scholarly journals Nematic alignment of self-propelled particles: From particle to macroscopic dynamics

2020 ◽  
Vol 30 (10) ◽  
pp. 1935-1986 ◽  
Author(s):  
Pierre Degond ◽  
Sara Merino-Aceituno
Keyword(s):  
Local Density ◽  
Particle Density ◽  
Mean Field ◽  
Collision Operator ◽  
Momentum Conservation ◽  
Diffusion System ◽  
Macroscopic Model ◽  
Particle Model ◽  
Cross Diffusion ◽  
Hilbert Expansion

Starting from a particle model describing self-propelled particles interacting through nematic alignment, we derive a macroscopic model for the particle density and mean direction of motion. We first propose a mean-field kinetic model of the particle dynamics. After diffusive rescaling of the kinetic equation, we formally show that the distribution function converges to an equilibrium distribution in particle direction, whose local density and mean direction satisfies a cross-diffusion system. We show that the system is consistent with symmetries typical of a nematic material. The derivation is carried over by means of a Hilbert expansion. It requires the inversion of the linearized collision operator for which we show that the generalized collision invariants, a concept introduced to overcome the lack of momentum conservation of the system, plays a central role. This cross-diffusion system poses many new challenging questions.

scholarly journals CONTINUUM LIMIT OF SELF-DRIVEN PARTICLES WITH ORIENTATION INTERACTION

2008 ◽  
Vol 18 (supp01) ◽  
pp. 1193-1215 ◽  
Author(s):  
PIERRE DEGOND ◽  
SÉBASTIEN MOTSCH
Keyword(s):  
Continuum Limit ◽  
Mean Field ◽  
Collision Operator ◽  
Conservation Equation ◽  
Macroscopic Model ◽  
Mean Velocity ◽  
Fish Schools ◽  
Orientation Interaction ◽  
The Mean ◽  
Animal Societies

The discrete Couzin–Vicsek algorithm (CVA), which describes the interactions of individuals among animal societies such as fish schools is considered. In this paper, a kinetic (mean-field) version of the CVA model is proposed and its formal macroscopic limit is provided. The final macroscopic model involves a conservation equation for the density of the individuals and a non-conservative equation for the director of the mean velocity and is proved to be hyperbolic. The derivation is based on the introduction of a non-conventional concept of a collisional invariant of a collision operator.

Diffusion limit of a Boltzmann–Poisson system with nonlinear equilibrium state

2019 ◽  
Vol 16 (01) ◽  
pp. 131-156
Author(s):  
Lanoir Addala ◽  
Mohamed Lazhar Tayeb
Keyword(s):  
Nonlinear Diffusion ◽  
Collision Operator ◽  
Nonlinear Diffusion Equation ◽  
One Dimension ◽  
Diffusion Limit ◽  
Poisson System ◽  
Nonlinear Relaxation ◽  
Contraction Property ◽  
Hilbert Expansion ◽  
Nonlinear Equilibrium

The diffusion approximation for a Boltzmann–Poisson system is studied. Nonlinear relaxation type collision operator is considered. A relative entropy is used to prove useful [Formula: see text]-estimates for the weak solutions of the scaled Boltzmann equation (coupled to Poisson) and to prove the convergence of the solution toward the solution of a nonlinear diffusion equation coupled to Poisson. In one dimension, a hybrid Hilbert expansion and the contraction property of the operator allow to exhibit a convergence rate.

On a cross-diffusion system arising in image denoising

2018 ◽  
Vol 76 (5) ◽  
pp. 984-996 ◽  
Author(s):  
Gonzalo Galiano ◽  
Julián Velasco
Keyword(s):  
Image Denoising ◽  
Diffusion System ◽  
Cross Diffusion

scholarly journals Stability of patterns and of constant steady states for a cross-diffusion system

2016 ◽  
Vol 293 ◽  
pp. 208-216 ◽  
Author(s):  
G. Svantnerné Sebestyén ◽  
István Faragó ◽  
Róbert Horváth ◽  
R. Kersner ◽  
M. Klincsik
Keyword(s):  
Steady States ◽  
Diffusion System ◽  
Cross Diffusion

scholarly journals Rigorous Derivation of Population Cross-Diffusion Systems from Moderately Interacting Particle Systems

