scholarly journals Modeling of a diffusion with aggregation: rigorous derivation and numerical simulation

2018 ◽  
Vol 52 (2) ◽  
pp. 567-593 ◽  
Author(s):  
Li Chen ◽  
Simone Göttlich ◽  
Stephan Knapp
Keyword(s):  
Particle System ◽  
Rigorous Derivation ◽  
Uniform Estimates ◽  
Estimates Of Solutions ◽  
Delta Potential ◽  
Intermediate Particle ◽  
Stochastic Particle System ◽  
Aggregation Equation ◽  
Discretization Schemes ◽  
Theoretical Results

In this paper, a diffusion-aggregation equation with delta potential is introduced. Based on the global existence and uniform estimates of solutions to the diffusion-aggregation equation, we also provide the rigorous derivation from a stochastic particle system while introducing an intermediate particle system with smooth interaction potential. The theoretical results are compared to numerical simulations relying on suitable discretization schemes for the microscopic and macroscopic level. In particular, the regime switch where the analytic theory fails is numerically analyzed very carefully and allows for a better understanding of the equation.

scholarly journals Asymptotic behavior for non-autonomous stochastic plate equation on unbounded domains

2019 ◽  
Vol 17 (1) ◽  
pp. 1281-1302 ◽  
Author(s):  
Xiaobin Yao ◽  
Xilan Liu
Keyword(s):  
Asymptotic Behavior ◽  
Additive Noise ◽  
Upper Semicontinuity ◽  
Unbounded Domains ◽  
Asymptotic Behavior Of Solutions ◽  
Plate Equation ◽  
Uniform Estimates ◽  
Estimates Of Solutions ◽  
Random Attractors

Abstract We study the asymptotic behavior of solutions to the non-autonomous stochastic plate equation driven by additive noise defined on unbounded domains. We first prove the uniform estimates of solutions, and then establish the existence and upper semicontinuity of random attractors.

scholarly journals Derivation of a kinetic model from a stochastic particle system

2008 ◽  
Vol 1 (4) ◽  
pp. 557-572 ◽  
Author(s):  
Pierre Degond ◽  
◽  
Simone Goettlich ◽  
Axel Klar ◽  
Mohammed Seaid ◽  
...  
Keyword(s):  
Kinetic Model ◽  
Particle System ◽  
Stochastic Particle System ◽  
Stochastic Particle

scholarly journals VARIATIONAL FORMULATION OF THE FOKKER–PLANCK EQUATION WITH DECAY: A PARTICLE APPROACH

2013 ◽  
Vol 15 (05) ◽  
pp. 1350017 ◽  
Author(s):  
MARK A. PELETIER ◽  
D. R. MICHIEL RENGER ◽  
MARCO VENERONI
Keyword(s):  
Variational Formulation ◽  
Large Deviation ◽  
Particle System ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Stochastic Particle System ◽  
Deviation Rate ◽  
The Many ◽  
Particle Approach ◽  
Fokker Planck

We introduce a stochastic particle system that corresponds to the Fokker–Planck equation with decay in the many-particle limit, and study its large deviations. We show that the large-deviation rate functional corresponds to an energy-dissipation functional in a Mosco-convergence sense. Moreover, we prove that the resulting functional, which involves entropic terms and the Wasserstein metric, is again a variational formulation for the Fokker–Planck equation with decay.

A FAMILY OF MIMETIC FINITE DIFFERENCE METHODS ON POLYGONAL AND POLYHEDRAL MESHES

2005 ◽  
Vol 15 (10) ◽  
pp. 1533-1551 ◽  
Author(s):  
FRANCO BREZZI ◽  
KONSTANTIN LIPNIKOV ◽  
VALERIA SIMONCINI
Keyword(s):  
Finite Difference ◽  
Material Properties ◽  
Numerical Experiments ◽  
Finite Difference Methods ◽  
Polyhedral Meshes ◽  
Discretization Schemes ◽  
Difference Methods ◽  
Theoretical Results ◽  
Diffusion Problems

A family of inexpensive discretization schemes for diffusion problems on unstructured polygonal and polyhedral meshes is introduced. The material properties are described by a full tensor. The theoretical results are confirmed with numerical experiments.

scholarly journals Spectral theory for the -Boson particle system

2014 ◽  
Vol 151 (1) ◽  
pp. 1-67 ◽  
Author(s):  
Alexei Borodin ◽  
Ivan Corwin ◽  
Leonid Petrov ◽  
Tomohiro Sasamoto
Keyword(s):  
Heat Equation ◽  
Spectral Theory ◽  
Initial Conditions ◽  
Particle System ◽  
Exclusion Process ◽  
Fixed Time ◽  
Bose Gas ◽  
Stochastic Heat Equation ◽  
Stochastic Particle System ◽  
Central Result

AbstractWe develop spectral theory for the generator of the $q$-Boson (stochastic) particle system. Our central result is a Plancherel type isomorphism theorem for this system. This theorem has various implications. It proves the completeness of the Bethe ansatz for the $q$-Boson generator and consequently enables us to solve the Kolmogorov forward and backward equations for general initial data. Owing to a Markov duality with $q$-TASEP ($q$-deformed totally asymmetric simple exclusion process), this leads to moment formulas which characterize the fixed time distribution of $q$-TASEP started from general initial conditions. The theorem also implies the biorthogonality of the left and right eigenfunctions. We consider limits of our $q$-Boson results to a discrete delta Bose gas considered previously by van Diejen, as well as to another discrete delta Bose gas that describes the evolution of moments of the semi-discrete stochastic heat equation (or equivalently, the O’Connell–Yor semi-discrete directed polymer partition function). A further limit takes us to the delta Bose gas which arises in studying moments of the stochastic heat equation/Kardar–Parisi–Zhang equation.

scholarly journals Propagation of chaos for the Vlasov–Poisson–Fokker–Planck equation with a polynomial cut-off

2019 ◽  
Vol 21 (04) ◽  
pp. 1850039 ◽  
Author(s):  
José A. Carrillo ◽  
Young-Pil Choi ◽  
Samir Salem
Keyword(s):  
Particle System ◽  
Planck Equation ◽  
Newtonian Potential ◽  
Empirical Measure ◽  
Propagation Of Chaos ◽  
Interaction Forces ◽  
Fokker Planck Equation ◽  
Stochastic Particle System ◽  
Stochastic Particle ◽  
Fokker Planck

We consider a [Formula: see text]-particle system interacting through the Newtonian potential with a polynomial cut-off in the presence of noise in velocity. We rigorously prove the propagation of chaos for this interacting stochastic particle system. Taking the cut-off like [Formula: see text] with [Formula: see text] in the force, we provide a quantitative error estimate between the empirical measure associated to that [Formula: see text]-particle system and the solutions of the [Formula: see text]-dimensional Vlasov–Poisson–Fokker–Planck (VPFP) system. We also study the propagation of chaos for the Vlasov–Fokker–Planck equation with less singular interaction forces than the Newtonian one.

Sign in / Sign up

Export Citation Format

Share Document