2003/12/12 by Balint Toth, Bálint Tóth, Toth, Balint +3
Materials Science · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Material Dynamics and Properties #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.AP #math.PR
paper · pdf · doi:10.48550/arxiv.math/0312256
69 pages, 3 figures
arxiv created 2003/12/12 · openalex publication_date 2003/12/12 · arxiv updated 2016/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider one-dimensional, locally finite interacting particle systems with two conservation laws which under Eulerian hydrodynamic limit lead to two-by-two systems of conservation laws: \pt ρ+\px Ψ(ρ, u)=0 \pt u+\px Φ(ρ,u)=0, with (ρ,u)∈\cal D⊂\R2, where \cal D is a convex compact polygon in \R2. The system is typically strictly hyperbolic in the interior of \cal D with possible non-hyperbolic degeneracies on the boundary ∂ \cal D. We consider the case of isolated singular (i.e. non hyperbolic) point on the interior of one of the edges of \cal D, call it (ρ0,u0)=(0,0) and assume \cal D⊂\ρ≥0\. This can be achieved by a linear transformation of the conserved quantities. We investigate the propagation of small nonequilibrium perturbations of the steady state of the microscopic interacting particle system, corresponding to the densities (ρ0,u0) of the conserved quantities. We prove that for a very rich class of systems, under proper hydrodynamic limit the propagation of these small perturbations are universally driven by the two-by-two system \ptρ+ \px(ρu)=0 \pt u + \px(ρ+ γu2) =0 where the parameter γ:=\frac12 Φuu(ρ0,u0) (with a proper choice of space and time scale) is the only trace of the microscopic structure. The proof is valid for the cases with γ>1. [truncated]