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Characterizing Full Nonequilibrium Dynamics of Simple Exclusion Processes

Zhimao Liu, Jing Liu, Pan Zhang, Ying Tang

arXiv:2608.25606v1cond-mat.stat-mechcond-mat.dis-nnstat.ML

TL;DR

Time-dependent joint distributions in simple exclusion processes are difficult to characterize across dimensions and transport regimes. The paper uses variational autoregressive networks to track finite-time dynamics, revealing shared activity, susceptibility, density, and scaling structures in one to three dimensions.

  • Problem

    Complete dynamical descriptions require fluctuations of time-integrated observables, but finite-time evolution spans an exponentially large configuration space and becomes harder with system size and dimension.

  • Method

    The study uses variational autoregressive probability representations with time-discretized evolution and blocked natural-gradient optimization to track tilted exclusion-process dynamics.

  • Results

    Across one to three dimensions, the framework reproduces prior benchmarks and reveals activity-phase organization, directional-density criteria, susceptibility trends, and finite-time scaling relations for exclusion processes.

  • Takeaways & Limitations

    The results establish a unified framework for characterizing nonequilibrium dynamics in representative simple exclusion processes across dimensions and transport regimes.

Abstract

from arXiv · show

The simple exclusion process (SEP) is a paradigmatic model for nonequilibrium transport, yet the rich dynamics of its time-dependent joint distribution over an exponentially large configuration space remain notoriously intractable. Here, we leverage variational autoregressive networks to systematically characterize the nonequilibrium dynamics of symmetric (SSEP), asymmetric (ASEP), and totally asymmetric (TASEP) cases from one to three dimensions. We first validate the approach by reproducing the previous finite-time results for the 1D SSEP and long-time tensor-network results for the 2D SSEP, and then provide richer finite-time dynamics of the SSEP, ASEP, and TASEP in 1D and 2D, and a new finite-time analysis in 3D. Specifically, in 1D, we reveal that finite-time dynamical-activity maps directly correspond to the classical three-phase TASEP steady-state organization, and, in the long-time limit, boundary and bulk effects separately govern the dynamical susceptibility during the crossover from diffusive to ballistic transport. In 2D, we establish a mean-field directional-density criterion, supported by our neural-network calculations, and show that long-time boundary and bulk effects mirror their 1D counterparts. In 3D, we uncover new finite-time scaling relations for the active-inactive phase transition of the SSEP, and reveal a broadly consistent scaling exponent of the phase-transition point versus system size, implying that the phase-transition point is asymptotically controlled by the characteristic length scale ($s_c\sim L^{-2}$) regardless of dimension. This work thus establishes a unified framework for characterizing the nonequilibrium dynamics of representative transport systems.

I. INTRODUCTION · II. NONEQUILIBRIUM STATISTICAL MECHANICS · A. Master equations

The paper develops a variational autoregressive framework for finite-time nonequilibrium dynamics in symmetric, asymmetric, and totally asymmetric exclusion processes from one to three dimensions. It addresses the difficulty of tracking transient distributions over exponentially large configuration spaces while characterizing fluctuations of time-integrated observables.

  • I. INTRODUCTION: SEP models stochastic particle hopping on lattices with hard-core exclusion, spanning symmetric, asymmetric, and totally asymmetric transport variants.Dynamical currents and activities probe complementary time-antisymmetric and time-symmetric sectors of nonequilibrium dynamics.
  • I. INTRODUCTION: A complete dynamical description must include fluctuations of time-integrated observables, not only typical densities or currents.
  • I. INTRODUCTION: Finite-time analysis is difficult because the dynamical partition function depends on the initial state and the full tilted-generator spectrum.The evolving distribution spans an exponentially large configuration space and retains multiple transient modes, with difficulty increasing with system size and spatial dimension.
  • I. INTRODUCTION: The study extends finite-time variational autoregressive networks with blocked natural-gradient optimization to SEP dynamics in one, two, and three dimensions.The VAN represents normalized configuration probabilities and variationally approximates tilted-distribution evolution, enabling evaluation of the dynamical partition function and derived observables.
  • I. INTRODUCTION: In 1D, finite-time dynamical-activity maps reflect the classical three-phase TASEP steady-state organization, while long-time susceptibility separates boundary-rate and bulk-driving effects.These effects differ across the crossover from diffusive to ballistic transport.
  • I. INTRODUCTION: In 2D, a directional-density criterion applies when boundary parameters along both directions select the same density under the 1D TASEP phase diagram.The supplied passage states that the bulk density converges to a corresponding value, but does not provide the remainder of that result.
  • A. Master equations: A d-dimensional exclusion process uses binary site variables x_i ∈ {0, 1}, with configurations represented by vectors |x⟩ and probabilities encoded in |P_t⟩.
  • A. Master equations: The probability vector evolves under a master equation governed by the Markov generator W, decomposed as W = K − R.K contains off-diagonal transition rates, while R is the diagonal escape-rate matrix ensuring probability conservation, ⟨−|W = 0.

