Continuum Approximations to Systems of Correlated Interacting Particles

Berlyand, Leonid; Creese, Robert C.; Jabin, Pierre‐Emmanuel; Potomkin, Mykhailo

doi:10.1007/s10955-018-2205-8

Cited by 8 publications

(10 citation statements)

References 43 publications

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“…Beyond the mean-field limit N → ∞, it would be interesting to quantify the fluctuation about the mean-field, for instance through a large deviation approach (see e.g. [6,7,11,30,46,67]).…”

Section: Discussionmentioning

confidence: 99%

Bulk topological states in a new collective dynamics model

Degond,

Diez,

2021

Preprint

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In this paper, we demonstrate the existence of topological states in a new collective dynamics model. This individual-based model (IBM) describes self-propelled rigid bodies moving with constant speed and adjusting their rigid-body attitude to that of their neighbors. In previous works, a macroscopic model has been derived from this IBM in a suitable scaling limit. In the present work, we exhibit explicit solutions of the macroscopic model characterized by a non-trivial topology. We show that these solutions are well approximated by the IBM during a certain time but then the IBM transitions towards topologically trivial states. Using a set of appropriately defined topological indicators, we reveal that the breakage of the non-trivial topology requires the system to go through a phase of maximal disorder. We also show that similar but topologically trivial initial conditions result in markedly different dynamics, suggesting that topology plays a key role in the dynamics of this system.

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Section: Discussionmentioning

confidence: 99%

Bulk topological states in a new collective dynamics model

Degond,

Diez,

2021

Preprint

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“…Note that for simplicity, here, we have implicitly assumed that Equation (11) has a strong solution: it may require precise assumptions on the coefficients b and σ (typically Lipschitz assumptions as in Proposition 1 below). This will also be the case for Equation (12) below. However we stress out that these assumptions are not satisfied in many important cases.…”

Section: Mckean-vlasov Diffusion Let Be Given Two Functionsmentioning

confidence: 83%

Propagation of chaos: A review of models, methods and applications. I. Models and methods

Chaintron

Diez

2022

KRM

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The notion of propagation of chaos for large systems of interacting particles originates in statistical physics and has recently become a central notion in many areas of applied mathematics. The present review describes old and new methods as well as several important results in the field. The models considered include the McKean-Vlasov diffusion, the mean-field jump models and the Boltzmann models. The first part of this review is an introduction to modelling aspects of stochastic particle systems and to the notion of propagation of chaos. The second part presents concrete applications and a more detailed study of some of the important models in the field.

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“…is the conditional distribution of s 3 and w 13 given s 1 . As argued for a similar class of particle systems (without the network weights) in [9,8], η 2 t seems to be unnecessary for a suitable approximation of the interaction integrals, since it only describes the relation of the two particles not interacting with each other at this instance. Hence their closure simplifies to…”

Section: Pair Closuresmentioning

confidence: 99%

Kinetic equations for processes on co-evolving networks

Burger¹

2022

KRM

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The aim of this paper is to derive macroscopic equations for processes on large co-evolving networks, examples being opinion polarization with the emergence of filter bubbles or other social processes such as norm development. This leads to processes on graphs (or networks), where both the states of particles in nodes as well as the weights between them are updated in time. In our derivation we follow the basic paradigm of statistical mechanics: We start from paradigmatic microscopic models and derive a Liouville-type equation in a high-dimensional space including not only the node states in the network (corresponding to positions in mechanics), but also the edge weights between them. We then derive a natural (finite) marginal hierarchy and pass to an infinite limit.We will discuss the closure problem for this hierarchy and see that a simple mean-field solution can only arise if the weight distributions between nodes of equal states are concentrated. In a more interesting general case we propose a suitable closure at the level of a two-particle distribution (including the weight between them) and discuss some properties of the arising kinetic equations. Moreover, we highlight some structure-preserving properties of this closure and discuss its analysis in a minimal model. We discuss the application of our theory to some agent-based models in literature and discuss some open mathematical issues.

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Continuum Approximations to Systems of Correlated Interacting Particles

Cited by 8 publications

References 43 publications

Bulk topological states in a new collective dynamics model

Bulk topological states in a new collective dynamics model

Propagation of chaos: A review of models, methods and applications. I. Models and methods

Kinetic equations for processes on co-evolving networks

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