5,209 research outputs found
Instabilities in the mean field limit
Consider a system of particles interacting through Newton's second law
with Coulomb interaction potential in one spatial dimension or a
smooth potential in any dimension. We prove that in the mean
field limit , the particles system displays instabilities
in times of order for some configurations approximately distributed
according to unstable homogeneous equilibria.Comment: minor typos corrected; Journal of Statistical Physics, accepte
Mean Field Limit for Coulomb-Type Flows
We establish the mean-field convergence for systems of points evolving along
the gradient flow of their interaction energy when the interaction is the
Coulomb potential or a super-coulombic Riesz potential, for the first time in
arbitrary dimension. The proof is based on a modulated energy method using a
Coulomb or Riesz distance, assumes that the solutions of the limiting equation
are regular enough and exploits a weak-strong stability property for them. The
method can handle the addition of a regular interaction kernel, and applies
also to conservative and mixed flows. In the appendix, it is also adapted to
prove the mean-field convergence of the solutions to Newton's law with Coulomb
or Riesz interaction in the monokinetic case to solutions of an Euler-Poisson
type system.Comment: Final version with expanded introduction, to appear in Duke Math
Journal. 35 page
Mean-field limit of systems with multiplicative noise
A detailed study of the mean-field solution of Langevin equations with
multiplicative noise is presented. Three different regimes depending on
noise-intensity (weak, intermediate, and strong-noise) are identified by
performing a self-consistent calculation on a fully connected lattice. The most
interesting, strong-noise, regime is shown to be intrinsically unstable with
respect to the inclusion of fluctuations, as a Ginzburg criterion shows. On the
other hand, the self-consistent approach is shown to be valid only in the
thermodynamic limit, while for finite systems the critical behavior is found to
be different. In this last case, the self-consistent field itself is broadly
distributed rather than taking a well defined mean value; its fluctuations,
described by an effective zero-dimensional multiplicative noise equation,
govern the critical properties. These findings are obtained analytically for a
fully connected graph, and verified numerically both on fully connected graphs
and on random regular networks. The results presented here shed some doubt on
what is the validity and meaning of a standard mean-field approach in systems
with multiplicative noise in finite dimensions, where each site does not see an
infinite number of neighbors, but a finite one. The implications of all this on
the existence of a finite upper critical dimension for multiplicative noise and
Kardar-Parisi-Zhang problems are briefly discussed.Comment: 9 Pages, 8 Figure
Mean-field limit for the stochastic Vicsek model
We consider the continuous version of the Vicsek model with noise, proposed
as a model for collective behavior of individuals with a fixed speed. We
rigorously derive the kinetic mean-field partial differential equation
satisfied when the number N of particles tends to infinity, quantifying the
convergence of the law of one particle to the solution of the PDE. For this we
adapt a classical coupling argument to the present case in which both the
particle system and the PDE are defined on a surface rather than on the whole
space. As part of the study we give existence and uniqueness results for both
the particle system and the PDE
Stochastic Mean-Field Limit: Non-Lipschitz Forces \& Swarming
We consider general stochastic systems of interacting particles with noise
which are relevant as models for the collective behavior of animals, and
rigorously prove that in the mean-field limit the system is close to the
solution of a kinetic PDE. Our aim is to include models widely studied in the
literature such as the Cucker-Smale model, adding noise to the behavior of
individuals. The difficulty, as compared to the classical case of globally
Lipschitz potentials, is that in several models the interaction potential
between particles is only locally Lipschitz, the local Lipschitz constant
growing to infinity with the size of the region considered. With this in mind,
we present an extension of the classical theory for globally Lipschitz
interactions, which works for only locally Lipschitz ones
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