Journal of Statistical Physics

, Volume 165, Issue 1, pp 115–125 | Cite as

Coexistence of Shocks and Rarefaction Fans: Complex Phase Diagram of a Simple Hyperbolic Particle System

  • Márton Balázs
  • Attila László Nagy
  • Bálint Tóth
  • István Tóth


This paper investigates the non-equilibrium hydrodynamic behavior of a simple totally asymmetric interacting particle system of particles, antiparticles and holes on \(\mathbb {Z}\). Rigorous hydrodynamic results apply to our model with a hydrodynamic flux that is exactly calculated and shown to change convexity in some region of the model parameters. We then characterize the entropy solutions of the hydrodynamic equation with step initial condition in this scenario which include various mixtures of rarefaction fans and shock waves. We highlight how the phase diagram of the model changes by varying the model parameters.


Asymmetric particle system Hydrodynamic equation Non-equilibrium behavior Non-convex flux Riemann problem Shock Rarefaction fan 



Márton Balázs and Bálint Tóth acknowledge support from the Grants OTKA K100473 and OTKA K109684. B. T. also thanks support from the Leverhulme International Network Grant “Laplacians, Random Walks and Quantum Spin Systems”. We are indebted to an anonymous referee whose detailed and helpful comments greatly improved the exposition of the paper.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Márton Balázs
    • 1
  • Attila László Nagy
    • 2
  • Bálint Tóth
    • 1
    • 3
  • István Tóth
    • 2
  1. 1.School of MathematicsUniversity of BristolBristolUK
  2. 2.Institute of MathematicsBudapest University of Technology and EconomicsBudapestHungary
  3. 3.MTA-BME Stochastics Research Group and Rényi Institute BudapestBudapestHungary

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