B. Dynamical partition function · C. System

The paper characterizes trajectory ensembles through a dynamical partition function and tilted generator, then specifies ASEP dynamics on one-, two-, and three-dimensional lattices with bulk and boundary rates. Finite-time behavior depends on the full tilted-generator spectrum, whereas long-time behavior is governed by its dominant eigenvalue.

  • B. Dynamical partition function: The trajectory ensemble contains all time-discretized configurations, with dynamical activity defined as the total number of configuration changes.The trajectory has time step δt, total time t = N_stepδt, and activity is time-reversal invariant, probing the frenetic sector.
  • B. Dynamical partition function: The dynamical partition function generates the statistics of a time-extensive observable K by weighting trajectories with counting field s.It is constructed from the trajectory weight p(ω_t) and the constraint on K.
  • B. Dynamical partition function: For activity, the tilted generator is W_s = e^−sK − R, and its largest eigenvalue controls long-time behavior but not arbitrary finite-time partition functions.At finite t, Z_t(s) depends on the full spectrum of W_s because W_s does not preserve probability.
  • B. Dynamical partition function: A nonanalyticity in the long-time SCGF ψ(s) marks a dynamical phase transition, such as a change between active and inactive trajectory phases.At s = 0, k(0) is the typical steady-state activity per site and χ(0) is the steady-state growth rate of activity variance.
  • C. System: ASEP is a continuous-time Markov process on a lattice with hard-core particles, so each occupation variable satisfies x_i ∈ {0, 1}.Its configuration evolves through local hopping events and boundary exchanges with reservoirs.
  • C. System: In one dimension, particles hop between vacant nearest-neighbor sites at rates p_x and q_x, while boundary insertion and removal use α_x, δ_x, γ_x, and β_x.The chain contains N = L sites, with left and right boundaries supplying the corresponding reservoir processes.
  • C. System: In two dimensions, the square lattice has N = L^2 sites with directional hopping rates p_x, p_y, q_x, q_y and four-boundary insertion and removal rates.The boundaries are left, top, right, and bottom, with rates indexed by their associated directions.
  • C. System: In three dimensions, the cubic lattice has N = L^3 sites with hopping along x, y, and z directions and six-boundary reservoir rates.Positive-direction rates are p_x, p_y, p_z; reverse rates are q_x, q_y, q_z, and each boundary has corresponding insertion and removal parameters.

III. NEURAL-NETWORK FRAMEWORK · A. Variational ansatz

The framework uses a variational autoregressive network to represent normalized probability distributions and evolve them under tilted dynamics. Masked autoregression, sequential normalization, and variational optimization together yield finite-time dynamical observables.

