Bias-Controlled Equilibrium Crossovers and Quench Relaxation in an Overdamped Bistable Langevin Model: Single-Source Computational Benchmark and Reproducibility Package

This repository (DOI: 10.5281/zenodo.21863317) contains the single-source Python computational framework, cached multi-level stochastic trajectory ensembles, machine-readable verification tables (outputs/*.csv), companion supplementary documentation (README.pdf and Supplementary_Material_ForReview.pdf), and publication-grade vector PDF figures supporting the revised manuscript: “Bias-Controlled Equilibrium Crossovers and Quench Relaxation in an Overdamped Bistable Langevin Model”. The codebase provides a fully reproducible analytical and numerical benchmark for examining how a dimensionless control parameter (λ ∈ [0, 1]) and effective linear tilt coefficient (h(λ) = 3(λ − 0.5)) govern equilibrium well occupation probabilities, analytical Kramers transition rates, and finite-time post-quench stochastic relaxation in a one-dimensional overdamped Langevin system with dimensionless effective potential Ueff(x; λ) = x4 − 4.5x2 + 3(λ − 0.5)x. Methodologies Included Closed-Form Viète Stationary Points & Spinodal Bounds: Exact evaluation of the depressed-cubic discriminant Δcubic(λ) = (243/16)[3 − (λ − 0.5)2] > 0 and closed-form Viète trigonometric roots xL(λ) < −√3/2 < xB(λ) < +√3/2 < xR(λ) in IEEE 754 double precision (eliminating iterative polynomial root-finder drift, with cubic force-balance residuals ≤ 10−12). Adaptive Gauss–Kronrod Quadrature vs. Kramers & Arrhenius Transition-Rate Theory: High-precision QUADPACK integration (scipy.integrate.quad, tolerance 10−12, stabilized via potential-minimum energy shifting) over the stationary Boltzmann density, benchmarked across λ ∈ [0, 1] against both the curvature-corrected Kramers escape-rate ratio (maximum 2.38% minority-well relative error) and a pure Arrhenius barrier-asymmetry approximation (maximum 17.74% minority-well relative error due to omitting the intrawell curvature ratio ωR(λ)/ωL(λ)). Post-Quench Relaxation Across λ ∈ {1.0, 0.5, 0.0} & Two-State Master Equation: Vectorized stochastic simulations (Ntraj = 1,000 independent trajectories partitioned into M = 10 batches of 100, initialized at x0 = xL(1.0) = −1.57726150 over T = 103) comparing maintained bias (λ = 1.0), the symmetric crossover quench (λ = 0.5), and full bias removal (λ = 0.0) against the analytical two-state Kramers master equation PRmaster(t; λ) = PRquad(λ)[1 − exp(−t/τKramers(λ))]. Statistical uncertainty is quantified via two-sided 95% batch Student-t intervals (t0.975,9 = 2.262) and exact 95% Clopper–Pearson binomial intervals. Four-Level Timestep Refinement & Drift-Tamed Euler–Maruyama Verification: Synchronized fine-to-coarse Brownian path integration across Δt ∈ {2 × 10−3, 10−3, 5 × 10−4, 10−4} alongside the explicit drift-tamed Euler–Maruyama scheme for non-globally Lipschitz cubic drift (0/1,000 blow-ups at |Xn| > 103; clean-level pathwise strong L2 convergence slope α̂ ∈ [1.07, 1.12]; explicit separation of baseline Δt = 10−3 numerical discretization distance DL1ref ≤ 0.024 from physical non-equilibrium arrest DL1eq = 0.833 at λ = 0.5). Repository Contents README.pdf: Master reproducibility package documentation, execution instructions, deterministic seed protocol, and verified numerical benchmark summary. Supplementary_Material_ForReview.pdf: Companion supplementary document detailing the file manifest (Table S1), coupled Brownian path construction, 9-point Viète stationary-root and occupation verification grids (Tables S2–S3), and post-quench statistical audit (Table S4). nls_d_26_00874_common.py: Shared numerical core implementing closed-form Viète roots, adaptive Boltzmann quadrature, Kramers/Arrhenius analytics, master-equation kinetics, coupled four-level explicit and drift-tamed Euler simulation engines, and statistical interval calculators. nls_d_26_00874_master_benchmark.py: Single entry-point driver script that regenerates all seven verification CSV tables in outputs/ and all four vector PDF figures from a single documented run. figure_1_potential.py → Fig1_potential.pdf: Dimensionless effective-potential landscapes Ueff(x; λ) at λ ∈ {0.0, 0.5, 1.0}. figure_2_equilibrium_occupations_kramers_arrhenius.py → Fig2_equilibrium_occupations_kramers_arrhenius.pdf: Equilibrium occupation crossover curves and minority-well relative error comparing Boltzmann quadrature against Kramers and Arrhenius approximations. figure_3_quench_relaxation_ci_boltzmann_refinement.py → Fig3_quench_relaxation_ci_boltzmann_refinement.pdf: Post-quench ensemble mean trajectories with 95% batch Student-t CIs, right-well occupation fractions vs. two-state Kramers master equation, and terminal empirical histograms vs. stationary Boltzmann densities. figure_4_effective_potential_and_spinodal_roots.py → Fig4_effective_potential_and_spinodal_roots.pdf: Continuous parametric family of effective potentials Ueff(x; λ), closed-form Viète stationary-point loci, and spinodal inflection lines (xinfl± = ±√3/2). quench_ensemble_results.npz: Cached multi-level simulation archive from the documented run (pass --force to re-simulate from scratch). outputs/*.csv: Machine-readable verification tables corresponding to Manuscript Tables 1–4 and Supplementary Tables S2–S3 (table_equilibrium.csv, table_kramers_comparison.csv, table_quench_summary.csv, table_refinement.csv, table_refinement_slopes.csv, supplementary_S2_viete_grid.csv, and supplementary_S3_occupation_comparison.csv). Dependencies & Execution Written in Python 3 (≥ 3.10 recommended). Requires numpy, scipy, and matplotlib. Run python nls_d_26_00874_master_benchmark.py to regenerate all tables and vector PDF figures.