The goal of this paper is to describe the various kinetic equations which arise from scaling limits of interacting particle systems. We provide a formalism which allows us to determine the kinetic equation for a given interaction potential and scaling limit. Our focus in this paper is on particle systems with long-range interactions. The derivation here is formal, but it provides an interpretation of particle systems as the motion of a particle in a random force field with a friction term which is due to the interaction with the surrounding particles. Some of the technical details of this method are discussed in the companion paper [47].
We propose and investigate general kinetic models %of Boltzmann type with transition probabilities that can describe the simultaneous change of multiple microscopic states of the interacting agents. These models can be applied to many problems in socio-economic sciences, where individuals may change both their compartment and their characteristic kinetic variable, as for instance kinetic models for epidemics or for international trade with possible transfers of agents. Mathematical properties of our kinetic model are proved, as existence and uniqueness of a solution for the Cauchy problem in s…
Weakly interacting quantum fluids allow for a natural kinetic theory description which takes into account the fermionic or bosonic nature of the interacting particles. In the simplest cases, one arrives at the Boltzmann-Nordheim equations for the reduced density matrix of the fluid. We discuss here two related topics: the kinetic theory of the fermionic Hubbard model, in which conservation of total spin results in an additional Vlasov type term in the Boltzmann equation, and the relation between kinetic theory and thermalization.
The kinetic equations describing irreversible aggregation and the scaling approach developed to describe them in the limit of large times and large sizes are tersely reviewed. Next, a system is considered in which aggregates can only react with aggregates of their own size. The existence of a scaling solution of the kinetic equations can then be shown rigorously in the case in which the total mass of the system is conserved. A large number of detailed properties of the solution, previously predicted by qualitative arguments, can be shown rigorously as well in this system. In the case in which …
We show how to "concatenate" variational principles over different bases into one over a single base, thereby providing a unified Lagrangian treatment of interacting systems. As an example we study a Klein-Gordon field interacting with a mesically charged particle. We employ our method to give a novel group-theoretic derivation of the kinetic stress-energy-momentum tensor density corresponding to the particle.
The derivation of the Nordheim-Boltzmann transport equation for weakly interacting quantum fluids is a longstanding problem in mathematical physics. Inspired by the method developed to handle classical dilute gases, a conventional approach is the use of the BBGKY hierarchy for the time-dependent reduced density matrices. In contrast, our contribution is motivated by the kinetic theory of the weakly nonlinear Schrodinger equation. The main observation is that the results obtained in the latter context carry over directly to weakly interacting quantum fluids provided one does not insist on norma…
We develop the idea of non-Markovian CTRW (continuous time random walk) approximation to the evolution of interacting particle systems, which leads to a general class of fractional kinetic measure-valued evolutions with variable order. We prove the well-posedness of the resulting new equations and present a probabilistic formula for their solutions. Though our method are quite general, for simplicity we treat in detail only the fractional versions of the interacting diffusions. The paper can be considered as a development of the ideas from the works of Belavkin and Maslov devoted to Markovian …
In this paper we continue the study of the derivation of different types of kinetic equations which arise from scaling limits of interacting particle systems. We began this study in \cite{NVW}. More precisely, we consider the derivation of the kinetic equations for systems with long range interaction. Particular emphasis is put on the fact that all the kinetic regimes can be obtained approximating the dynamics of interacting particle systems, as well as the dynamics of Rayleigh Gases, by a stochastic Langevin-type dynamics for a single particle. We will present this approximation in detail and…
Levy flights and fractional Brownian motion (fBm) have become exemplars of the heavy tailed jumps and long-ranged memory widely seen in physics. Natural time series frequently combine both effects, and linear fractional stable motion (lfsm) is a model process of this type, combining alpha-stable jumps with a memory kernel. In contrast complex physical spatiotemporal diffusion processes where both the above effects compete have for many years been modelled using the fully fractional (FF) kinetic equation for the continuous time random walk (CTRW), with power laws in the pdfs of both jump size a…
These lecture notes provide the material for a short introductory course on effective equations for classical particle systems. They concern the basic equations in kinetic theory, written by Boltzmann and Landau, describing rarefied gases and weakly interacting plasmas respectively. These equations can be derived formally, under suitable scaling limits, taking classical particle systems as a starting point. A rigorous proof of this limiting procedure is difficult and still largely open. We discuss some mathematical problems arising in this context.
