An arXiv preprint reports that one computational framework can follow dynamical changes in simple exclusion models across one, two and three dimensions. Its clearest cross-dimensional signal concerns the transition point: the critical counting field, written as s_c, broadly followed an inverse-square relation with system size, with the reported value falling as the lattice length grew. That is numerical support for a shared pattern, not evidence that s_c is an exact or universal law.
One model, several dimensions
The models are SSEP, ASEP and TASEP, three versions of the simple exclusion process, within the same cross-dimensional setup. The method uses a variational autoregressive network, or VAN, to calculate a variational free energy and the scaled cumulant generating function, or SCGF. The SCGF summarizes the statistics of trajectories after they are reweighted by a counting field. Parameters at successive time slices are optimized with blocked natural-gradient descent. The pipeline is applied to both finite-time and long-time dynamics.
The one-dimensional map
In one dimension, the finite-time activity maps directly matched the classical three-phase organization of TASEP steady states. Low-density, coexistence and maximal-current boundary regimes appeared as distinct activity landscapes in the computed results. For a general reader, activity here means the amount of hopping counted along the model's trajectories.
The response to a change in the counting field was not equally prominent in every regime. In the plotted 1D SSEP cases, low-activity boundaries showed the pronounced susceptibility peak. In TASEP, the dominant peak appeared at high-activity boundaries instead. The authors describe this as a qualitative pattern for the parameter choices shown, so it should be read as a map of the tested regimes rather than a universal rule.
Direction and bulk driving
The two-dimensional results extended the directional logic of the one-dimensional model. In 2D TASEP, when the densities selected by 1D TASEP matched along both directions, the analysis associated that match with convergence of the bulk density towards a shared value in the thermodynamic limit. The criterion organizes the bulk sectors, but it does not determine every finite-size boundary or interface feature.
The two-dimensional activity and susceptibility picture varied with bulk asymmetry as well. Stronger driving was associated with smoother density profiles near half filling and with a shift in the dominant susceptibility response towards high-activity boundary conditions. In the reported maps, activity regions and susceptibility patterns differed across the tested bulk and boundary choices. Because the comparison used finite lattices and transient density snapshots, the result describes computed regime patterns rather than a complete account of all finite-size behavior.
A scaling signal, tested against benchmarks
Scaling fits in 2D SSEP gave a finite-size exponent alpha of about 0.99 and a temporal exponent beta of about 0.97, with rescaled transition lines collapsing. In 3D SSEP, the reported values were about 0.53 for finite-size scaling and 1.18 for temporal scaling, and the scaled curves collapsed within the accessible ranges. The 3D analysis carried stronger finite-time and finite-size corrections, so its exponents are less settled than the compact collapse might suggest.
Several benchmark comparisons tested how closely the VAN calculation tracked other methods. In 1D TASEP, its maximum relative deviation from exact phase-dependent values was about 8% for mean density and 7% for mean current. In the finite-time 1D SSEP benchmark, beta was about 0.92 and the long-time size exponent alpha about 1.99, with L=20 and L=40 data collapsing onto a common curve. For steady-state 2D SSEP, the VAN reproduced the SCGF, activity, susceptibility peak and critical-field scaling; its fitted alpha of about 1.02 was close to a tensor-network estimate of about 0.92.
An additional exact check followed the finite-time partition function, the quantity that sums the tilted trajectory weights. The reported relative error was on the order of 10^-4 for the tested 2D systems at L=3 and below 10^-5 for the 3D SSEP benchmark at L=2. These are small-system benchmark cases, however, rather than exact validation of every lattice size or of the full larger-system calculation.
What the calculation cannot settle
Scale is the main unresolved issue. The calculations use finite lattices, with the 3D results especially dependent on the accessible ranges; the authors say larger 3D systems are needed to quantify the remaining finite-size corrections. The variational calculation also relies on selected exact checks, while the 2D coexistence analysis does not resolve the position, width or finite-size fluctuations of an interface. The inverse-square relation is therefore best treated as a numerical pattern to test further, not a finished law.
Publication and data
The document is identified as arXiv:2608.25606v1 and dated 26 August 2026 in the supplied record. The authors state that supporting data are available within the paper. Its acknowledgments report support from Chinese national and Sichuan provincial science-funding bodies, a Chinese Academy of Sciences program and the University of Electronic Science and Technology of China HPC center.
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
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