Phase transitions in Schloegl's second model for autocatalysis on a Bethe lattice

Area of Science:

• Statistical Mechanics
• Complex Systems Modeling

Background:

• Schloegl's second model, or quadratic contact process, involves particle annihilation and autocatalytic creation.
• Analyzing stochastic models on infinite lattices often relies on finite simulations, but boundary effects complicate Bethe lattices.

Purpose of the Study:

• To analyze Schloegl's quadratic contact process on a Bethe lattice.
• To investigate methods for predicting infinite lattice behavior despite boundary effects on Bethe lattices.

Main Methods:

• Analysis of Schloegl's model on a Bethe lattice with coordination number z=3.
• Exploration of various boundary conditions and unconventional simulation ensembles.
• Prediction of behavior for infinite lattice size.

Main Results:

• The model exhibits a discontinuous transition to the vacuum state.
• A threshold annihilation rate (p) of approximately 0.053 was identified for z=3.
• Boundary effects on Bethe lattices can induce spatially heterogeneous states.

Conclusions:

• The study predicts a critical annihilation rate for the transition to a vacuum state on an infinite Bethe lattice.
• Unconventional simulation approaches are necessary to overcome boundary effects on Bethe lattices.
• The findings provide insights into phase transitions in complex systems.