A computational study of particle motion on lattices reports a broad inverse-square relationship between system size and a parameter used to track changes in dynamical activity. The analysis follows activity, density and response in exclusion models from one to three dimensions.
The work is a preprint, not a report of an experiment or a study of a real transport system. Its data come from continuous-time Markov exclusion processes on lattices whose sites are either occupied or empty. The models are the symmetric simple exclusion process, or SSEP, the asymmetric version, ASEP, and the totally asymmetric version, TASEP.
A way to follow rare and typical motion
The researchers represented the evolving probability distribution with a variational autoregressive network. In plain terms, the network provides a compact way to describe the many possible arrangements of occupied and empty sites as the system evolves. The calculations varied the counting field, a parameter that reweights trajectories according to how active they are, along with time, system size, boundary rates and forward and backward hopping rates.
The calculation was tested against several smaller or more structured benchmarks, including exact one-dimensional results, tensor-network calculations and exact evolution for selected two- and three-dimensional systems. The reported errors were within 1.2% in the one-dimensional checks, below 10−4 for the two-dimensional benchmark and below 10−5 for the three-dimensional benchmark.
A central computational obstacle is memory. The study used blocked natural-gradient optimization, which keeps the full natural-gradient direction while changing how the calculation is arranged in memory and across computation steps. The benchmarks remain finite-size checks and do not guarantee accuracy beyond the ranges examined.
Activity followed the model’s transport regime
In one-dimensional TASEP, the maps of activity followed the organization of low-density, coexistence and maximal-current regimes. As the system moved toward the maximal-current regime, the active region expanded while the inactive region narrowed. The result links the space-time pattern of activity with the steady-state density phase of the model.
The strongest response to changes in activity depended on both the bulk motion and the boundaries. In diffusive SSEP, low-activity boundaries were associated with a pronounced susceptibility response, while in ballistic TASEP the dominant response appeared at high-activity boundaries. The calculation therefore points to a shift in which part of the system is most responsive as the transport regime changes.
The same broad pattern appeared in two dimensions. Under maximal-current boundary conditions, increasing the bulk drive from SSEP through ASEP to TASEP was associated with a progressively larger active region. The susceptibility response was stronger in low-activity regimes for diffusive SSEP and in high-activity regimes near the ballistic TASEP limit.
A proposed two-dimensional density rule
The study also proposes a way to organize two-dimensional TASEP densities by looking separately at the two lattice directions. When both directions selected the same steady-state density according to the one-dimensional TASEP phase diagram, the two-dimensional bulk density was expected to approach that shared value in the thermodynamic limit. The numerical test supports the criterion in two dimensions, but it does not settle the position, width or finite-size fluctuations of coexistence interfaces.
The scaling results are suggestive rather than a universal law. In two-dimensional SSEP, the reported finite-size and temporal exponents were approximately 0.99 and 0.97, with a common finite-time scaling collapse. In three-dimensional SSEP, the corresponding exponents were approximately 0.60 and 1.18, with the same type of collapse.
Across the dimensions studied, the critical counting field was broadly consistent with an inverse-square dependence on the system’s linear size. The authors interpret this as evidence that a characteristic diffusive length scale may organize the transition-like response. Larger three-dimensional systems would still be needed to determine how much of the pattern survives finite-size corrections.
What remains open
The calculations describe selected finite lattices and accessible trajectory times, so long-time and thermodynamic-limit conclusions depend in part on extrapolation. The reported activity relation is qualitative and does not account fully for correlations, boundary terms, density gradients or finite-counting-field effects. The two-dimensional density criterion has not yet been shown to determine all coexistence-interface properties or to generalize to higher dimensions.
That makes the study most relevant as a computational framework for researchers working on nonequilibrium transport and many-body probability distributions. It offers model-based hypotheses about activity, susceptibility and scaling, rather than evidence about human, biological or real-world transport outcomes.
The document is identified as arXiv version 2, dated 29 September 2026. The authors say supporting data are available within the paper and describe a PyTorch implementation as available on GitHub, but the supplied record retains a placeholder instead of a usable repository link.
The authors report support from Chinese national and provincial science foundations, the Chinese Academy of Sciences and the University of Electronic Science and Technology of China HPC center, with discussion support from Online Club Nanothermodynamica.
Paper data and sources
Original title: Characterizing Full Nonequilibrium Dynamics of Simple Exclusion Processes
Authors: Zhimao Liu, Jing Liu, Pan Zhang, Ying Tang
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text