  • A. Variational ansatz: The VAN serves as a variational ansatz for the exact time-dependent probability vector, approximating the distribution Pt(x) with learnable neural-network parameters.The approximation is normalized and depends on the fixed counting field s unless explicitly shown.
  • A. Variational ansatz: The normalized distribution is represented through autoregressive factorization, with each variable conditioned only on its predecessors.This ordering excludes dependence on successors and supports normalized sampling.
  • A. Variational ansatz: At each discretized time step tj = jδt, a masked autoencoder for distribution estimation implements the VAN representation.The architecture is a MADE applied to the variational state at each step.
  • A. Variational ansatz: Binary masks convert feed-forward propagation into ordered conditional representations by enforcing connections consistent with hidden-unit and input order numbers.The resulting ansatz preserves causal ordering while remaining expressive for high-dimensional lattice distributions.
  • A. Variational ansatz: The computational pipeline represents the normalized distribution, advances it with the tilted operator, and updates parameters using variational free energy.Iterating the pipeline produces Zt(s), the SCGF ψ(s), and related dynamical observables.
  • A. Variational ansatz: Sequential application of Ts = I + δtWs advances the tilted dynamics, while accumulated one-step normalization factors construct the full dynamical partition function.Because tilted evolution produces an unnormalized vector, each next VAN state is normalized before optimization.
  • A. Variational ansatz: The update minimizes the KL divergence between the VAN prediction and the normalized evolved distribution, equivalently optimizing the variational free-energy objective.The one-step normalization factor is independent of the next-step network parameters.
  • A. Variational ansatz: RE-INFORCE estimates gradients from samples drawn from the current VAN, making updates scalable but potentially ill-conditioned in high-dimensional parameter space.Consequently, first-order optimizers such as Adam may converge slowly.

B. Blocked natural-gradient descent

Natural-gradient descent adapts optimization to the probabilistic geometry of the variational autoregressive network by minimizing free energy under a local KL-distance constraint. Damping and exact blocked batch-space computation stabilize and reduce memory costs without changing the natural-gradient direction.

  • Natural-gradient geometry: Natural-gradient descent measures update distance in probability-distribution space, better matching the VAN than ordinary first-order optimization.It minimizes variational free energy under a fixed local KL-distance constraint.
  • Damped update: Damping regularizes the natural-gradient update when the parameter-space Fisher matrix may be singular, improving numerical stability without requiring invertibility.This addresses cases where the number of parameters exceeds the batch size.
  • Batch-space formulation: The Woodbury reformulation replaces inversion of the NP × NP parameter-space matrix with a linear solve in NB × NB batch space.This shifts the main linear-algebra cost to the batch dimension.
  • Blocked implementation: Exact block accumulation includes every cross-block product, so the assembled matrix equals the full OO⊤ matrix rather than a block-diagonal approximation.The implementation divides the batch into sub-batches and accumulates their gradient-matrix products.
  • Blocked implementation: Blocking changes only memory layout and computation schedule, preserving algebraic equivalence to the full Woodbury update and the same natural-gradient direction.Peak memory is controlled by the chosen block sizes.

IV. APPLICATIONS

This section applies the VAN framework to physical results for SEPs in one, two, and three dimensions. It examines 1D activity and susceptibility behavior, a 2D directional-density criterion, and finite-time active-inactive transition scaling in 2D and 3D SSEP.

  • Overview: The VAN framework is applied to symmetric, asymmetric, and totally asymmetric exclusion processes in one, two, and three dimensions.The section presents physical results obtained with VAN across these dimensions and SEP variants.
  • One-dimensional SEPs: In 1D, finite-time activity maps and long-time susceptibility responses examine how TASEP’s boundary-selected phase organization appears across SSEP, ASEP, and TASEP.The finite-time activity analysis includes crossover between active and inactive phases across counting field s and trajectory time t.
  • Two-dimensional TASEP: In 2D, a directional-density criterion connects bulk-density organization in the 2D TASEP to the classical 1D TASEP phase diagram.The criterion is formulated using mean-field theory and supported numerically in Appendix E and Table III.
  • Two- and three-dimensional SSEP: Finite-time scaling of the active-inactive transition is analyzed for the 2D and 3D SSEP.Corresponding benchmarks are documented in Appendices C, F, G, and H.

A. 1D: boundary and bulk effects

In one dimension, the steady-state TASEP phase is reflected in finite-time activity landscapes, while boundary rates and bulk driving separately control long-time susceptibility. A mean-field activity relation explains these trends through density and density-variance effects.

