Irreversible Local Markov Chains with Rapid Convergence towards Equilibrium.

Area of Science:

• Statistical Mechanics
• Computational Physics
• Dynamical Systems

Background:

• Reversible Markov chains like heat bath and Metropolis algorithms are standard for simulating physical models.
• These algorithms can exhibit slow mixing times, limiting simulation efficiency.
• The one-dimensional hard-sphere model is a fundamental system in statistical mechanics.

Purpose of the Study:

• To develop and analyze irreversible local Markov chains for the continuous one-dimensional hard-sphere model.
• To demonstrate that these new chains achieve faster mixing time scales compared to reversible methods.
• To explore the universality classes of mixing times for these irreversible chains.

Main Methods:

• Introduction of irreversible local Markov chains for the hard-sphere model.
• Analysis of mixing time scales and identification of universality classes.
• Connection to lattice-gas models via the symmetric simple exclusion process (SEP).
• Proposal of a 'lifted TASEP' variant for faster irreversible dynamics.

Main Results:

• Irreversible Markov chains exhibit faster mixing times than reversible ones.
• Two distinct universality classes for mixing times were identified, both faster than reversible chains.
• The event-chain algorithm, an infinitesimal limit, belongs to the fastest class.
• The totally asymmetric SEP (TASEP) and a novel lifted TASEP realize these universality classes.

Conclusions:

• Irreversible Markov chains offer significant speedups for simulating hard-sphere models.
• The proposed lifted Markov chains and factorized Metropolis rule generalize to higher dimensions and other interactions.
• This work provides a new algorithmic framework for efficient simulation in statistical physics.