Hydrodynamic Limit of Condensing Two-Species Zero Range Processes with Sub-critical Initial Profiles.

Area of Science:

• Statistical Mechanics
• Probability Theory
• Mathematical Physics

Background:

• Two-species condensing zero-range processes (ZRPs) model interacting particles with phase separation tendencies.
• Understanding the large-scale behavior (hydrodynamic limit) of such systems is crucial for statistical physics.
• Previous studies focused on simpler cases, leaving the hydrodynamic limit for two-species ZRPs under specific conditions unaddressed.

Purpose of the Study:

• To rigorously prove the hydrodynamic limit for a two-species condensing ZRP with nearest-neighbor interactions and mean-zero jumps.
• To analyze the system's behavior for sub-critical initial density profiles.
• To investigate the global existence of solutions for the hydrodynamic equation in the species-blind case.

Main Methods:

• Application of H.T. Yau's relative entropy method.
• Analysis relies on the existence of sufficiently regular solutions to the associated hydrodynamic equation.
• Focus on nearest-neighbor interactions and bounded local jump rates.

Main Results:

• The hydrodynamic limit is proven for two-species condensing ZRPs with sub-critical initial profiles.
• Phase separation behavior is confirmed outside the domain of sub-critical densities.
• For the species-blind ZRP, global existence of solutions to the hydrodynamic equation is established.

Conclusions:

• The study provides a rigorous mathematical framework for understanding the macroscopic behavior of two-species ZRPs.
• The results validate the phase separation phenomenon in these systems under specific conditions.
• The global existence of solutions in the species-blind case extends the validity of the hydrodynamic limit to all times.