  • A. 1D: boundary and bulk effects: Finite-time activity landscapes correspond directly to the steady-state TASEP phases, with LD and HD showing lower activity than MC despite distinct scarcity and jamming mechanisms.The LD phase reflects particle scarcity, whereas the HD phase reflects particle jamming.
  • A. 1D: boundary and bulk effects: The activity maps compare SSEP, ASEP, and TASEP across representative LD, coexistence, and MC boundary-rate pairs: (0.2, 0.8), (0.3, 0.3), and (0.6, 0.6).The corresponding bulk rates are SSEP (px = qx = 0.5), ASEP (px = 0.8, qx = 0.2), and TASEP (px = 1.0, qx = 0.0).
  • A. 1D: boundary and bulk effects: In diffusive SSEP, low-activity boundary rates (αx, βx) = (0.2, 0.8) produce the pronounced susceptibility peak, whereas ballistic TASEP peaks under high-activity rates (0.6, 0.6).Increasing bulk driving changes which boundary conditions dominate the long-time susceptibility response.
  • A. 1D: boundary and bulk effects: At t = 10^3 and s = 10^-2, the mean-field relation kMF ≃ 0.25 − (¯ρ − 0.5)^2 − Var(ρ) reproduces the VAN activity ordering across SSEP, ASEP, and TASEP.It predicts higher activity when mean density is closer to half filling and density variance is smaller.
  • A. 1D: boundary and bulk effects: Overall, boundary rates and bulk driving govern distinct aspects of susceptibility, while density homogeneity near ¯ρ = 0.5 qualitatively explains the activity trends.These results motivate extending the boundary-and-bulk analysis to coupled transport in two dimensions.

B. 2D: boundary and bulk effects · C. 2D: dynamical phase transitions

In 2D, a directional-density criterion connects boundary-selected 1D phases to bulk density and organizes finite-time activity across SSEP, ASEP, and TASEP. For the 2D SSEP, VAN calculations capture the active-inactive transition and support finite-size and temporal scaling through data collapse.

  • B. 2D: boundary and bulk effects: The directional-density criterion predicts that the 2D bulk density converges to the shared boundary-selected 1D TASEP density when both directions select the same phase.Low-density, maximal-current, and high-density phases appear successively along the diagonal of the two directional densities.
  • B. 2D: boundary and bulk effects: For L = 5, low-activity boundaries produce the strongest long-time susceptibility response in SSEP, whereas high-activity boundaries produce the pronounced peak in TASEP.The SSEP low-activity set is (0.2, 0.8, 0.2, 0.8), while the TASEP high-activity set is (0.6, 0.6, 0.6, 0.6).
  • B. 2D: boundary and bulk effects: VAN density and current estimates closely match mean-field predictions for representative low-density, coexistence, maximal-current, and high-density boundary conditions.The representative rate sets are (0.1, 0.4, 0.1, 0.9), (0.4, 0.4, 0.4, 0.4), (0.6, 0.7, 0.8, 0.9), and (0.4, 0.3, 0.6, 0.3), respectively.
  • B. 2D: boundary and bulk effects: In 2D TASEP, increasing density from the low-density phase toward the maximal-current phase progressively expands the active region, paralleling 1D TASEP behavior.This correspondence links density phases with the active-inactive dynamical phase.
  • B. 2D: boundary and bulk effects: Under maximal-current boundary conditions, the finite-time active region expands progressively as bulk dynamics changes from diffusive SSEP through driven ASEP to ballistic TASEP.The corresponding density profiles also change across these bulk dynamics.
  • B. 2D: boundary and bulk effects: For low-activity boundary sets, the susceptibility peak first increases and then decreases with stronger bulk drive, while high-activity sets increase toward the TASEP limit.These trends show that boundary and bulk effects separately govern the dynamical susceptibility response.
  • C. 2D: dynamical phase transitions: The VAN accurately captures the active-inactive dynamical phase transition in the 2D SSEP on an open square lattice with px = qx = py = qy = 1.0 and boundary rates 0.5.For linear size L, the configuration space contains 2^N states with N = L^2.
  • C. 2D: dynamical phase transitions: The critical point sc(N, t) is identified from the peak position of χt(s), and rescaling sc(N, t) by sc(N) with time N^αt−β produces data collapse.The collapse is shown in Fig. 5(c).

