[Paper Review] Second order Boltzmann-Gibbs principle for polynomial functions and applications

Patrícia Gonçalves, Milton Jara+1 more|LillOA (Université de Lille (University Of Lille))|Jul 22, 2015

Stochastic processes and statistical mechanics6 references4 citations

TL;DR

This paper presents a new proof of the second-order Boltzmann-Gibbs principle for polynomial functions in conservative particle systems, eliminating the need for spectral gap inequalities by using a polynomial decomposition of the antisymmetric current. The key contribution is the rigorous derivation of equilibrium fluctuations converging to a trivial process in super-diffusive systems and to the energy solution of the stochastic Burgers equation in weakly asymmetric diffusive systems.

ABSTRACT

In this paper we give a new proof of the second order Boltzmann-Gibbs principle. The proof does not impose the knowledge on the spectral gap inequality for the underlying model and it relies on a proper decomposition of the antisymmetric part of the current of the system in terms of polynomial functions. In addition, we fully derive the convergence of the equilibrium fluctuations towards 1) a trivial process in case of supper-diffusive systems, 2) an Ornstein-Uhlenbeck process or the unique energy solution of the stochastic Burgers 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.

Motivation & Objective

• To provide a new proof of the second-order Boltzmann-Gibbs principle without relying on spectral gap inequalities.
• To extend the applicability of the principle to systems where spectral gap estimates are unknown or inapplicable, such as kinetically constrained exclusion processes and certain zero-range processes.
• To establish the convergence of the equilibrium fluctuation field to either a trivial process (super-diffusive) or the energy solution of the stochastic Burgers equation (weakly asymmetric diffusive).
• To generalize the method to higher-degree polynomial functions beyond quadratic forms.
• To provide a unified framework for analyzing fluctuations in interacting particle systems via polynomial decomposition of current components.

Proposed method

• Decompose the antisymmetric part of the current into polynomial functions of local particle configurations.
• Apply a multiscale analysis based on one-block and two-blocks estimates, adapted to polynomial structures.
• Use Cauchy-Schwarz and Young’s inequality to control error terms arising from local averages and fluctuations.
• Employ exchangeability arguments via particle swaps (e.g., $ au^{x,x+1}$) to rewrite and bound non-local terms.
• Leverage the invariance of the product measure $ u_ ho$ to decouple expectations and control variance via $ u_ ho$-expectations.
• Bound the quadratic variation of the fluctuation field using $L^2$-norms and scaling parameters $n$, $L$, and $t$.

Experimental results

Research questions

• RQ1 Can the second-order Boltzmann-Gibbs principle be proven without assuming a spectral gap inequality for the underlying dynamics?
• RQ2 How do equilibrium fluctuations behave in super-diffusive systems under the new framework?
• RQ3 What is the limiting process for the fluctuation field in weakly asymmetric diffusive systems when the spectral gap is not available?
• RQ4 Can the method be extended to higher-degree polynomial functions beyond quadratic forms?
• RQ5 To what extent can this approach be applied to models like kinetically constrained exclusion processes or zero-range processes where spectral gap is unknown?

Key findings

• The second-order Boltzmann-Gibbs principle is proven without requiring a spectral gap inequality, relying instead on polynomial decomposition of the current.
• For super-diffusive systems, such as the asymmetric simple exclusion process, the density fluctuation field does not evolve on a certain time scale, converging to a trivial process.
• In weakly asymmetric diffusive systems, the fluctuation field converges to the unique energy solution of the stochastic Burgers equation as defined by Gubinelli and Perkowski.
• The convergence is established by showing that all limit points of the fluctuation field are concentrated on energy solutions, leveraging the uniqueness result from GubPer.
• The error bounds in the proof are controlled by $C( ho)rac{tL}{n^{a-1}} orm{v}_{2,n}^2$, ensuring tightness and convergence under appropriate scaling.
• The method applies to models including weakly and partially asymmetric exclusion processes, weakly asymmetric speed change exclusion processes, and Hamiltonian systems with exponential interactions.