Exact phase diagram for an asymmetric avalanche process

Area of Science:

• Statistical physics
• Complex systems
• Non-equilibrium dynamics

Background:

• Avalanche processes are fundamental in various complex systems, from sandpiles to neural networks.
• Understanding the dynamics of particle flow, especially in asymmetric systems, is crucial for modeling emergent behaviors.
• Previous studies often relied on simulations; exact analytical solutions are less common for such systems.

Purpose of the Study:

• To derive exact results for an asymmetric avalanche process on a ring.
• To investigate the transition in particle flow behavior as a function of particle density and toppling probabilities.
• To determine the critical exponent governing the divergence of average velocity.

Main Methods:

• Utilizing the Bethe ansatz method for exact solutions.
• Employing an iterative procedure based on detailed balance.
• Deriving the average velocity (v) as a function of toppling probabilities and particle density (rho).

Main Results:

• A transition from intermittent to continuous particle flow is observed as rho increases.
• The average velocity (v) diverges at a critical particle density (rho(c)) with exponent alpha.
• The exact phase diagram of the transition is determined.
• The critical exponent alpha is found to be dependent on the specific toppling rules.

Conclusions:

• The study provides an exact analytical solution for an asymmetric avalanche process on a ring.
• The findings elucidate the critical behavior and phase transitions in non-equilibrium systems.
• The dependence of the critical exponent on toppling rules highlights the sensitivity of system dynamics to local interactions.