D. 3D: dynamical phase transitions · V. DISCUSSION

The study extends finite-time active-inactive transition analysis to 3D SSEP and presents a variational framework for nonequilibrium SEP dynamics. It reports 3D scaling estimates, implementation details, and extensions toward finite-time current statistics and broader nonequilibrium systems.

  • D. 3D: dynamical phase transitions: For 3D SSEP, the VAN resolves finite-time activity landscapes on open cubic lattices despite configuration spaces growing as 2^N with N = L^3.Unit bulk hopping rates and boundary rates 0.5 are used in each direction.
  • D. 3D: dynamical phase transitions: The 3D critical point obeys sc(N, t) − sc(N) ∼ t^−β with β ≳ 1, while finite-time curves collapse under N^α/t^β rescaling.The fitted spatial exponent is α ≈ 0.53 over the simulated ranges.
  • D. 3D: dynamical phase transitions: Finite-size estimates give α ≈ 1.99 in 1D, α ≈ 0.99 in 2D, and α ≈ 0.53 for the finite-time 3D fit over L ∈ {3, 4}.The steady-state 3D VMC estimate is α ≈ 0.61, indicating stronger finite-time and finite-size corrections.
  • D. 3D: dynamical phase transitions: The numerical estimates support relating the critical counting field to the characteristic diffusive length scale and universal finite-size scaling of activity fluctuations.This is also consistent with analytical studies of the activity-biased SSEP.
  • V. DISCUSSION: The variational framework tracks finite-time tilted SEP dynamics and evaluates the dynamical partition function without enumerating the full configuration space.It combines an autoregressive probability representation with blocked natural-gradient descent.
  • V. DISCUSSION: Finite-time calculations use a fixed MADE network of depth 4 and width 16, with NB = 1000, η = 1, λd = 10^−5, and up to N = 64 lattice sites.The time step is δt = 0.05 in 1D and 0.005 in 2D and 3D; larger systems may require greater network capacity.
  • V. DISCUSSION: A proposed extension uses directional counting fields to track finite-time SCGF, current, and current susceptibility across 1D–3D SEPs.The resulting tilted distributions could support efficient sampling of rare-current trajectories, with possible extensions beyond SEPs.

APPENDIX … Appendix D: Mean-field estimate of the 1D activity

The appendices establish benchmarks for VAN representations of boundary-driven TASEP and finite-time SSEP dynamics, then derive a qualitative mean-field estimate for 1D activity. They also summarize notation and the TASEP phase organization used for comparison.

  • Appendix A: Principal notation and conventions: Table V summarizes the principal notation and conventions used throughout the paper.
  • Appendix A: Principal notation and conventions: The 1D TASEP phase diagram contains maximal-current, low-density, and high-density phases, with a first-order transition along αx = βx < 1/2.The maximal-current phase requires αx > 1/2 and βx > 1/2; low density requires αx < 1/2 and βx > αx; high density requires βx < 1/2 and αx > βx.
  • Appendix B: 1D TASEP density and current benchmark: VAN calculations benchmark the open 1D TASEP using L = 20, s = 10−2, and t = 103 in a weakly tilted, long-time regime.The exact comparison uses steady-state density and current values known from the matrix product ansatz.
  • Appendix B: 1D TASEP density and current benchmark: Maximum relative deviations are approximately 8% for mean density and 7% for mean current across low-density, high-density, and maximal-current regimes.
  • Appendix B: 1D TASEP density and current benchmark: Close agreement with unbiased phase-dependent observables shows that VAN represents weakly tilted boundary-driven distributions near the unbiased limit.Exact equality is not expected because the calculation uses finite L, s, and t.
  • Appendix C: 1D SSEP benchmark: For 1D SSEP, the VAN reproduces the finite-time activity crossover and scaling exponents β ≈0.92 and α ≈1.99 obtained against MPS results.The critical field follows sc(N, t) −sc(N) ∼t−β, while its long-time value scales as sc(N) ∼N −α; data for L = 20 and L = 40 collapse using these exponents.
  • Appendix C: 1D SSEP benchmark: Agreement in activity landscapes, transition locations, and finite-time and finite-size scaling supports VAN representation of evolving tilted distributions throughout the finite-time regime.
  • Appendix D: Mean-field estimate of the 1D activity: The 1D mean-field activity estimate factorizes nearest-neighbor occupations as ⟨ni(1−nj)⟩≃ρi(1−ρj), neglects an order L−1 boundary contribution, and assumes slowly varying density.For the 1D calculations, px+qx = 1.

