Dynamical simulations of classical stochastic systems using matrix product states

Area of Science:

• Statistical Physics
• Computational Physics
• Complex Systems

Background:

• The time-evolving block decimation (TEBD) algorithm is a powerful tool for simulating one-dimensional quantum systems.
• Simulating nonequilibrium stochastic systems presents significant computational challenges.

Purpose of the Study:

• To adapt the TEBD algorithm for simulating the time evolution of nonequilibrium stochastic systems.
• To evaluate the efficacy of matrix product states (MPS) in representing probability distributions for stochastic systems.

Main Methods:

• Representing a system's probability distribution using a finite-dimension matrix product state (MPS).
• Employing TEBD for efficient time evolution simulation via repeated updates and refactorization of the MPS.
• Comparing non-negative matrix factorization (NMF) and singular value decomposition (SVD) for MPS representation accuracy.

Main Results:

• Singular value decomposition (SVD) demonstrates substantially greater accuracy than current non-negative matrix factorization (NMF) algorithms.
• TEBD accurately reproduces the steady state of the totally asymmetric simple exclusion process (TASEP) for large systems.
• Negligible errors in expectation values are achievable even with highly compressed MPS representations.

Conclusions:

• The adapted TEBD algorithm provides an accurate and efficient method for simulating nonequilibrium stochastic systems.
• MPS, particularly when factorized using SVD, offers a robust approach for approximating probability distributions in stochastic dynamics.
• The method shows promise for studying complex systems dynamics where traditional methods face limitations.