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Section: New Results

Macroscopic behaviors of large interacting particle systems

A vast amount of physical phenomena were first described at the macroscopic scale, in terms of the classical partial differential equations (PDEs) of mathematical physics. Over the last decades the scientific community has pursued part of its research towards the following universality principle, which is well known in statistical physics: “the qualitative behavior of physical systems depend on the microscopic details of the system only through some large-scale variables”. Typically, the microscopic systems are composed of a huge number of atoms and one looks at a very large time scale with respect to the typical frequency of atom vibrations. Mathematically, this corresponds to a space-time scaling limit procedure.

The macroscopic laws that can arise from microscopic systems can either be partial differential equations (PDEs) or stochastic PDEs (SPDEs) depending on whether one is looking at the convergence to the mean or at the fluctuations around that mean. Therefore, it is a natural problem in the field of interacting particle systems to obtain the macroscopic laws of the relevant thermodynamical quantities, using an underlying microscopic dynamics, namely particles that move according to some prescribed stochastic law. Probabilistically speaking, these systems are continuous time Markov processes.

Anomalous diffusion

First, one can imagine that at the microscopic scale, the population is well modeled by stochastic differential equations (SDEs). Then, the macroscopic description of the population densities is provided by partial differential equations (PDEs), which can be of different types. All these systems may characterize the collective behavior of individuals in biology models, but also agents in economics and finance. In [14] M. Simon in collaboration with C. Olivera has obtained a limit process which belongs to the family of non-local PDEs, and is related to anomalous diffusions. More precisely, they study the asymptotic behavior of a system of particles which interact moderately, i.e. an intermediate situation between weak and strong interaction, and which are submitted to random scattering. They prove a law of large numbers for the empirical density process, which in the macroscopic limit follows a fractional conservation law. The latter is a generalization of convection-diffusion equations, and can appear in physical models (e.g. over-driven detonation in gases [38], or semiconductor growth [55]), but also in areas like hydrodynamics and molecular biology.

Another approach which aims at understanding this abnormally diffusive phenomena is to start from deterministic system of Newtonian particles, and then perturb this system with a stochastic component which will provide enough ergodicity to the dynamics. It is already well known that these stochastic chains model correctly the behavior of the conductivity [35]. In two published papers [18][32], and another submitted one [19], M. Simon with her coauthors C. Bernardin, P. Gonçalves, M. Jara, T. Komorowski, S. Olla and M. Sasada have observed both behaviors, normal and anomalous diffusion, in the context of low dimensional asymmetric systems. They manage to describe the microscopic phenomena at play which are responsible for each one of these phenomena, and they go beyond the predictions that have recently been done in [51], [52].

Towards the weak KPZ universality conjecture

Among the classical SPDEs is the Kardar-Parisi-Zhang (KPZ) equation which has been first introduced more than thirty years ago in [46] as the universal law describing the fluctuations of randomly growing interfaces of one-dimensional stochastic dynamics close to a stationary state (as for example, models of bacterial growth, or fire propagation). In particular, the weak KPZ universality conjecture [52] states that the fluctuations of a large class of one-dimensional microscopic interface growth models are ruled at the macroscopic scale by solutions of the KPZ equation. Thanks to the recent result of M. Jara and P. Gonçalves [45], one has now all in hands to establish that conjecture. In their paper, the authors introduce a new tool, called the second order Boltzmann-Gibbs principle, which permits to replace certain additive functionals of the dynamics by similar functionals given in terms of the density of the particles. In [13], M. Simon in collaboration with P. Gonçalves and M. Jara give a new proof of that principle, which does not impose the knowledge on the spectral gap inequality for the underlying model and relies on a proper decomposition of the antisymmetric part of the current of the system in terms of polynomial functions. In addition, they fully derive the convergence of the equilibrium fluctuations towards (1) a trivial process in case of super-diffusive systems, (2) an Ornstein-Uhlenbeck process or the unique energy solution of the stochastic Burgers equation (SBE) (and its companion, the KPZ equation), in case of weakly asymmetric diffusive systems. Examples and applications are presented for weakly and partial asymmetric exclusion processes, weakly asymmetric speed change exclusion processes and Hamiltonian systems with exponential interactions.

In [30], M. Simon together with P. Gonçalves and N. Perkowski go beyond the weak KPZ universality conjecture to derive a new SPDE, namely, the KPZ equation with boundary conditions, from an interacting particle system in contact with stochastic reservoirs. They legitimate the choice done at the macroscopic level for the KPZ/SBE equation from the microscopic description of the system. For that purpose, they prove two main theorems: first, they extend the notion of energy solutions to the stochastic Burgers equation by adding Dirichlet boundary conditions. Second, they construct a microscopic model (based on weakly asymmetric exclusion processes) from which the energy solution naturally emerges as the macroscopic limit of its stationary density fluctuations. This gives a physical justification for the Dirichlet boundary conditions the SBE equation. They also prove existence and uniqueness of energy solutions to two related SPDEs: the KPZ equation with Neumann boundary conditions and the SHE with Robin boundary conditions, and they rigorously establish the formal links between the equations. This is more subtle than expected, because the boundary conditions do not behave canonically. Finally, they associate an interface growth model to the microscopic model, roughly speaking by integrating it in the space variable, and show that it converges to the energy solution of the KPZ equation, thereby giving a physical justification of the Neumann boundary conditions.