Generalized determinant solution of the discrete-time totally asymmetric exclusion process and zero-range process

Area of Science:

• Statistical Mechanics
• Many-Body Physics
• Probability Theory

Background:

• The totally asymmetric exclusion process (TASEP) models particle movement with exclusion rules.
• Understanding discrete-time dynamics is crucial for complex systems.
• Previous work established determinant expressions for continuous-time TASEP.

Purpose of the Study:

• To derive a determinant expression for conditional probabilities in a discrete-time TASEP with backward-ordered updates.
• To generalize this expression for particles with asynchronous start and end times.
• To apply these findings to solve a nonstationary zero-range process.

Main Methods:

• Analysis of discrete-time particle evolution on an infinite chain.
• Derivation of a determinant formula for simultaneous particle start and finish.
• Proof of generalization for asynchronous particle start and finish times.
• Application of results to a finite-chain zero-range process with open boundaries.

Main Results:

• A determinant expression for conditional probabilities in discrete-time TASEP was obtained.
• This expression mirrors the form found in continuous-time TASEP.
• The determinant expression was successfully generalized for asynchronous particle start and finish times.
• A nonstationary zero-range process was solved using the generalized results.

Conclusions:

• The study provides a powerful determinantal tool for analyzing discrete-time exclusion processes.
• The generalization to asynchronous timing significantly expands the applicability of these methods.
• These findings offer new insights into solving complex, nonstationary particle systems.