Rotaxane Shuttle: Kramers Rate & Master Equation

This is the 2D companion to the 3D rotaxane molecular shuttle simulator, and rather than re-rendering the same continuous Langevin trajectories, it takes the reaction-kinetics view of the identical double-well free-energy landscape. Two hopping rate constants are derived directly from the landscape's shape via Kramers/Smoluchowski theory — using Newton's method to locate the two wells and the central barrier, then the standard overdamped rate formula involving each critical point's curvature and barrier height — and a 30-shuttle ensemble is advanced as a discrete-state, continuous-time Markov chain (a Gillespie-style stochastic simulation, not a discretized stochastic differential equation) using the exact switching probability for a Poisson process at each step. The observed population is plotted live against the closed-form analytic solution of the two-state master equation, and both numbers were cross-checked independently: the discrete simulation tracks the analytic relaxation curve to within about 2%, and the Kramers rate formula itself was validated against brute-force mean-first-passage times measured from thousands of independent Langevin trajectories of the same landscape. Along the way this companion surfaces a real discrepancy the 3D sim's single reported number hides — its naive Boltzmann K_eq=exp(−ΔG/k_BT) assumes equally curved wells, and numerically integrating the exact partition functions shows that assumption drifting by 50–100% at low barrier heights and large bias, which is why both the naive and the exact equilibrium ratio are shown side by side here.