2021 ◽  
Vol 31 (6) ◽  
Author(s):  
Li Chen ◽  
Esther S. Daus ◽  
Alexandra Holzinger ◽  
Ansgar Jüngel
Keyword(s):  
Interacting Particle Systems ◽  
Mean Field ◽  
Strong Solutions ◽  
Particle Systems ◽  
Mean Field Limit ◽  
Diffusion Term ◽  
Small Initial Data ◽  
Dirac Delta ◽  
Cross Diffusion ◽  
Diffusion Systems

AbstractPopulation cross-diffusion systems of Shigesada–Kawasaki–Teramoto type are derived in a mean-field-type limit from stochastic, moderately interacting many-particle systems for multiple population species in the whole space. The diffusion term in the stochastic model depends nonlinearly on the interactions between the individuals, and the drift term is the gradient of the environmental potential. In the first step, the mean-field limit leads to an intermediate nonlocal model. The local cross-diffusion system is derived in the second step in a moderate scaling regime, when the interaction potentials approach the Dirac delta distribution. The global existence of strong solutions to the intermediate and the local diffusion systems is proved for sufficiently small initial data. Furthermore, numerical simulations on the particle level are presented.

scholarly journals Study of charged-particle multiplicity fluctuations in pp collisions with Monte Carlo event generators at the LHC

2020 ◽  
Vol 29 (09) ◽  
pp. 2050074
Author(s):  
E. Shokr ◽  
A. H. El-Farrash ◽  
A. De Roeck ◽  
M. A. Mahmoud
Keyword(s):  
Charged Particle ◽  
Particle Production ◽  
Local Density ◽  
Particle Density ◽  
Hadron Collider ◽  
Center Of Mass ◽  
Monte Carlo Event ◽  
Particle Multiplicity ◽  
Charged Particle Multiplicity ◽  
Proton Proton

Proton–Proton ([Formula: see text]) collisions at the Large Hadron Collider (LHC) are simulated in order to study events with a high local density of charged particles produced in narrow pseudorapidty windows of [Formula: see text] = 0.1, 0.2, and 0.5. The [Formula: see text] collisions are generated at center of mass energies of [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] TeV, i.e., the energies at which the LHC has operated so far, using PYTHIA and HERWIG event generators. We have also studied the average of the maximum charged-particle density versus the event multiplicity for all events, using the different pseudorapidity windows. This study prepares for the multi-particle production background expected in a future search for anomalous high-density multiplicity fluctuations using the LHC data.

scholarly journals REALISTIC MODELING OF STRONGLY CORRELATED ELECTRON SYSTEMS: AN INTRODUCTION TO THE LDA+DMFT APPROACH

2001 ◽  
Vol 15 (19n20) ◽  
pp. 2611-2625 ◽  
Author(s):  
K. HELD ◽  
I. A. NEKRASOV ◽  
N. BLÜMER ◽  
V. I. ANISIMOV ◽  
D. VOLLHARDT
Keyword(s):  
Quantum Monte Carlo ◽  
Local Density ◽  
Mean Field Theory ◽  
Mean Field ◽  
Structure Theory ◽  
Strongly Correlated Electron ◽  
Correlated Electron Systems ◽  
Correlated Electron ◽  
Basic Ideas ◽  
Set Up

The LDA+DMFT approach merges conventional band structure theory in the local density approximation (LDA) with a state-of-the-art many-body technique, the dynamical mean-field theory (DMFT). This new computational scheme has recently become a powerful tool for ab initio investigations of real materials with strong electronic correlations. In this paper an introduction to the basic ideas and the set-up of the LDA+DMFT approach is given. Results for the photoemission spectra of the transition metal oxide La 1-x Sr x TiO 3, obtained by solving the DMFT equations by quantum Monte Carlo (QMC) simulations, are presented and are found to be in very good agreement with experiment. The numerically exact DMFT(QMC) solution is compared with results obtained by two approximative solutions, i.e. the iterative perturbation theory and the non-crossing approximation.

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