We consider hydrodynamic scaling limits for a class of reversible interacting particle systems, which includes the symmetric simple exclusion process and certain zero-range processes. We study a (non-quadratic) microscopic action functional for these systems. We analyse the behaviour of this functional in the hydrodynamic limit and we establish conditions under which it converges to the (quadratic) action functional of Macroscopic Fluctuation Theory. We discuss the implications of these results for rigorous analysis of hydrodynamic limits.
A mean-field approach (filtering out subgrid scales) is applied to the Boltzmann equation in order to derive a subgrid turbulence model based on kinetic theory. It is demonstrated that the only Smagorinsky type model which survives in the hydrodynamic limit on the viscosity time scale is the so-called tensor-diffusivity model. Scaling of the filter-width with Reynolds number and Knudsen number is established. This sets the first rigorous step in deriving turbulence models from kinetic theory.
Using the renormalization method introduced in \cite{GJ}, we prove what we call the {\em local} Boltzmann-Gibbs principle for conservative, stationary interacting particle systems in dimension $d=1$. As applications of this result, we obtain various scaling limits of additive functionals of particle systems, like the occupation time of a given site or extensive additive fields of the dynamics. As a by-product of these results, we also construct a novel process, related to the stationary solution of the stochastic Burgers equation.
We study a family of conservative interacting particle systems with degenerate rates called noncooperative kinetically constrained lattice gases. We prove for all models in this family the diffusive scaling of the relaxation time, the positivity of the diffusion coefficient, and the positivity of the self-diffusion coefficient.
An accurate non-gradient-expansion based correction to Thomas--Fermi is developed using solvable model. The used model is a system of $N$ non-interacting electrons moving independently in the Coulomb field of the nuclear charge. The presented correction is applicable for atoms and should be extendable beyond that. The method exploits the fact that the difference between the Thomas--Fermi approximation and the non-interacting kinetic energy is comparable to the difference between the same values inside the proposed solvable model. The numerical experiments show that by adding this correction fa…
In self-interacting scalar field theories kinetic expansion is an alternative way of calculating the generating functional for Green's functions where the zeroth order non-Gaussian path integral becomes diagonal in x-space and reduces to the product of an ordinary integral at each point which can be evaluated exactly. We discuss how to deal with such functional integrals and propose a new perturbative expansion scheme which combines the elements of the kinetic expansion with that of usual perturbation theory. It is then shown that, when the cutoff dependent bare parameters in the potential are…
We consider a system of infinitely many interacting Brownian motions that models the height of a one-dimensional interface between two bulk phases. We prove that the large scale fluctuations of the system are well approximated by the solution to the KPZ equation provided the microscopic interaction is weakly asymmetric. The proof is based on the martingale solutions of Goncalves and Jara and the corresponding uniqueness result of Gubinelli and Perkowski.
The scaling of the spatio-temporal response of coarsening systems is studied through simulations of the 2D and 3D Ising model with Glauber dynamics. The scaling functions agree with the prediction of local scale invariance, extending dynamical scaling to a space-time dynamical symmetry.
This paper presents our study of the asymptotic behavior of a two-component system of Brownian motions undergoing certain singular interactions. In particular, the system is a combination of two different types of particles and the mechanical properties and interaction parameters depend on the corresponding type of particles. We prove that the hydrodynamic limit of the empirical densities of two types is the solution of a certain quasi-linear parabolic partial differential equation known as the Maxwell-Stefan equation.
The phase-ordering kinetics of the ferromagnetic two-dimensional Ising model with uniform disorder is investigated by intensive Monte Carlo simulations. Taking into account finite-time corrections to scaling, simple ageing behaviour is observed in the two-time responses and correlators. The dynamical exponent z and the form of the scaling functions only depend on the ratio eps/T, where eps describes the width of the distribution of the disorder. The agreement of the predictions of local scale-invariance generalised to z<> 2 for the two-time scaling functions of response and correlations with t…
