Self-Maintained Order and Hysteretic Collapse in a Non-Equilibrium Rotational Lattice

Abstract

Version 11, prepared for submission to Physica A (version 11 merges a professional language edit; the nearest-neighbour and BKT conclusions are stated for the parameters and resolution tested). A two-dimensional rotor lattice whose ordering couplings are removed by an order-dependent flux and restored by repair. The mean-field reduction is bistable above a computed feedback threshold; it includes a kinetic Monte Carlo study designed to separate static from rate-dependent hysteresis, and finds that the bistability survives on the lattice only when degradation senses order over an extended region. What the Monte Carlo shows. The annealed mean-field reduction of the model is bistable (window 3.92 < ρ < 11.30 at h = 6, K = 0.5, β = 20; threshold βc = 2.42 at h = 6, and no bistability for h ≤ 1). On the lattice with nearest-neighbour feedback it is not. The loop area of ρ sweeps (L = 16–64, nine rates, eight runs) peaks at 2.5 for fast sweeps and is zero within error for dwells of 64 steps or more at every size and in a colder field-free variant; the mean-field static area is 4.30. Ordered and collapsed starts inside the window converge to the same state (within 0.004 in all 40 cases), in 15–115 steps independent of N. The cause is the closure: the lattice applies the average of the steep feedback over a broad distribution of local disorder, which flattens it. With the kill rate sensing wider blocks, the slow-sweep loop area is 0.01, 0.03, 0.21, 1.00 and 1.93 for 4, 8, 24, 80 sites and global sensing, and the two starts split at 80 sites and globally. Bistability is realized when degradation senses order over an extended range. Quenched-dilution finite-size scaling (L = 16–96, eight realizations) places the field-free loss of stiffness at fKT = 0.205 (1/ln2L extrapolation, 95% interval 0.193–0.216) and 0.205 (Weber–Minnhagen, 0.198–0.213), far below site percolation (0.407). In size and time (v1.1.0, manuscript v10). Two-start runs at L = 32–96 for 15 000 steps: with global sensing none of 128 runs left its branch at any size (strong numerical evidence of bistability); with 80-site sensing the branches persist mid-window, while at the window edges metastable states decay by nucleation, sooner in larger systems, so the finite-range case is consistent with a discontinuous transition with long-lived metastability. The five two-start intervals that exclude zero differ by at most 0.0021 and none survives Holm correction. The stationary maintenance current peaks near ρ = 4, where order is most fragile; the entropy production of Section 6 is that of an auxiliary cycle with added reverse rates and is illustrative. Every figure of the manuscript, including the analytic ones, is generated by scripts in this archive. Also: an exact order–entropy bridge (dS/dU = −κ for the von Mises law) and the Schnakenberg entropy production of the maintenance cycle. Code, raw data and analysis: https://github.com/sandlerleon/rotational-lattice-maintenance, archived at 10.5281/zenodo.23131432. Since version 9 it also corrects the discrete-time statement of the β = 0 baseline and the positivity condition of the entropy production (λR > ε2).