[Paper Review] Hydrodynamic limit for particle systems with degenerate rates without exclusive constraints

Makiko Sasada|arXiv (Cornell University)|Aug 23, 2010

Theoretical and Computational Physics3 references3 citations

TL;DR

This paper establishes the hydrodynamic limit for conservative particle systems with degenerate jump rates on a d-dimensional discrete torus, without exclusive constraints on particle numbers per site. Using the relative entropy method, it proves that the macroscopic density profile evolves according to a modified porous medium equation (MPME), where diffusion vanishes as density approaches zero, under smooth initial profiles bounded away from zero.

ABSTRACT

We consider the hydrodynamic behavior of some conservative particle systems with degenerate jump rates without exclusive constraints. More precisely, we study the particle systems without restrictions on the total number of particles per site with nearest neighbor exchange rates which vanish for certain configurations. Due to the degeneracy of the rates, there exists blocked configurations which do not evolve under the dynamics and all of the hyperplanes of configurations with a fixed number particles can be decomposed into different irreducible sets. We show that, for initial profiles smooth enough and bounded away from zero, the macroscopic density profile evolves under the diffusive time scaling according to a nonlinear diffusion equation (which we call the modified porous medium equation). The proof is based on the Relative Entropy method but it cannot be straightforwardly applied because of the degeneracy.

Motivation & Objective

• To derive the hydrodynamic limit for conservative particle systems with degenerate jump rates on a d-dimensional torus without exclusion constraints on particle numbers per site.
• To show that the macroscopic density profile evolves according to a modified porous medium equation (MPME) under diffusive time scaling.
• To extend previous results on the porous medium equation to systems without particle number restrictions per site, addressing degeneracy in transition rates.
• To establish the validity of the relative entropy method in the presence of degenerate rates, where standard assumptions fail.

Proposed method

• Models the system as a continuous-time Markov process on the discrete torus $\mathbb{T}^d_N$ with state space $\mathbb{N}^{\mathbb{T}^d_N}$, where $\eta(x)$ denotes the number of particles at site $x$.
• Defines the generator $L_N$ with degenerate rates $c(x,y,\eta)g(\eta(x))$, where $g(k) = 0$ iff $k = 0$, and $c(x,y,\eta)$ depends on local configuration and nearest-neighbor structure.
• Uses the relative entropy method to prove the hydrodynamic limit, adapting it to handle degeneracy by introducing a one-block estimate and a block-fitting procedure.
• Applies a logarithmic Sobolev-type inequality and exponential moment bounds via a coupling with grand canonical Gibbs measures $\nu_{\rho}$ to control entropy production.
• Employs a large-deviation principle for the empirical measure under $\nu_{\rho}$, with rate function $J_\beta(\lambda)$, to control the entropy relative to the local equilibrium measure.
• Introduces a truncation procedure via local averages $\eta^l(x)$ and uses a localization argument to control the entropy production in the presence of degeneracy.

Experimental results

Research questions

• RQ1 Can the hydrodynamic limit be established for conservative particle systems with degenerate jump rates when there are no constraints on the number of particles per site?
• RQ2 Does the macroscopic density profile still evolve according to a nonlinear diffusion equation in the absence of exclusion constraints?
• RQ3 How can the relative entropy method be adapted to handle degenerate rates where the generator is not uniformly elliptic?
• RQ4 What conditions on the rate function $g$ are necessary to ensure convergence to the modified porous medium equation?

Key findings

• The hydrodynamic limit holds for initial profiles that are smooth and bounded away from zero, leading to a macroscopic evolution governed by the modified porous medium equation (MPME).
• The MPME is given by $\partial_t \rho = \Delta(\Phi(\rho)^m)$, where $m \in \mathbb{N} \setminus \{0,1\}$, and $\Phi(\rho)$ is a smooth, strictly increasing function with $\Phi(0) = 0$.
• The diffusion coefficient $D(\rho) = m\Phi(\rho)^{m-1}\Phi'(\rho)$ vanishes as $\rho \to 0$, causing the equation to lose parabolicity in the degenerate regime.
• The relative entropy method is successfully adapted to degenerate systems by introducing a one-block estimate and a localization argument that controls entropy production via exponential moment bounds.
• A key technical condition $(G)$ on $g$ is required, which depends on $m$, and ensures the necessary moment bounds for the entropy estimate.
• The proof establishes that the entropy production vanishes in the hydrodynamic limit, with the limiting profile satisfying the MPME for all $t \in [0,T]$.