Bethe ansatz solution of zero-range process with nonuniform stationary state

Area of Science:

• Statistical Mechanics
• Many-Body Physics
• Nonlinear Dynamics

Background:

• The zero-range process is a fundamental model in statistical mechanics.
• Understanding particle dynamics and emergent phenomena in interacting systems is crucial.
• Exact solutions are valuable for validating approximations and uncovering universal behaviors.

Purpose of the Study:

• To find exact solutions for the master equation of a zero-range process with asymmetric dynamics on a ring.
• To investigate the role of interaction strength (parameter q) on particle dynamics.
• To analyze the large-time scaling behavior and identify universality classes.

Main Methods:

• Exact solution using the Bethe ansatz, incorporating stationary weights of particle configurations.
• Generalization of hopping rates to q-numbers [n](q), encompassing noninteracting and interacting cases.
• Analysis of the partition function and generating function for total particle distance.

Main Results:

• Exact eigenfunctions and eigenvalues of the master equation were derived.
• The noninteracting case (q=1) and specific limits (q=0, infinity) corresponding to known models were recovered.
• For interacting cases (q≠1), the generating function exhibits universal scaling behavior characteristic of the Kardar-Parizi-Zhang class.

Conclusions:

• The Bethe ansatz provides an exact method for studying the zero-range process with generalized interactions.
• The model demonstrates a transition to Kardar-Parizi-Zhang universality for interacting particles.
• This work offers insights into the statistical mechanics of driven diffusive systems.