrlich Straße 31, 67663 Kaiserslautern, Germany

^* Corresponding author: Steffen Plunder
^* Corresponding author: Steffen Plunder

Received: March 2022

Revised: February 2023

Early access: March 2023

Published: December 2023

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Abstract

We consider a particle system with uniform coupling between a macroscopic component and individual particles. The constraint for each particle is of full rank, which implies that each movement of the macroscopic component leads to a movement of all particles and vice versa. Skeletal muscle tissues share a similar property which motivates this work.

We prove convergence of the mean-field limit, well-posedness and a stability estimate for the mean-field PDE. This work generalises our previous results from [25] to the case of nonlinear constraints.

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Mathematics Subject Classification: Primary: 70F45; Secondary: 34A09.

Citation:

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Figure 1. Skeletal muscle tissue contains arrays of aligned actin-myosin filaments. The myosin heads (red) attach to the actin filament (purple). Millions of such actin-myosin filaments repeatedly perform a cycle of "attaching, pulling and detaching". The resulting accumulated force leads to muscle contraction. In article, we ignore the repeated cycling. Instead, we focus on the mathematical implication of the coupling to the macroscopic component

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Figure 2. Example for the deformation map and its action on cross-bridges

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Figure 3. Simplified model for a straight muscle fiber (orange) with attached cross-bridges (purple) inside

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