Appendix E: Directional-density criterion for 2D TASEP · 1. Homogeneous low-density phase

The appendix develops a mean-field directional-density mapping for homogeneous 2D TASEP phases, showing that 1D-selected bulk densities solve the interior equations while asymmetric boundaries affect only boundary layers. Finite-size calculations closely match VAN estimates and approach the predicted densities and currents.

  • Appendix E: Directional-density criterion for 2D TASEP: The directional-density criterion uses the exact 1D open-boundary TASEP phase diagram to select bulk densities separately along the x and y directions.The 2D TASEP has open boundaries on an L × L square lattice, with boundary parameters (αx, βx) and (αy, βy).
  • Appendix E: Directional-density criterion for 2D TASEP: The matching construction yields a homogeneous interior mean-field solution that fixes the leading thermodynamic-limit bulk density, while allowing finite boundary layers.The coexistence line is treated separately because its 1D state contains a shock rather than a single homogeneous bulk density.
  • Appendix E: Directional-density criterion for 2D TASEP: At L = 5, mean-field and VAN estimates differ by at most 0.008 in density and 0.007 in directional currents.The VAN values are evaluated at weak tilt s = 10^-3 and finite time t = 50.
  • 1. Homogeneous low-density phase: In the homogeneous low-density phase, equal injections αx = αy with αx < 1/2 and αx < βx, βy select the common 1D bulk state ρbulk = αx.The exit rates may remain asymmetric under these conditions.
  • 1. Homogeneous low-density phase: Substituting ρx,y = αx into the steady-state continuity equation makes both sides equal to 2αx(1 − αx), proving an exact interior mean-field solution.This establishes the 1D-selected low-density profile away from boundaries.
  • 1. Homogeneous low-density phase: Asymmetric extraction changes the local right-boundary density but not the bulk density, because transverse self-coupling terms cancel at the boundary.The boundary analysis assumes an adjacent interior density αx and translational invariance along y.
  • 1. Homogeneous low-density phase: The low-density boundary-layer decay length is ξ = −1/ ln[αx/(1 − αx)], independent of L, so its fraction vanishes as L →∞.Corner effects remain confined to an O(1) region and do not alter bulk convergence.

2. Homogeneous high-density phase

When both directions are in the homogeneous high-density phase with equal extraction rates, the common exit rate selects the bulk density, while asymmetric injection affects only boundary layers. Perturbations from the entrance decay exponentially over an O(1) length scale.

  • 2. Homogeneous high-density phase: For equal extraction rates βx = βy with βx < 1/2 and βx < αx, αy, the target bulk density is ρbulk = 1 −βx.The homogeneous profile ρx,y = 1 −βx satisfies Eq. (E3).
  • 2. Homogeneous high-density phase: Perturbations away from the entrance boundary decay exponentially into the bulk over an O(1) length scale.
  • 2. Homogeneous high-density phase: Asymmetric injection rates modify only boundary layers and do not change the macroscopic high-density bulk selected by the common exit rate.

3. Maximal-current phase · 4. Coexistence line · Appendix F: 2D SSEP and ASEP benchmarks

The maximal-current criterion gives a bulk density of 1/2 but leaves two-dimensional boundary scaling unresolved, while the coexistence line suggests phase separation with an undetermined shock interface. Appendix benchmarks show that variational autoregressive networks reproduce two-dimensional SSEP observables and capture tilted-ensemble phase-transition signatures for diffusive and driven dynamics.

