Source Code and Simulation Data for: Bias-Controlled Equilibrium Crossovers in a Minimal Overdamped Bistable Model

This repository contains the computational framework, discrete Python solver code, and high-resolution publication figures supporting the manuscript: "Bias-Controlled Equilibrium Crossovers in a Minimal Overdamped Bistable Model with a Phenomenological Membrane-Voltage Mapping." The codebase provides a fully reproducible mathematical model for examining how the amplitude of a static conservative bias modifies occupation probabilities and transition statistics in an asymmetric bistable system. The model uses a single overdamped bistable coordinate as a minimal phenomenological surrogate for a collective membrane-state variable, explicitly distinguishing between analytical equilibrium occupation distributions and finite-time stochastic relaxation. Methodologies Included: Direct Numerical Quadrature (Equilibrium): Adaptive Gauss-Kronrod integration over the stationary Boltzmann distribution to analytically calculate exact equilibrium crossovers, partition functions, and voltage moments. Stabilized against exponential underflow via localized energy shifting. Euler-Maruyama Integration (Finite-Time Relaxation): High-performance, vectorized stochastic simulations tracking 1,000 independent Langevin trajectories. Simulates an instantaneous quench from a baseline biased state (λ=1.0) to an unbiased intrinsic state (λ=0.0). Repository Contents: rigorous_euler_maruyama_and_boltzmann_equilibrium_solver.py: Establishes the intrinsic asymmetric quartic double-well potential and evaluates the effective potential landscapes under varying conservative bias parameters (Generates Figure 1). two_panel_quench_relaxation_simulation.py: Executes the finite-time Euler-Maruyama integration over 109 discrete operations, plotting ensemble phenomenological voltage means, standard deviations, and right-well occupation fractions (Generates Figure 2). figure_3_equilibrium_quadrature.py: Computes exact stationary distributions via the Fokker-Planck equation and plots the equilibrium crossover occupation probabilities (Generates Figure 3). High-resolution generated publication figures (.png and .pdf formats corresponding to the scripts above). Dependencies: numpy, scipy, matplotlib