T1 Relaxation from First Principles: Stochastic BPP Simulation

The 3D version of this simulator shows the Bloembergen-Purcell-Pound (BPP) picture as a rendered scene: a lattice of tumbling dipoles and a recovering magnetization arrow, both driven by plugging your slider settings straight into the closed-form 1/T₁ formula. This 2D companion takes the harder, more honest route: it never evaluates that formula directly. Instead it simulates a stochastic Ornstein-Uhlenbeck process as a stand-in for the randomly tumbling local field, measures that trace's own autocorrelation to confirm it really does decay as exp(-t/τ_c), numerically integrates that autocorrelation's Fourier transform via quadrature to obtain the Lorentzian spectral density J(ω) at the Larmor frequency and its second harmonic, and finally runs a Monte Carlo ensemble of independently-flipping two-level spins whose emergent ensemble average is compared, live, against the analytic inversion-recovery curve. Every panel reports its own verification error against the theory it is supposed to reproduce.