Construction and application of an algebraic dual basis and the Fine-Scale Greens’ Function for computing projections and reconstructing unresolved scales

In this paper, we build on the work of Hughes and Sangalli (2007) dealing with the explicit computation of the Fine-Scale Greens’ function. The original approach chooses a set of functionals associated with a projector to compute the Fine-Scale Greens’ function. The construction of these functionals, however, does not generalise to arbitrary...

A simple QP modification of the OC update to permit treatment of the topology design problem of self-weight

This note communicates a simple modification of the optimality criteria (OC) design update—as found in well-known Matlab implementations of the classical topology design problem—to an update based on a quadratic program (QP) with a single linear constraint. This QP update is a special case of the dual of Falk, which in general accommodates...

Ergodic Theory of Multi-layer Interacting Particle Systems

We consider a class of multi-layer interacting particle systems and characterize the set of ergodic probability measures with finite moments. The main technical tool is duality combined with successful coupling.

Finite-Dimensional Approximation in Dual Domain: with Applications in Opinion Dynamics and Dynamic Programming

This thesis is comprised of two main parts. In the first part of the thesis, we study the nonlinear Fokker-Planck (FP) equation that arises as a mean-field (macroscopic) approximation of the bounded confidence opinion dynamics, where opinions are influenced by environmental noises and opinions of radicals (stubborn individuals). The distribution...

Switching Interacting Particle Systems: Scaling Limits, Uphill Diffusion and Boundary Layer

This paper considers three classes of interacting particle systems on Z: independent random walks, the exclusion process, and the inclusion process. Particles are allowed to switch their jump rate (the rate identifies the type of particle) between 1 (fast particles) and ϵ∈ [0 , 1] (slow particles). The switch between the two jump rates...

Fluctuations for Interacting Particle Systems with Duality

This thesis is concerned with fluctuations of interacting particle systems that<br/>enjoy the property of duality. The main contributions of this work are divided<br/>in two main parts. In the first part we study some of the advantages of looking<br/>at the density fluctuation field through the lenses of orthogonal self-dualities. In<br/>the...

Consistent particle systems and duality

We consider consistent particle systems, which include independent random walkers, the symmetric exclusion and inclusion processes, as well as the dual of the Kipnis-Marchioro-Presutti model. Consistent systems are such that the distribution obtained by first evolving n particles and then removing a particle at random is the same as the one...

Hydrodynamics for the partial exclusion process in random environment

In this paper, we introduce a random environment for the exclusion process in Z^d obtained by assigning a maximal occupancy to each site. This maximal occupancy is allowed to randomly vary among sites, and partial exclusion occurs. Under the assumption of ergodicity under translation and uniform ellipticity of the environment, we...

Exact formulas for two interacting particles and applications in particle systems with duality

We consider two particles performing continuous-time nearest neighbor random walk on Z and interacting with each other when they are at neighboring positions. The interaction is either repulsive (partial exclusion process) or attractive (inclusion process). We provide an exact formula for the Laplace-Fourier transform of the transition...

Symmetric interacting particle systems: Self-duality and hydrodynamics in dynamic random environment

In this thesis, we study scaling and detailed properties of a class of conservative interacting particle systems. In particular, in the first part we derive the hydrodynamic equation for the symmetric exclusion process in presence of dynamic random environment. The second part of the thesis focuses on a detailed property of conservative particle...

Quantitative Boltzmann–Gibbs Principles via Orthogonal Polynomial Duality

We study fluctuation fields of orthogonal polynomials in the context of particle systems with duality. We thereby obtain a systematic orthogonal decomposition of the fluctuation fields of local functions, where the order of every term can be quantified. This implies a quantitative generalization of the Boltzmann–Gibbs principle. In the...

Factorized Duality, Stationary Product Measures and Generating Functions

We find all self-duality functions of the form (Formula presented.)for a class of interacting particle systems. We call these duality functions of simple factorized form. The functions we recover are self-duality functions for interacting particle systems such as zero-range processes, symmetric inclusion and exclusion processes, as well as...