  • 3. Maximal-current phase: When αx, αy ≥1/2 and βx, βy ≥1/2, both directions select the maximal-current state with ρbulk = 1/2.Substitution into the interior equation is satisfied in this regime.
  • 3. Maximal-current phase: The degenerate characteristic root λ = 1 prevents simple mean-field theory from producing an exponentially localized boundary layer.VAN profiles remain near bulk density 1/2 while resolving finite-size boundary structure beyond this description.
  • 4. Coexistence line: On the coexistence line αx = βx = λ < 1/2 and αy = βy = λ < 1/2, a delocalized shock separates densities λ and 1 −λ.The extended directional-density criterion therefore suggests a phase-separated two-dimensional profile rather than homogeneous convergence.
  • 4. Coexistence line: In symmetric square geometry, the phase-separated construction is consistent with a diagonal shock-like interface between injection-dominated and exit-dominated regions.The criterion does not fix the interface position, width, or finite-size fluctuations.
  • Appendix F: 2D SSEP and ASEP benchmarks: Finite-time VAN evolution in 2D approaches long-time SSEP, ASEP, and TASEP SCGF benchmarks, while spatial density profiles stabilize visibly after t = 12.5.The comparison uses finite-time results at t = 50, long-time extrapolation to t = 135, and independent steady-state VMC results.
  • Appendix F: 2D SSEP and ASEP benchmarks: For steady-state 2D SSEP, VAN reproduces the SCGF, activity, susceptibility peak, and critical-field scaling sc(N) ∼N −α, with fitted α ≈1.02 versus tensor-network α ≈0.92.Here N = L2 for the finite system sizes considered.
  • Appendix F: 2D SSEP and ASEP benchmarks: For 2D ASEP, VAN estimates of the SCGF, horizontal current, and current susceptibility along a selected counting-field line are broadly consistent with PEPS calculations.The benchmarks demonstrate representation of two-dimensional tilted ensembles and dynamical phase-transition signatures under driven dynamics.
  • Appendix F: 2D SSEP and ASEP benchmarks: Steady-state activity k(s) varies with boundary-rate sets at fixed bulk dynamics and with bulk dynamics changing from SSEP through ASEP to TASEP.These activity trends complement boundary- and bulk-dependent susceptibility trends.

Appendix G: Steady state of 3D SSEP via VMC · Appendix H: Tracking the dynamical partition function · Appendix I: Computational cost

The appendices validate VAN finite-time dynamics against exact and steady-state benchmarks, extend steady-state 3D SSEP analysis, and document computational settings and costs. The 3D steady-state susceptibility sharpens with system size, while the critical-field scaling is close to the finite-time estimate.

  • Appendix G: Steady state of 3D SSEP via VMC: The 3D SSEP susceptibility peak grows and sharpens with system size, with critical-field scaling sc(N) ∼ N^-α and α ≈ 0.61.The steady-state VMC estimate is close to the finite-time exponent α ≈ 0.53, indicating finite-time and finite-size corrections over accessible sizes.
  • Appendix G: Steady state of 3D SSEP via VMC: Steady-state VMC calculates the SCGF, dynamical activity, and susceptibility for 3D SSEP sizes L ∈ {3, 4, 5}.These calculations complement the finite-time 3D results.
  • Appendix H: Tracking the dynamical partition function: For 2D SSEP, ASEP, and TASEP with L = 3, VAN tracking agrees with exact numerical results to relative error on the order of 10^-4 throughout the tested evolution.The comparison holds across different boundary conditions.
  • Appendix I: Computational cost: Table VII summarizes the VAN architecture and wall-clock time for finite-time evolution and steady-state VMC calculations.Reported computational time is measured for a single counting-field value.
  • Appendix H: Tracking the dynamical partition function: The 3D SSEP benchmark achieves error below 10^-5 for L = 2.For L = 3 and L = 4, finite-time long-time SCGF extrapolations are broadly consistent with optimized steady-state VMC results.
  • Appendix H: Tracking the dynamical partition function: For L = 3 and L = 4, finite-time 3D SSEP SCGF extrapolations are broadly consistent with optimized steady-state VMC results.The comparison supports extrapolation toward the steady state for larger systems.
  • Appendix I: Computational cost: The calculations use δt = 0.05 in 1D, δt = 0.005 in 2D and 3D, batch size NB = 1000, learning rate η = 1, and damping λd = 10^-5.Different lattice lengths and numbers of time steps are used across one-, two-, and three-dimensional systems.
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